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Projectile motion (part 6)

https://www.youtube.com/watch?v=bl2DvFn8LjM

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WEBVTT Kind: captions Language: en 00:00:01.000 --> 00:00:01.910 Welcome back. 00:00:01.910 --> 00:00:04.330 Actually, before I teach you how to figure out how high the 00:00:04.330 --> 00:00:06.300 ball went-- and you might already be able to figure it 00:00:06.300 --> 00:00:08.450 out-- I want to show you really kind of a more 00:00:08.450 --> 00:00:11.080 intuitive way for figuring out how fast the ball went. 00:00:11.080 --> 00:00:13.470 I used the equation this first time, just to really show you 00:00:13.470 --> 00:00:17.810 that this equation can be useful, but I personally 00:00:17.810 --> 00:00:21.430 always forget equations, so I find it very useful to have 00:00:21.430 --> 00:00:23.750 kind of a common sense way of figuring it out, as well. 00:00:23.750 --> 00:00:26.650 And really, this equation was derived by coming up with a 00:00:26.650 --> 00:00:29.440 common sense way-- I don't know if that last statement 00:00:29.440 --> 00:00:31.440 made sense, but let's move on. 00:00:31.440 --> 00:00:34.200 Let's say that same problem, but let's just think about it 00:00:34.200 --> 00:00:37.170 without our equations, because that's always a good fall back 00:00:37.170 --> 00:00:39.520 when you're panicking in the middle of the exam, and you 00:00:39.520 --> 00:00:43.190 can't remember if an equation had a 1/2, or a 2, or a minus, 00:00:43.190 --> 00:00:46.820 or a plus, or t, or t squared. 00:00:46.820 --> 00:00:49.470 It's good just to think about what's happening. 00:00:49.470 --> 00:00:53.770 So when I throw a ball straight up-- I have a, let's 00:00:53.770 --> 00:00:56.770 say it's a baseball, it looks like a baseball-- and I throw 00:00:56.770 --> 00:01:04.050 it straight up where my velocity initial is equal to-- 00:01:04.050 --> 00:01:06.790 well, let's say that this the variable. 00:01:06.790 --> 00:01:09.300 This is my initial velocity, v sub i. 00:01:09.300 --> 00:01:10.380 What's going to happen? 00:01:10.380 --> 00:01:12.250 As soon as I throw it up, it's going to start 00:01:12.250 --> 00:01:13.760 decellerating, right? 00:01:13.760 --> 00:01:19.110 Because I have the force of gravity decelerating it 00:01:19.110 --> 00:01:22.660 immediately-- so, with gravity, we're saying minus 10 00:01:22.660 --> 00:01:26.640 meters per second squared, right? 00:01:26.640 --> 00:01:30.360 This ball is going to keep decelerating until its 00:01:30.360 --> 00:01:32.560 velocity goes to 0, right? 00:01:32.560 --> 00:01:40.460 The ball, if we were to graph time and distance, where this 00:01:40.460 --> 00:01:44.182 is time, and then this is distance. 00:01:44.182 --> 00:01:44.600 Right? 00:01:44.600 --> 00:01:47.620 At time zero, we're on the ground. 00:01:47.620 --> 00:01:52.510 The ball starts off going really fast, then starts 00:01:52.510 --> 00:01:57.170 slowing down, then its velocity goes to 0, and then 00:01:57.170 --> 00:01:59.856 it starts accelerating in the negative direction, and it 00:01:59.856 --> 00:02:02.890 starts going fast, and bam-- it hits the ground again. 00:02:02.890 --> 00:02:05.340 What happens is that the ball starts fast, and starts going 00:02:05.340 --> 00:02:08.699 slower, slower, slower, slower, and slower, until its 00:02:08.699 --> 00:02:15.180 velocity is 0, and then it starts-- you could say 00:02:15.180 --> 00:02:17.750 reaccelerating in the opposite direction-- or decellerating, 00:02:17.750 --> 00:02:20.185 really-- but reaccelerating in the opposite direction, and 00:02:20.185 --> 00:02:21.520 then it hits the ground. 00:02:21.520 --> 00:02:26.350 Actually, we know-- assuming nothing about air resistance, 00:02:26.350 --> 00:02:30.170 et cetera et cetera-- that the velocity that it hits the 00:02:30.170 --> 00:02:32.390 ground with is the same velocity that it left your 00:02:32.390 --> 00:02:35.220 hand with, just in the opposite direction. 00:02:35.220 --> 00:02:37.410 So there's a couple of interesting things here. 00:02:37.410 --> 00:02:43.300 The time at which its velocity is at 0-- so that point right 00:02:43.300 --> 00:02:49.080 there-- that's going to be at t equals 2. 00:02:49.080 --> 00:02:52.410 We know that this shape is actually a parabola, if you 00:02:52.410 --> 00:02:53.430 remember that from algebra two. 00:02:53.430 --> 00:02:54.350 Why is that a parabola? 00:02:54.350 --> 00:02:55.460 What was the equation for it? 00:02:55.460 --> 00:02:57.300 We figured out the equation using that previous formula. 00:02:57.300 --> 00:02:59.510 I don't want to use it this time, but what was that 00:02:59.510 --> 00:03:00.490 previous formula? 00:03:00.490 --> 00:03:06.120 It was change in distance is equal to vit plus at 00:03:06.120 --> 00:03:07.230 squared over 2. 00:03:07.230 --> 00:03:09.570 It's a parabola, but I think if you had thought about it, 00:03:09.570 --> 00:03:11.850 you would have realized also it's a parabola. 00:03:11.850 --> 00:03:15.050 And it points downward because a is negative, so the t 00:03:15.050 --> 00:03:18.710 squared term is negative-- that's why it opens to the 00:03:18.710 --> 00:03:20.166 downward side. 00:03:20.166 --> 00:03:21.850 I think that might make a little sense to you. 00:03:21.850 --> 00:03:25.600 So what we could figure out is, if we're given a t, we 00:03:25.600 --> 00:03:30.200 could say well, half of that number-- let's say that t 00:03:30.200 --> 00:03:34.230 equals 10 seconds. 00:03:34.230 --> 00:03:37.380 So we know that in 10 seconds, the ball left my hand, went up 00:03:37.380 --> 00:03:40.650 some distance, and then came back down and hit the ground. 00:03:40.650 --> 00:03:46.490 What we also know then, though, is that t over 2 at 5 00:03:46.490 --> 00:03:49.860 seconds, the ball was essentially stationary for 00:03:49.860 --> 00:03:52.890 just a moment-- its velocity had decelerated, decelerated, 00:03:52.890 --> 00:03:57.560 decelerated, and hit 0, and then right before it started 00:03:57.560 --> 00:04:00.670 reaccelerating again, or reaccelerating downwards, its 00:04:00.670 --> 00:04:03.835 velocity was 0 at the time t equals 0. 00:04:07.820 --> 00:04:10.340 The fact that the ball decelerated from my initial 00:04:10.340 --> 00:04:15.510 velocity to 0 in 5 seconds-- what does that tell us? 00:04:15.510 --> 00:04:20.360 Well, we have the very simple equation, you know, change in 00:04:20.360 --> 00:04:22.960 velocity is equal to 00:04:22.960 --> 00:04:26.380 acceleration times time, right? 00:04:26.380 --> 00:04:29.010 You probably knew that before watching any of these videos. 00:04:29.010 --> 00:04:30.615 And the change in the acceleration, well, that's 00:04:30.615 --> 00:04:35.690 just the final velocity minus the initial velocity is equal 00:04:35.690 --> 00:04:38.460 to the acceleration times time. 00:04:38.460 --> 00:04:40.610 In this situation, what's the final velocity? 00:04:40.610 --> 00:04:43.120 Remember, we're not going to go all the way here-- we're 00:04:43.120 --> 00:04:46.900 just figuring out from here to time equals 2, right? 00:04:46.900 --> 00:04:48.900 So what's the final velocity? 00:04:48.900 --> 00:04:51.870 We're saying that point is where the ball is not going 00:04:51.870 --> 00:04:54.865 up, and it's not going down, so its final velocity is 0. 00:04:54.865 --> 00:04:59.160 So, 0 minus initial velocity is equal to acceleration-- 00:04:59.160 --> 00:05:01.700 acceleration is the acceleration of gravity-- 00:05:01.700 --> 00:05:05.805 minus 10 meters per second squared. 00:05:13.090 --> 00:05:14.100 I know it's a little confusing, because I'm using 00:05:14.100 --> 00:05:20.610 the same t, but let's say that this time is t sub 0, just to 00:05:20.610 --> 00:05:23.490 kind of make sure it's not a variable, it's actual time. 00:05:23.490 --> 00:05:26.820 So this is t sub naught over 2, right? 00:05:26.820 --> 00:05:31.480 Because the ball is motionless right at the peak of its-- 00:05:31.480 --> 00:05:35.700 we're not an arc, but right at the peak of its travel. 00:05:35.700 --> 00:05:39.370 So, it's acceleration times time, but at this time, the 00:05:39.370 --> 00:05:41.365 time is going to be t sub naught over 2. 00:05:47.120 --> 00:05:51.980 Once again, the 0 doesn't matter, and we can multiply 00:05:51.980 --> 00:05:53.920 both sides times negative 1. 00:05:53.920 --> 00:06:00.290 We get plus vi, and we get vi is equal to 10 divided by 2-- 00:06:00.290 --> 00:06:05.165 5 meters per second squared t sub naught. 00:06:05.165 --> 00:06:09.730 Which was exactly what we got in the previous video when we 00:06:09.730 --> 00:06:10.800 used this formula. 00:06:10.800 --> 00:06:14.000 And I think it makes sense to you -- hopefully, this was 00:06:14.000 --> 00:06:17.620 kind of an intuitive way of thinking about what happened. 00:06:17.620 --> 00:06:19.850 Before actually I do the distance, I actually want to 00:06:19.850 --> 00:06:22.080 graph what's happening, because I think it just dawned 00:06:22.080 --> 00:06:24.620 on me that that might be something that will give you 00:06:24.620 --> 00:06:25.770 more intuition. 00:06:25.770 --> 00:06:27.785 I'm all about giving you intuition so you never forget 00:06:27.785 --> 00:06:30.030 this stuff. 00:06:30.030 --> 00:06:34.140 So this is, if we were to graph-- that's an ugly looking 00:06:34.140 --> 00:06:37.800 axes, but I think you'll get the point-- this is distance, 00:06:37.800 --> 00:06:40.000 this is time. 00:06:40.000 --> 00:06:43.486 We already said it's going to be like a parabola, like that. 00:06:43.486 --> 00:06:44.890 Right? 00:06:44.890 --> 00:06:50.610 Where this is t sub naught over 2, this t sub naught. 00:06:50.610 --> 00:06:54.330 It launches really fast, then it slows down, and then it's 00:06:54.330 --> 00:06:56.360 motionless right here, and then it starts 00:06:56.360 --> 00:06:57.580 reaccelerating downwards. 00:06:57.580 --> 00:06:59.470 If that's the distance, what does the 00:06:59.470 --> 00:07:02.790 velocity graph look like? 00:07:02.790 --> 00:07:05.530 The velocity graph, I'll draw right below-- I'll draw it in 00:07:05.530 --> 00:07:09.144 another color, just for variety. 00:07:09.144 --> 00:07:10.394 That's a bold. 00:07:13.840 --> 00:07:17.440 So over-- actually, that's not how I want to draw it. 00:07:17.440 --> 00:07:20.400 I have to draw the negative side, too. 00:07:25.060 --> 00:07:33.950 So this is time, and then this axis is velocity, so we start 00:07:33.950 --> 00:07:35.480 off at a positive velocity, right? 00:07:35.480 --> 00:07:43.450 We start off at v sub i, and what's going to happen here is 00:07:43.450 --> 00:07:46.180 the velocity decreases at a constant rate, right? 00:07:46.180 --> 00:07:48.797 And that rate is just the rate of acceleration. 00:07:48.797 --> 00:07:57.440 The velocity decreases until at t sub naught-- let me 00:07:57.440 --> 00:08:03.190 switch back to yellow-- at t sub naught-- woops, I'm using 00:08:03.190 --> 00:08:04.850 the wrong tool, it actually looked like I was drawing 00:08:04.850 --> 00:08:10.740 something-- at t sub naught, the velocity now is 00:08:10.740 --> 00:08:12.130 negative vi, right? 00:08:12.130 --> 00:08:14.930 Remember we said, when the ball comes back down, it's 00:08:14.930 --> 00:08:16.500 going at the same velocity, just at 00:08:16.500 --> 00:08:18.150 the opposite direction. 00:08:18.150 --> 00:08:23.040 This point right here, which is t sub naught over 2, that 00:08:23.040 --> 00:08:25.560 corresponds to this point, right? 00:08:25.560 --> 00:08:28.550 Which makes sense, because that's the point at which the 00:08:28.550 --> 00:08:29.915 ball has no velocity. 00:08:29.915 --> 00:08:32.200 And look, the velocity is 0. 00:08:32.200 --> 00:08:36.100 So the ball starts going up really fast, slows down at a 00:08:36.100 --> 00:08:38.960 constant rate, and what is the slope of this line? 00:08:38.960 --> 00:08:41.130 Well, the slope is just the acceleration, right? 00:08:41.130 --> 00:08:43.419 Because velocity is the acceleration times time. 00:08:43.419 --> 00:08:46.390 And then it's stationary for just a moment, because its 00:08:46.390 --> 00:08:51.060 velocity is 0, and then it starts accelerating-- or you 00:08:51.060 --> 00:08:53.150 could say decelerating, or accelerating in the negative 00:08:53.150 --> 00:08:54.780 direction-- until the point that it's 00:08:54.780 --> 00:08:58.240 going at v sub i down. 00:08:58.240 --> 00:09:02.634 And of course, if you were to graph acceleration-- if I were 00:09:02.634 --> 00:09:06.540 to graph acceleration over time-- 00:09:06.540 --> 00:09:10.930 acceleration is constant. 00:09:10.930 --> 00:09:11.705 It's right here. 00:09:11.705 --> 00:09:14.890 Let me just get a line tool. 00:09:14.890 --> 00:09:17.850 Acceleration is just a constant minus 10 meters per 00:09:17.850 --> 00:09:19.420 second, so it's going to look like that. 00:09:19.420 --> 00:09:21.420 And it's just the slope of this line. 00:09:21.420 --> 00:09:25.370 If you know calculus, it'll make sense to you that this 00:09:25.370 --> 00:09:28.475 line is the derivative of this line, or this curve. 00:09:28.475 --> 00:09:30.570 This line is the derivative of this curve. 00:09:30.570 --> 00:09:32.990 And even if you don't know calculus, I think it makes 00:09:32.990 --> 00:09:38.010 sense to you that this is the slope of this line. 00:09:38.010 --> 00:09:39.920 And just so, if you haven't learned calculus, a derivative 00:09:39.920 --> 00:09:43.460 is just to a way of figuring out a slope at any point along 00:09:43.460 --> 00:09:45.770 the curve, so it's nothing too fancy. 00:09:45.770 --> 00:09:47.770 I'll see you in the next presentation.

Projectile motion (part 8)

https://www.youtube.com/watch?v=oDcPHWX2Nv0

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https://www.youtube.com/api/timedtext?v=oDcPHWX2Nv0&ei=c2eUZffJOOuIp-oPhKOnmAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249828&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1A1AF9A261306C2ED4A003F576367CF5D7E79FCC.56586B21EF0EAE9A8D5FA2785646FD2A2140C280&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.800 --> 00:00:01.330 Welcome back. 00:00:01.330 --> 00:00:04.280 I got too excited, and I went 20 seconds over time-- I hope 00:00:04.280 --> 00:00:06.960 YouTube still lets me publish this video, so I'll try my 00:00:06.960 --> 00:00:07.790 best. 00:00:07.790 --> 00:00:12.710 What I was saying is how related these two formulas 00:00:12.710 --> 00:00:15.885 are, and that makes sense because at this point the ball 00:00:15.885 --> 00:00:19.060 is stationary-- it's completely the same as a ball 00:00:19.060 --> 00:00:22.090 just being dropped, a stationary ball being dropped 00:00:22.090 --> 00:00:23.710 from height h. 00:00:23.710 --> 00:00:26.100 We can use this formula to do the same thing that we did 00:00:26.100 --> 00:00:29.010 intuitively in the last video. 00:00:29.010 --> 00:00:31.875 If we're just dropping a ball, the initial velocity is zero, 00:00:31.875 --> 00:00:34.110 so this term goes to zero, and this term is the 00:00:34.110 --> 00:00:36.480 only one that matters. 00:00:36.480 --> 00:00:39.150 Once again, if we're dropping it, we started at height h and 00:00:39.150 --> 00:00:42.900 we go to zero, so the change in distance is now minus h. 00:00:42.900 --> 00:00:46.780 We have at squared over 2, and that equals gravity times-- 00:00:46.780 --> 00:00:48.000 what's the time? 00:00:48.000 --> 00:00:56.960 The time is not this full t0, it's t0 over 2, so g t0 over 2 00:00:56.960 --> 00:00:59.170 squared over 2. 00:00:59.170 --> 00:01:01.780 You multiply both sides times negative 1, and you get this 00:01:01.780 --> 00:01:03.610 exact same thing. 00:01:03.610 --> 00:01:04.530 That's pretty cool. 00:01:04.530 --> 00:01:07.720 What I love about physics it that there's so many ways to 00:01:07.720 --> 00:01:10.280 approach the problems. You can actually start off with any 00:01:10.280 --> 00:01:13.720 one of the formulas that we've been looking at, and as long 00:01:13.720 --> 00:01:15.990 as you kind of keep playing with it and keep thinking 00:01:15.990 --> 00:01:18.680 about it, you might not find the fastest route to the 00:01:18.680 --> 00:01:21.390 solution, but as long as you don't do anything incorrect, 00:01:21.390 --> 00:01:23.330 you will get the solution. 00:01:23.330 --> 00:01:25.950 Hopefully, this gives you a little bit of an intuition of 00:01:25.950 --> 00:01:27.230 projectile motion. 00:01:27.230 --> 00:01:30.400 In the next set of videos, we're going to make it 00:01:30.400 --> 00:01:33.970 slightly more complicated, but also slightly more realistic. 00:01:33.970 --> 00:01:35.700 I'm going to introduce you to what happens in 00:01:35.700 --> 00:01:37.850 two-dimensional motion-- it's when you're not throwing 00:01:37.850 --> 00:01:40.360 something straight up, but you actually throw something at an 00:01:40.360 --> 00:01:43.080 angle like you normally do. 00:01:43.080 --> 00:01:46.400 I'll see you in the next video.

Projectile motion (part 7)

https://www.youtube.com/watch?v=-uAfg0t6NmM

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en

WEBVTT Kind: captions Language: en 00:00:00.730 --> 00:00:01.680 Welcome back. 00:00:01.680 --> 00:00:05.470 Now I think that hopefully you have some intuition, and maybe 00:00:05.470 --> 00:00:07.090 you can figure this out on your own, but I'll 00:00:07.090 --> 00:00:07.760 show how to do it. 00:00:07.760 --> 00:00:10.920 We spent a lot of time on how fast did I throw the ball-- 00:00:10.920 --> 00:00:13.050 how fast was it when it left my hand-- so let's figure out 00:00:13.050 --> 00:00:14.970 how high did it go, because that's kind of a cool thing to 00:00:14.970 --> 00:00:17.110 know, as well. 00:00:17.110 --> 00:00:23.070 We figured out that the final velocity-- actually, the 00:00:23.070 --> 00:00:27.840 initial velocity-- is just equal to minus the 00:00:27.840 --> 00:00:33.530 acceleration of gravity times time over 2. 00:00:33.530 --> 00:00:35.650 If we say that the acceleration of gravity is 00:00:35.650 --> 00:00:38.530 minus 10 meters per second squared, then this becomes 00:00:38.530 --> 00:00:43.970 5t-- so that's initial velocity, and the final 00:00:43.970 --> 00:00:46.695 velocity is just the negative of the initial velocity. 00:00:46.695 --> 00:00:50.140 It's going back to what we were doing before. 00:00:50.140 --> 00:00:51.260 The initial velocity-- we want to figure out 00:00:51.260 --> 00:00:53.140 how high did we go? 00:00:53.140 --> 00:00:55.970 Let's just go back to our basic-- we want to figure out 00:00:55.970 --> 00:00:57.700 this point, and we want to figure out how high 00:00:57.700 --> 00:00:59.820 did this thing go. 00:00:59.820 --> 00:01:01.650 That's actually the for vertex of this parabola, and if you 00:01:01.650 --> 00:01:04.739 know the formula for a vertex of parabola, you could do it-- 00:01:04.739 --> 00:01:07.010 once again, I like to do it intuitively, as opposed to 00:01:07.010 --> 00:01:08.440 memorizing formulas. 00:01:08.440 --> 00:01:10.200 We want to figure out how high did this go-- 00:01:10.200 --> 00:01:12.440 let's call that h. 00:01:12.440 --> 00:01:14.030 So what happens? 00:01:14.030 --> 00:01:20.880 I'm going from an initial velocity of vi, and I'm going 00:01:20.880 --> 00:01:24.620 to a final velocity-- well, not a final velocity, you can 00:01:24.620 --> 00:01:28.940 call this an intermediate velocity-- a velocity of 0. 00:01:28.940 --> 00:01:33.020 It takes a time of t sub naught over 2-- that doesn't 00:01:33.020 --> 00:01:44.660 look too good-- it takes t over 2, or t 00:01:44.660 --> 00:01:46.150 sub naught over 2. 00:01:46.150 --> 00:01:48.520 I know it gets confusing when it's t sub naught-- t sub 00:01:48.520 --> 00:01:51.260 naught just says this is an actual constant t as opposed 00:01:51.260 --> 00:01:53.220 to just the variable t, and that's all it is. 00:01:53.220 --> 00:01:55.810 I know it can get confusing sometimes. 00:01:55.810 --> 00:01:59.710 So, the initial velocity is equal to minus g times 00:01:59.710 --> 00:02:00.990 whatever we time, or whatever time your 00:02:00.990 --> 00:02:02.790 friend gets on the timer. 00:02:02.790 --> 00:02:08.060 But we also know that between time 0 and t sub naught over 00:02:08.060 --> 00:02:11.350 2-- so half the time that your friend has on the timer-- the 00:02:11.350 --> 00:02:16.100 velocity went from this velocity to 0. 00:02:16.100 --> 00:02:20.070 The average velocity over that time-- not the average 00:02:20.070 --> 00:02:22.780 velocity over the entire time, the average velocity over just 00:02:22.780 --> 00:02:30.650 this first interval-- the average velocity here would 00:02:30.650 --> 00:02:39.770 just be v sub i plus 0, because we're kind of this 00:02:39.770 --> 00:02:45.070 intermediate velocity, because we're stationary up here, so 00:02:45.070 --> 00:02:48.440 we divide that by 2. 00:02:48.440 --> 00:02:51.760 v sub i is just this, so if we take this, and substitute 00:02:51.760 --> 00:02:59.270 here, we get their average velocity over this time 00:02:59.270 --> 00:03:04.810 interval is equal to minus g times t sub 00:03:04.810 --> 00:03:08.440 naught over 4, now. 00:03:08.440 --> 00:03:11.830 Remember, this is only the average velocity over this 00:03:11.830 --> 00:03:14.760 time period-- the average velocity over the entire time 00:03:14.760 --> 00:03:18.310 period over t sub naught is actually 0, because we start 00:03:18.310 --> 00:03:20.200 and end at the same place. 00:03:20.200 --> 00:03:22.630 This is kind of an interesting point as to how speed and 00:03:22.630 --> 00:03:24.250 velocity is different. 00:03:24.250 --> 00:03:25.560 You're probably thinking-- hey, Sal, how could the 00:03:25.560 --> 00:03:28.530 average velocity over the entire time be 0, when the 00:03:28.530 --> 00:03:32.450 ball was clearly going quite fast most of the time? 00:03:32.450 --> 00:03:33.820 That's the difference between speed and velocity. 00:03:33.820 --> 00:03:36.960 It turns out that the speed was, I agree with you, quite 00:03:36.960 --> 00:03:41.080 fast the entire time, but when you take the average velocity, 00:03:41.080 --> 00:03:44.490 all the times that the velocity was positive, is 00:03:44.490 --> 00:03:47.760 balanced off by the times that the velocity is negative. 00:03:47.760 --> 00:03:52.580 Over the entire period, the average of the velocity is 0, 00:03:52.580 --> 00:03:56.960 and that makes sense because, the ball didn't go anywhere-- 00:03:56.960 --> 00:03:59.500 it just came back to where it started, and so that's an 00:03:59.500 --> 00:04:03.030 interesting distinction between speed and velocity. 00:04:03.030 --> 00:04:04.840 Anyway-- going back, and I hope I'm not confusing you-- 00:04:04.840 --> 00:04:11.440 the average velocity over this time period is minus g times t 00:04:11.440 --> 00:04:12.540 sub naught over 4. 00:04:12.540 --> 00:04:15.250 If we were to graph it here, that would be right here-- it 00:04:15.250 --> 00:04:21.574 would be essentially this, at this point. 00:04:24.930 --> 00:04:29.140 This is our average velocity over half the time period, and 00:04:29.140 --> 00:04:31.270 it would be equal to v sub i over 2, 00:04:31.270 --> 00:04:34.050 which is equal to this. 00:04:34.050 --> 00:04:37.360 We know our average velocity over this time period, and we 00:04:37.360 --> 00:04:40.500 also know the time-- the time is t sub naught over 2, so we 00:04:40.500 --> 00:04:44.710 can just say change in distance is equal to average 00:04:44.710 --> 00:04:49.030 velocity times time. 00:04:49.030 --> 00:04:51.610 What's the average velocity? 00:04:51.610 --> 00:05:05.660 It's all this stuff minus gt sub naught over 4. 00:05:09.970 --> 00:05:15.090 The average velocity is this, and what's the time? 00:05:15.090 --> 00:05:17.250 We're saying the average velocity over t sub naught 00:05:17.250 --> 00:05:21.960 over 2, not the whole t sub naught-- maybe if I do some 00:05:21.960 --> 00:05:24.625 examples with numbers, this won't be as confusing. 00:05:24.625 --> 00:05:28.150 So, the time is t sub naught over 2. 00:05:28.150 --> 00:05:35.350 So t sub naught over 2. 00:05:35.350 --> 00:05:37.470 That's the time. 00:05:37.470 --> 00:05:43.235 And so the distance we traveled, h, is equal to-- if 00:05:43.235 --> 00:05:54.840 we multiply it out-- minus gt sub naught squared over 8. 00:05:54.840 --> 00:05:56.170 That's interesting. 00:05:56.170 --> 00:06:00.950 If I throw a ball, and so this is h-- I know I'm making this 00:06:00.950 --> 00:06:09.360 really crowded-- if I throw a ball, and it takes 5 seconds 00:06:09.360 --> 00:06:14.560 to go all the way up, and go all the way down. 00:06:14.560 --> 00:06:17.330 If I just substituted in this equation so that it takes 5 00:06:17.330 --> 00:06:22.420 seconds to go all the way up, and all the way down-- once 00:06:22.420 --> 00:06:25.950 again, this is minus g, so if gravity is actually a minus 10 00:06:25.950 --> 00:06:29.190 meters per second, so this becomes a positive. 00:06:29.190 --> 00:06:34.840 Going back, if I say h is equal to-- so minus g is 00:06:34.840 --> 00:06:38.850 positive 10 meters per second times 5 seconds. 00:06:41.725 --> 00:06:52.630 It's 10 times 5 over 8, so 40 over 8 equals 50-- 00:06:52.630 --> 00:06:53.755 oh, sorry, t squared. 00:06:53.755 --> 00:06:55.180 I was about to say something, because that's 00:06:55.180 --> 00:06:58.300 not right-- t squared. 00:06:58.300 --> 00:07:02.230 It's minus g t naught squared over 8. 00:07:02.230 --> 00:07:04.050 So what's 10 times 25? 00:07:04.050 --> 00:07:13.880 It's 250 over 8, so that's 30 something meters-- 00:07:13.880 --> 00:07:15.430 it's like 31 meters. 00:07:15.430 --> 00:07:18.420 So if I can throw a ball, and it stays in the air for a 00:07:18.420 --> 00:07:22.150 total of 5 seconds, I threw it 31 meters into the air-- that 00:07:22.150 --> 00:07:28.280 might not sound high, but 31 meters is 93 feet. 00:07:28.280 --> 00:07:30.490 That's equivalent to a nine story building, so that 00:07:30.490 --> 00:07:32.120 actually is pretty high. 00:07:32.120 --> 00:07:36.260 If you can if you can keep a ball in the air for 5 seconds, 00:07:36.260 --> 00:07:38.240 you're pretty strong. 00:07:38.240 --> 00:07:39.800 I want to show you something else that's kind of 00:07:39.800 --> 00:07:42.020 interesting, or at least interesting to me. 00:07:42.020 --> 00:07:43.880 Let me erase some of this, just because I think it's 00:07:43.880 --> 00:07:46.315 getting really crazy now. 00:07:46.315 --> 00:07:48.200 Let me erase this stuff. 00:07:54.164 --> 00:07:58.030 See, I'll even erase this stuff. 00:07:58.030 --> 00:08:00.340 I think you get it now, and you can just back up and 00:08:00.340 --> 00:08:02.990 rewatch it-- that's the beauty of these videos. 00:08:02.990 --> 00:08:06.710 You don't really have to take notes, but just back it up and 00:08:06.710 --> 00:08:07.960 rewatch the video. 00:08:11.650 --> 00:08:12.230 OK. 00:08:12.230 --> 00:08:15.350 I think I have a clean space now. 00:08:15.350 --> 00:08:18.080 So let me just do a nice color, and 00:08:18.080 --> 00:08:20.260 back to the pen tool. 00:08:20.260 --> 00:08:28.250 We got that the height is equal to minus gt naught 00:08:28.250 --> 00:08:35.059 squared over 8, but we could have also written this as-- 00:08:35.059 --> 00:08:37.710 I'm just playing around with this algebraically-- this is 00:08:37.710 --> 00:08:49.870 the same thing as minus g over 2 times t naught squared-- t 00:08:49.870 --> 00:08:51.370 naught over 2 squared. 00:08:51.370 --> 00:08:54.220 This is just the exact same thing. 00:08:54.220 --> 00:09:00.030 I was wondering, does this concept look familiar? 00:09:00.030 --> 00:09:00.600 Well, sure. 00:09:00.600 --> 00:09:04.130 This is in our distance equation that we did before, 00:09:04.130 --> 00:09:10.500 where we just say that that change in distance is equal to 00:09:10.500 --> 00:09:17.460 the initial velocity times time plus at squared over 2. 00:09:21.170 --> 00:09:26.780 This term right here looks an awfully lot like this term, 00:09:26.780 --> 00:09:30.170 and this would be equivalent to this equation if this 00:09:30.170 --> 00:09:32.670 initial velocity was 0. 00:09:32.670 --> 00:09:36.090 So how does this relate? 00:09:36.090 --> 00:09:40.440 Another way to think about it is this distance that you're 00:09:40.440 --> 00:09:48.990 traveling up, it's also the same distance-- maybe just the 00:09:48.990 --> 00:09:52.045 negative-- of the distance that you travel down. 00:09:56.520 --> 00:10:02.520 If we said that we're starting at this point, at this point 00:10:02.520 --> 00:10:05.540 the ball is stationary, so it's identical to just taking 00:10:05.540 --> 00:10:09.660 that ball, and dropping it. 00:10:09.660 --> 00:10:13.300 If we say that we're going to drop a ball from height h, how 00:10:13.300 --> 00:10:15.000 long does it take? 00:10:15.000 --> 00:10:17.240 Then we can use this formula. 00:10:17.240 --> 00:10:21.200 If I drop a ball from height h, the change in distance is 00:10:21.200 --> 00:10:24.710 going to be minus h, because it's going to go from height h 00:10:24.710 --> 00:10:29.460 to 0, and so we could say that minus h is equal to-- the 00:10:29.460 --> 00:10:32.930 initial velocity is going to be 0. 00:10:32.930 --> 00:10:33.260 Whoops! 00:10:33.260 --> 00:10:34.860 I'm over time.

Projectile motion (part 5)

https://www.youtube.com/watch?v=dlpmllTx5MY

vtt

https://www.youtube.com/api/timedtext?v=dlpmllTx5MY&ei=c2eUZbfVNLXNhcIPzbyRqAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249827&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=152EDBB29FC2F3E6BDB6C7FC03F6569C1F2D6882.63220DD0CD6ED50FB8DE39E75359E876C4799292&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.120 --> 00:00:01.920 Welcome back. 00:00:01.920 --> 00:00:05.180 Let's continue doing projectile motion problems. I 00:00:05.180 --> 00:00:07.380 think this video will be especially entertaining, 00:00:07.380 --> 00:00:09.370 because I will teach you a game that you can play with a 00:00:09.370 --> 00:00:13.370 friend, and it's called let's see how fast and how high I 00:00:13.370 --> 00:00:15.110 can throw this ball. 00:00:15.110 --> 00:00:18.270 You'd be surprised-- it's actually quite 00:00:18.270 --> 00:00:20.020 a stimulating game. 00:00:20.020 --> 00:00:22.910 Let me just write down everything 00:00:22.910 --> 00:00:24.100 we've learned so far. 00:00:24.100 --> 00:00:32.119 Change in distance is equal to the average 00:00:32.119 --> 00:00:36.540 velocity times time. 00:00:36.540 --> 00:00:43.170 We know that change in velocity is equal to 00:00:43.170 --> 00:00:45.340 acceleration times time. 00:00:45.340 --> 00:00:48.600 We also know that average velocity is equal to the final 00:00:48.600 --> 00:00:53.800 velocity plus the initial velocity over 2. 00:00:53.800 --> 00:00:56.400 We know the change in velocity, of course, is equal 00:00:56.400 --> 00:00:58.960 to the final velocity minus the initial velocity. 00:00:58.960 --> 00:01:00.775 This should hopefully be intuitive to you, because it's 00:01:00.775 --> 00:01:02.660 just how fast you're going at the end, minus how fast you're 00:01:02.660 --> 00:01:05.080 going at the beginning, divided-- oh no, no division, 00:01:05.080 --> 00:01:08.080 it's just that I got stuck in a pattern. 00:01:08.080 --> 00:01:09.900 It's just vf minus vi, of course. 00:01:14.500 --> 00:01:17.560 You probably already knew this before you even stumbled upon 00:01:17.560 --> 00:01:21.340 my videos, but-- the two non-intuitive ones that we've 00:01:21.340 --> 00:01:23.980 learned, they're really just derived from what I've just 00:01:23.980 --> 00:01:24.470 written up here. 00:01:24.470 --> 00:01:26.120 If you ever forget them, you should try to derive them. 00:01:26.120 --> 00:01:28.130 Actually, you should try to derive them, even if you don't 00:01:28.130 --> 00:01:29.780 forget them, so that you when you do forget it, 00:01:29.780 --> 00:01:31.000 you can derive it. 00:01:31.000 --> 00:01:37.020 It's change in distance-- let me change it to lowercase d, 00:01:37.020 --> 00:01:42.680 just to confuse you-- is equal to the initial velocity times 00:01:42.680 --> 00:01:52.760 time plus at squared over 2, and that's one of what I'll 00:01:52.760 --> 00:01:55.110 call the non-intuitive formulas. 00:01:55.110 --> 00:01:59.600 The other one is the final velocity squared is equal to 00:01:59.600 --> 00:02:03.200 the initial velocity squared plus 2ad. 00:02:03.200 --> 00:02:05.090 We've derived all of these, and I encourage you to try 00:02:05.090 --> 00:02:07.060 rederive them. 00:02:07.060 --> 00:02:12.070 But using these two formulas you can play my fun game of 00:02:12.070 --> 00:02:14.630 how fast and how high did I throw this ball? 00:02:14.630 --> 00:02:19.890 All you need is your arm, a ball, a stopwatch, and maybe 00:02:19.890 --> 00:02:23.200 some friends to watch you throw the ball. 00:02:23.200 --> 00:02:24.460 So how do we play this game? 00:02:24.460 --> 00:02:28.110 We take a ball, and we throw it as high as we can. 00:02:28.110 --> 00:02:31.760 We see how long does the ball stay in the air? 00:02:31.760 --> 00:02:32.810 What do we know? 00:02:32.810 --> 00:02:37.120 We know the time for the ball to leave your hand, to 00:02:37.120 --> 00:02:39.970 essentially leave the ground and come back to the ground. 00:02:39.970 --> 00:02:48.240 We are given time, and what else do we know? 00:02:48.240 --> 00:02:51.570 We know acceleration-- we know acceleration is this minus 10 00:02:51.570 --> 00:02:52.670 meters per second. 00:02:52.670 --> 00:02:55.130 If you're actually playing this game for money, or 00:02:55.130 --> 00:02:57.770 something, you would probably want to use a more accurate 00:02:57.770 --> 00:03:00.550 acceleration-- you could look it up on Wikipedia. 00:03:00.550 --> 00:03:07.400 It's minus 9.81 meters per second squared. 00:03:07.400 --> 00:03:09.680 Do we know the change in distance? 00:03:09.680 --> 00:03:12.060 At first, you're saying-- Sal, I don't know how high this 00:03:12.060 --> 00:03:14.540 ball went, but we're talking about the change in distance 00:03:14.540 --> 00:03:17.700 over the entire time. 00:03:17.700 --> 00:03:19.430 It starts at the ground-- essentially at the ground, 00:03:19.430 --> 00:03:21.266 because I'm assuming that you're not 100 feet tall, and 00:03:21.266 --> 00:03:24.050 so you're essentially at the ground-- so it starts at the 00:03:24.050 --> 00:03:26.610 ground, and it ends of the ground, so the change in 00:03:26.610 --> 00:03:33.260 distance of delta d is 0. 00:03:33.260 --> 00:03:35.180 It starts with at the ground and ends at the ground. 00:03:35.180 --> 00:03:38.080 This is interesting-- this is a vector quantity, because the 00:03:38.080 --> 00:03:38.880 direction matters. 00:03:38.880 --> 00:03:41.460 If I told you how far did the ball travel, then you'd have 00:03:41.460 --> 00:03:43.720 to look at its path, and say how high did it go, and how 00:03:43.720 --> 00:03:45.690 high would it come back? 00:03:45.690 --> 00:03:48.000 Actually, if you want to be really exact, the change in 00:03:48.000 --> 00:03:50.930 distance would be the height from your hand when the ball 00:03:50.930 --> 00:03:54.880 left your hand, to the ground-- so, if you're 6 feet 00:03:54.880 --> 00:03:57.440 tall, or 2 meters tall, the change in distance would 00:03:57.440 --> 00:03:59.490 actually be minus 2 meters, but we're not going to do 00:03:59.490 --> 00:04:02.870 that, because that would just be too much, but you could do 00:04:02.870 --> 00:04:06.190 it if there's ever a close call between you and a friend, 00:04:06.190 --> 00:04:08.280 and you're betting for money. 00:04:08.280 --> 00:04:10.830 You're given these things, and we want to figure 00:04:10.830 --> 00:04:13.300 out a couple of things. 00:04:13.300 --> 00:04:16.410 The first thing I want to figure out is how fast did I 00:04:16.410 --> 00:04:19.560 throw the ball, because that's what's interesting-- that 00:04:19.560 --> 00:04:22.800 would be a pure test of testosterone. 00:04:22.800 --> 00:04:23.450 How fast? 00:04:23.450 --> 00:04:30.470 I want to figure out vi-- vi equals question mark. 00:04:30.470 --> 00:04:31.640 Which of these formulas can be used? 00:04:31.640 --> 00:04:33.300 Actually, I'm going do it first with the formulas, and 00:04:33.300 --> 00:04:35.750 then I'm going to show you almost an easier way to do it, 00:04:35.750 --> 00:04:36.640 where it's more intuitive. 00:04:36.640 --> 00:04:40.440 I want to show you that these formulas can be used for fun 00:04:40.440 --> 00:04:42.310 with your friends. 00:04:42.310 --> 00:04:46.060 We know time, we know acceleration, we know the 00:04:46.060 --> 00:04:49.930 change in distance, so we could just solve for vi-- 00:04:49.930 --> 00:04:51.350 let's do that. 00:04:51.350 --> 00:04:55.310 In this situation, change in distance is 0-- let me change 00:04:55.310 --> 00:05:00.730 colors again just to change colors-- so change in distance 00:05:00.730 --> 00:05:09.450 is 0 is equal to vi times time. 00:05:09.450 --> 00:05:13.420 Let me put the g in for here, so it's minus 10 meters per 00:05:13.420 --> 00:05:17.850 second squared divided by 2, and it's minus 5 meters per 00:05:17.850 --> 00:05:24.720 second squared-- so it's minus 5t squared. 00:05:24.720 --> 00:05:27.170 All I did it is that I took minus 10 meters per second 00:05:27.170 --> 00:05:29.400 squared for a, divided it by 2, and that's how I 00:05:29.400 --> 00:05:30.070 got the minus 5. 00:05:30.070 --> 00:05:32.510 If you used 9.81 or whatever, this would be 00:05:32.510 --> 00:05:35.610 4.905 something something. 00:05:35.610 --> 00:05:37.630 Anyway, let's get back to the problem. 00:05:37.630 --> 00:05:42.000 If we wanted to solve this equation for vi, 00:05:42.000 --> 00:05:44.860 what could we do? 00:05:44.860 --> 00:05:46.900 This is pretty interesting, because we 00:05:46.900 --> 00:05:49.910 could factor a t out. 00:05:49.910 --> 00:05:51.700 What's cool about these physics equations is that 00:05:51.700 --> 00:05:55.190 everything we do actually has kind of a real reasoning 00:05:55.190 --> 00:05:57.590 behind it in the real world, so let me actually flip the 00:05:57.590 --> 00:06:01.600 sides, and factor out a t, just to make it confusing. 00:06:01.600 --> 00:06:12.210 I get t times vi minus 5t is equal to 0. 00:06:12.210 --> 00:06:14.770 All I did is that I factored out a t, and I could do this-- 00:06:14.770 --> 00:06:16.710 I didn't have to use a quadratic equation, or factor, 00:06:16.710 --> 00:06:18.930 because there wasn't a constant term here. 00:06:18.930 --> 00:06:22.320 So I have this expression, and if I were to solve it, 00:06:22.320 --> 00:06:25.110 assuming that you know vi is some positive number, I know 00:06:25.110 --> 00:06:29.160 that there's two times where this equation is true. 00:06:29.160 --> 00:06:41.810 Either t equals 0, or this term equals 0-- vi minus 5t is 00:06:41.810 --> 00:06:48.460 equal to 0, or since I'm solving for velocity, we know 00:06:48.460 --> 00:06:54.323 that vi is equal to 5t. 00:06:57.198 --> 00:06:58.880 That's interesting. 00:06:58.880 --> 00:07:01.070 So what does this say? 00:07:01.070 --> 00:07:03.050 If we knew the velocity, we could solve it the other way. 00:07:03.050 --> 00:07:08.580 We could say that t is equal to vi divided by 5-- these are 00:07:08.580 --> 00:07:11.370 the same things, just solving for a different variable. 00:07:11.370 --> 00:07:16.790 But that's cool, because there are two times when the change 00:07:16.790 --> 00:07:20.000 in distance is zero-- at time equals 0, of course, the 00:07:20.000 --> 00:07:22.430 change in distance is zero, because I haven't thrown the 00:07:22.430 --> 00:07:27.150 ball yet, and then, at a later time, or my initial velocity 00:07:27.150 --> 00:07:29.530 divided by 5, it'll also hit the ground again. 00:07:29.530 --> 00:07:30.930 Those are the two times that the change 00:07:30.930 --> 00:07:31.740 in distance is zero. 00:07:31.740 --> 00:07:32.330 That's pretty cool. 00:07:32.330 --> 00:07:34.810 This isn't just math-- everything we're doing in math 00:07:34.810 --> 00:07:38.550 has kind of an application in the real world. 00:07:38.550 --> 00:07:42.570 We've solved our equation-- vi is equal to 5t. 00:07:42.570 --> 00:07:49.150 So, if you and a friend go outside and throw a ball, and 00:07:49.150 --> 00:07:50.710 you try to throw it straight up--0 although we'll learn 00:07:50.710 --> 00:07:53.540 when we do the two dimensional projectile motion that it 00:07:53.540 --> 00:07:56.000 actually doesn't matter if you have a little bit of an angle 00:07:56.000 --> 00:07:59.340 on it, because the vertical motion and the horizontal 00:07:59.340 --> 00:08:01.390 motions are actually independent, or can be viewed 00:08:01.390 --> 00:08:06.070 as independent from each other-- this velocity you're 00:08:06.070 --> 00:08:09.160 going to get if you play this game is going to be just the 00:08:09.160 --> 00:08:12.670 component of your velocity that goes straight up. 00:08:12.670 --> 00:08:14.645 I know that might be a little confusing, and hopefully it 00:08:14.645 --> 00:08:16.270 will make a little more sense in a couple of videos from now 00:08:16.270 --> 00:08:17.520 when I teach you vectors. 00:08:20.030 --> 00:08:24.940 If you were to throw this ball straight up, and time when it 00:08:24.940 --> 00:08:27.120 hits the ground, then this velocity would literally the 00:08:27.120 --> 00:08:28.990 speed-- actually, the velocity-- at which 00:08:28.990 --> 00:08:30.230 you throw the ball. 00:08:30.230 --> 00:08:31.410 So what would it be? 00:08:31.410 --> 00:08:39.950 If I threw a ball, and it took two seconds to go up hit the 00:08:39.950 --> 00:08:42.400 ground, then I could use this formula. 00:08:42.400 --> 00:08:44.890 This is actually 5 meters per second 00:08:44.890 --> 00:08:51.490 squared times t seconds. 00:08:51.490 --> 00:08:57.710 If it took 2 seconds-- so if t is equal to 2-- then my 00:08:57.710 --> 00:09:01.100 initial velocity is equal to 10 meters per second. 00:09:01.100 --> 00:09:03.160 You could convert that to miles per hour-- we've 00:09:03.160 --> 00:09:05.980 actually done that in previous videos. 00:09:05.980 --> 00:09:16.380 If you throw a ball that stays up in the air for 10 seconds, 00:09:16.380 --> 00:09:19.780 then you threw it at 50 meters per second, which is 00:09:19.780 --> 00:09:24.170 extremely, extremely fast. Hopefully, I've taught you a 00:09:24.170 --> 00:09:25.560 little bit about a fun game. 00:09:25.560 --> 00:09:28.040 In the next video, I'll show you how to figure out-- how 00:09:28.040 --> 00:09:29.380 high did the ball go? 00:09:29.380 --> 00:09:30.790 I'll see you soon.

Projectile motion (part 4)

https://www.youtube.com/watch?v=-W3RkgvLrGI

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WEBVTT Kind: captions Language: en 00:00:01.060 --> 00:00:05.500 We'll now use that equation we just derived to go back and 00:00:05.500 --> 00:00:07.980 solve-- or at least address-- that same problem we were 00:00:07.980 --> 00:00:10.510 doing before, so let's write that equation down again. 00:00:10.510 --> 00:00:11.660 Actually, let's write the problem down. 00:00:11.660 --> 00:00:15.140 Lets say I have the cliff again, and so my initial 00:00:15.140 --> 00:00:18.750 distance is 0, but it goes down 500 meters. 00:00:18.750 --> 00:00:20.230 I'm not going to redraw the cliff, because it takes a lot 00:00:20.230 --> 00:00:23.260 of space up on my limited chalkboard. 00:00:23.260 --> 00:00:27.440 We know that the change in distance is equal 00:00:27.440 --> 00:00:32.659 to minus 500 meters. 00:00:32.659 --> 00:00:35.100 I'm still going to use the example where I don't just 00:00:35.100 --> 00:00:38.070 drop the ball, or the penny, or whatever I'm throwing off 00:00:38.070 --> 00:00:40.270 the cliff, but I actually throw it straight up, so it's 00:00:40.270 --> 00:00:43.145 going to go up and slow down from gravity, and then it will 00:00:43.145 --> 00:00:46.370 go to 0 velocity, and start accelerating downwards. 00:00:46.370 --> 00:00:47.275 You could even say decelerating 00:00:47.275 --> 00:00:48.930 in the other direction. 00:00:48.930 --> 00:00:59.780 The initial velocity, vi, is equal to 30 meters per second, 00:00:59.780 --> 00:01:02.730 and of course, we know that the acceleration is equal to 00:01:02.730 --> 00:01:06.530 minus 10 meters per second squared-- it's because 00:01:06.530 --> 00:01:09.200 acceleration gravity is always pulling downwards, or towards 00:01:09.200 --> 00:01:11.260 the center of our planet. 00:01:11.260 --> 00:01:13.150 If we wanted to figure out the final velocity, we could have 00:01:13.150 --> 00:01:15.150 just used the formula, and we did this in the last video-- 00:01:15.150 --> 00:01:21.340 vf squared is equal to vi squared plus 2ad. 00:01:21.340 --> 00:01:24.900 Now what I want to do is use the formula that we learned in 00:01:24.900 --> 00:01:27.260 the very last video to figure out-- how long does it take to 00:01:27.260 --> 00:01:30.680 get to the bottom, and to hit the ground? 00:01:30.680 --> 00:01:37.350 Let's use that formula: we derived that the change in 00:01:37.350 --> 00:01:43.720 distance is equal to the initial velocity times time 00:01:43.720 --> 00:01:48.570 plus acceleration time squared over 2, and 00:01:48.570 --> 00:01:50.200 that's initial velocity. 00:01:50.200 --> 00:01:55.550 The change in distance is minus 500, and that's equal to 00:01:55.550 --> 00:01:59.410 the initial velocity-- that's positive, going upwards, 30 00:01:59.410 --> 00:02:02.300 meters per second, 30t. 00:02:02.300 --> 00:02:04.600 I'm not going to write the units right now, because I'll 00:02:04.600 --> 00:02:07.030 run out of space, but you can redo it with the units, and 00:02:07.030 --> 00:02:08.550 see that the units do work out. 00:02:08.550 --> 00:02:11.009 When you square time, you have to square the time units, 00:02:11.009 --> 00:02:12.792 although we're solving for time. 00:02:12.792 --> 00:02:18.250 Then, plus acceleration, and acceleration is minus 10, and 00:02:18.250 --> 00:02:25.850 we're going to divided it by 2, so it's minus 5t squared. 00:02:25.850 --> 00:02:30.520 We have minus 500 is equal to 30t plus minus 5t, and we 00:02:30.520 --> 00:02:32.000 could just say minus 5t squared, and 00:02:32.000 --> 00:02:34.270 get rid of this plus. 00:02:34.270 --> 00:02:37.150 At first, you say, Sal-- there's a t, that's t to the 00:02:37.150 --> 00:02:39.600 first, and t to the second, how do I saw solve this? 00:02:39.600 --> 00:02:43.310 Hopefully, you've taken algebra two or algebra one, in 00:02:43.310 --> 00:02:45.430 some places, and you remember how to solve this. 00:02:45.430 --> 00:02:47.310 Otherwise, you're about to learn the quadratic equation, 00:02:47.310 --> 00:02:49.150 although I recommend you go back, and learn about 00:02:49.150 --> 00:02:51.430 factoring in the quadratic equation, which there are 00:02:51.430 --> 00:02:54.280 videos on that I've put on Youtube. 00:02:54.280 --> 00:02:56.570 I hope you watch those first if you don't remember. 00:02:56.570 --> 00:02:59.080 We can do this-- let's put these two right terms on the 00:02:59.080 --> 00:03:01.810 left hand side, and then we'll use the quadratic equation to 00:03:01.810 --> 00:03:03.625 solve, because I don't think this is easy to factor. 00:03:03.625 --> 00:03:14.020 We'll get 5t squared minus 30t minus 500 is equal to 0-- I 00:03:14.020 --> 00:03:17.060 just took these terms and put them on the left side. 00:03:17.060 --> 00:03:20.580 We could divide both sides by 5, just to simplify things, 00:03:20.580 --> 00:03:27.740 and so we get t squared minus 6t minus 100 is equal to 0. 00:03:27.740 --> 00:03:30.160 I could do that, because 0 divided by 5 is just five, so 00:03:30.160 --> 00:03:32.220 I just cleaned it up a little bit. 00:03:32.220 --> 00:03:35.130 Let's use the quadratic equation, and for those of us 00:03:35.130 --> 00:03:37.840 who need a refresher, I'll write it down. 00:03:37.840 --> 00:03:43.740 The roots of any quadratic-- in this case, it's t we're 00:03:43.740 --> 00:03:48.060 solving for-- t will equal negative b plus or minus the 00:03:48.060 --> 00:03:56.770 square root of b squared minus 4ac over 2a, where a is a 00:03:56.770 --> 00:03:59.990 coefficient on this term, b is the coefficient on this term, 00:03:59.990 --> 00:04:03.900 negative 6, and c is the constant, so minus 100. 00:04:03.900 --> 00:04:05.830 Let's just solve. 00:04:05.830 --> 00:04:11.190 We get t is equal to negative b-- so negative this term. 00:04:11.190 --> 00:04:14.850 This term is negative 6, so if we make it a negative, it 00:04:14.850 --> 00:04:16.380 becomes plus 6. 00:04:16.380 --> 00:04:23.305 It becomes 6 plus or minus the square root of b squared, so 00:04:23.305 --> 00:04:33.130 it's minus 6 squared, 36, minus 4 times a, and the 00:04:33.130 --> 00:04:36.826 coefficient on a is here, and that's just times 1. 00:04:36.826 --> 00:04:42.810 With 4ac, c is a constant term, minus 100, minus 4 times 00:04:42.810 --> 00:04:51.250 1 times minus 100, and all of that is over 2a-- a is 1 00:04:51.250 --> 00:04:54.010 agains, so all of that is over 2. 00:04:54.010 --> 00:04:59.140 That just equals 6 plus or minus the square root-- this 00:04:59.140 --> 00:05:02.470 is minus 4 times minus 100, and these become pluses, so it 00:05:02.470 --> 00:05:05.490 becomes 36 plus 400. 00:05:05.490 --> 00:05:15.180 So, 6 plus or minus 436 divided by 2. 00:05:15.180 --> 00:05:17.680 This is not a clean number, and if you type into a 00:05:17.680 --> 00:05:22.050 calculator, it's something on the order of about 20.9. 00:05:22.050 --> 00:05:24.730 We can just say approximately 21-- you might want to get the 00:05:24.730 --> 00:05:26.710 exact number, if you're actually doing this on a test, 00:05:26.710 --> 00:05:30.880 or trying to send something to Mars, but for our purposes, I 00:05:30.880 --> 00:05:32.420 think you'll get the point. 00:05:32.420 --> 00:05:34.080 I'll say it approximately now, because we're going to be a 00:05:34.080 --> 00:05:36.210 little off, but just to have clean numbers, this is 00:05:36.210 --> 00:05:39.540 approximately 21-- it's like 20.9. 00:05:39.540 --> 00:05:43.260 We'll say 6 plus or minus-- let me just write 00:05:43.260 --> 00:05:52.410 20.9-- 20.9 over 2. 00:05:52.410 --> 00:05:58.250 Let me ask you a question: if I do 6 minus 00:05:58.250 --> 00:06:00.940 20.9, what do I get? 00:06:00.940 --> 00:06:02.770 I get a negative number, and does a 00:06:02.770 --> 00:06:05.340 negative time make sense? 00:06:05.340 --> 00:06:08.120 No, it does not, and that means that somehow in the 00:06:08.120 --> 00:06:11.100 past-- I don't want to get philosophical-- the negative 00:06:11.100 --> 00:06:13.360 time in this context will not make sense. 00:06:13.360 --> 00:06:16.342 Really, we can just stick to the plus, because 6 minus 20 00:06:16.342 --> 00:06:20.540 is negative, so there's only one time that will solve this 00:06:20.540 --> 00:06:22.570 in a meaningful way. 00:06:22.570 --> 00:06:28.900 Time is approximately equal to 6 plus 20.9, so that's 26.9 00:06:28.900 --> 00:06:39.690 over 2, and that equals 13.45 seconds. 00:06:39.690 --> 00:06:40.670 That's interesting. 00:06:40.670 --> 00:06:43.490 I think if you remember way back, maybe four or five 00:06:43.490 --> 00:06:47.110 videos ago, when we first did this problem, we just dropped 00:06:47.110 --> 00:06:49.530 the penny straight from the height. 00:06:49.530 --> 00:06:52.095 Actually, in that problem, I gave you the time-- I said it 00:06:52.095 --> 00:06:54.790 took 10 seconds to hit the ground, and we worked 00:06:54.790 --> 00:06:58.660 backwards to figure out that the cliff was 500 meters high. 00:06:58.660 --> 00:07:02.660 Now, if you're here at the top of a 500 cliff, or building, 00:07:02.660 --> 00:07:04.970 and you drop something that has air resistance-- like a 00:07:04.970 --> 00:07:07.700 penny, that has very air resistance-- it would take 10 00:07:07.700 --> 00:07:10.060 seconds to reach the ground, assuming all of our 00:07:10.060 --> 00:07:12.000 assumptions about gravity. 00:07:12.000 --> 00:07:15.280 But if you were to throw the penny straight up, off the 00:07:15.280 --> 00:07:19.470 edge of the cliff, at 30 meters per second right here, 00:07:19.470 --> 00:07:23.720 it's going to take 13.5-- roughly, 13 and 1/2 seconds-- 00:07:23.720 --> 00:07:24.680 to reach the ground. 00:07:24.680 --> 00:07:27.460 It takes a little bit longer, and that should make sense 00:07:27.460 --> 00:07:33.230 because-- I have time to draw a little picture. 00:07:33.230 --> 00:07:43.040 In the first case, I just took the penny, and its motion just 00:07:43.040 --> 00:07:45.220 went straight down. 00:07:45.220 --> 00:07:51.310 In the second case, I took the penny-- it first went up, and 00:07:51.310 --> 00:07:52.930 then it went down. 00:07:55.440 --> 00:07:59.600 It had all the time when it went up, and then it went down 00:07:59.600 --> 00:08:03.310 a longer distance, so it makes sense that this time-- this 00:08:03.310 --> 00:08:09.950 was 10 seconds, while this time was 13.45 five seconds. 00:08:09.950 --> 00:08:13.310 You can kind of say that it took-- well, you actually 00:08:13.310 --> 00:08:15.920 can't say that. 00:08:15.920 --> 00:08:20.000 I don't want to get too involved, but I hopefully this 00:08:20.000 --> 00:08:20.710 make sense to you. 00:08:20.710 --> 00:08:23.620 If you have a smaller number here, you should have gone and 00:08:23.620 --> 00:08:25.600 checked your work, because why would it take less time when I 00:08:25.600 --> 00:08:30.030 throw the object straight up? 00:08:30.030 --> 00:08:33.370 Hopefully, that gave you a little bit more intuition, and 00:08:33.370 --> 00:08:36.620 you really do have in your arsenal now all of the 00:08:36.620 --> 00:08:40.049 equations-- and hopefully, the intuition you need-- to solve 00:08:40.049 --> 00:08:41.480 basic projectile problems. 00:08:41.480 --> 00:08:43.960 I'll now probably do a couple more videos where I just do a 00:08:43.960 --> 00:08:47.580 bunch of problems, just to really drive the points home. 00:08:47.580 --> 00:08:50.570 Then, I'll expand these problems to two 00:08:50.570 --> 00:08:52.700 dimensions and angles. 00:08:52.700 --> 00:08:54.510 Before we get there, you might want to refresh your 00:08:54.510 --> 00:08:55.440 trigonometry. 00:08:55.440 --> 00:08:57.060 I'll see you soon.

Projectile motion (part 3)

https://www.youtube.com/watch?v=Y5cSGxdDHz4

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WEBVTT Kind: captions Language: en 00:00:01.140 --> 00:00:05.910 In the last video, I said that we started off with the change 00:00:05.910 --> 00:00:08.119 in distance, so we said that we know 00:00:08.119 --> 00:00:09.369 the change in distance. 00:00:11.490 --> 00:00:13.350 These are the things that we are given. 00:00:13.350 --> 00:00:17.610 We're given the acceleration, we're given the initial 00:00:17.610 --> 00:00:20.910 velocity, and I asked you how do we figure out what the 00:00:20.910 --> 00:00:22.070 final velocity is? 00:00:22.070 --> 00:00:24.290 In the last video-- if you don't remember it, go watch 00:00:24.290 --> 00:00:28.960 that last video again-- we derived the formula that vf 00:00:28.960 --> 00:00:34.640 squared, the final velocity squared, is equal to the 00:00:34.640 --> 00:00:39.180 initial velocity squared plus 2 times 00:00:39.180 --> 00:00:40.680 the change in distance. 00:00:40.680 --> 00:00:43.320 You'll sometimes just see it written as 2 times distance, 00:00:43.320 --> 00:00:47.570 because we assume that the initial distance is at point 00:00:47.570 --> 00:00:49.470 0, so the change in distance would just 00:00:49.470 --> 00:00:51.540 be the final distance. 00:00:51.540 --> 00:00:53.620 We could write it either way, and hopefully, at this point, 00:00:53.620 --> 00:00:55.600 you see why I keep switching between change in 00:00:55.600 --> 00:00:57.060 distance and distance. 00:00:57.060 --> 00:00:59.740 It's just so you're comfortable when you see it 00:00:59.740 --> 00:01:01.630 either way. 00:01:01.630 --> 00:01:04.330 This is for the situation when we didn't 00:01:04.330 --> 00:01:06.230 know what the vf was. 00:01:06.230 --> 00:01:11.330 Let's say we want to solve for time instead. 00:01:11.330 --> 00:01:13.470 Once we solve for the final velocity, we could actually 00:01:13.470 --> 00:01:15.620 solve for time, and I'll show you how to do that, but let's 00:01:15.620 --> 00:01:17.460 say we didn't want to go through this step-- how can we 00:01:17.460 --> 00:01:20.560 solve for time directly, given the change in distance, the 00:01:20.560 --> 00:01:23.140 acceleration, and the initial velocity? 00:01:23.140 --> 00:01:27.620 Let's go back once again to the most basic distance 00:01:27.620 --> 00:01:30.040 formula-- not the distance formula, but how distance 00:01:30.040 --> 00:01:32.290 relates to velocity. 00:01:32.290 --> 00:01:36.010 We know that-- I'll write it slightly different this time-- 00:01:36.010 --> 00:01:44.130 the change in distance over the change in time is equal to 00:01:44.130 --> 00:01:47.520 the average velocity. 00:01:47.520 --> 00:01:56.030 We could have rewritten this as the change in distance is 00:01:56.030 --> 00:02:00.780 equal to the average velocity times the change in time. 00:02:00.780 --> 00:02:04.130 This is change in time and change in distance. 00:02:04.130 --> 00:02:07.510 Sometimes we'll just see this written as d equals-- let me 00:02:07.510 --> 00:02:09.440 write this in a different color, so we have some 00:02:09.440 --> 00:02:16.640 variety-- velocity times time, or d equals rate times time. 00:02:16.640 --> 00:02:19.010 The reason why I have change in distance here, or change in 00:02:19.010 --> 00:02:22.310 time, is that I'm not assuming necessarily that we're 00:02:22.310 --> 00:02:24.920 starting off at the point 0 or at time 0. 00:02:24.920 --> 00:02:28.340 If we do, then it just turns out to the final distance is 00:02:28.340 --> 00:02:31.920 equal to the average velocity times the final time, but 00:02:31.920 --> 00:02:32.950 let's stick to this. 00:02:32.950 --> 00:02:37.720 We want to figure out time given this set of inputs. 00:02:45.090 --> 00:02:48.200 Let's go from this equation. 00:02:48.200 --> 00:02:51.670 If we want to solve for time, or the change in time, we 00:02:51.670 --> 00:02:55.730 could just could divide both sides by the average 00:02:55.730 --> 00:02:58.600 velocity-- actually, no, let's not do that. 00:02:58.600 --> 00:03:01.180 Let's just stay in terms of change in distance. 00:03:05.266 --> 00:03:08.740 I've wasted space too fast, so let me clear 00:03:08.740 --> 00:03:12.670 this and start again. 00:03:12.670 --> 00:03:23.750 We're given change in distance, initial velocity, 00:03:23.750 --> 00:03:27.780 and acceleration, and we want to figure out what the time 00:03:27.780 --> 00:03:31.810 is-- it's really the change in time, but let's just assume 00:03:31.810 --> 00:03:34.070 that we start time 0, so it's kind of the final time. 00:03:37.150 --> 00:03:40.230 Let's just start with the simple formula: distance, or 00:03:40.230 --> 00:03:44.790 change in distance-- I'll use them interchangeably, with a 00:03:44.790 --> 00:03:48.990 lower case d this time-- is equal to the average velocity 00:03:48.990 --> 00:03:51.320 times time. 00:03:51.320 --> 00:03:52.570 What's the average velocity? 00:03:56.600 --> 00:04:01.730 The average velocity is just the initial velocity plus the 00:04:01.730 --> 00:04:04.230 final velocity over 2. 00:04:04.230 --> 00:04:07.565 The only reason why we can just average the initial and 00:04:07.565 --> 00:04:09.410 the final is because we're assuming constant 00:04:09.410 --> 00:04:12.240 acceleration, and that's very important, but in most 00:04:12.240 --> 00:04:14.840 projectile problems, we do have constant acceleration-- 00:04:14.840 --> 00:04:16.890 downwards-- and that's gravity. 00:04:16.890 --> 00:04:18.899 We can assume, and we can do this-- we can say that the 00:04:18.899 --> 00:04:22.070 average of the initial and the final velocity is the average 00:04:22.070 --> 00:04:25.080 velocity, and then we multiply that times time. 00:04:29.050 --> 00:04:30.700 Can we use this equation directly? 00:04:30.700 --> 00:04:33.310 No. we know acceleration, but don't know final velocity. 00:04:33.310 --> 00:04:37.040 If we can write this final velocity in terms of the other 00:04:37.040 --> 00:04:41.060 things in this equation, then maybe we can solve for time. 00:04:41.060 --> 00:04:47.960 Let's try to do that: distance is equal to-- let me take a 00:04:47.960 --> 00:04:49.250 little side here. 00:04:49.250 --> 00:04:51.870 What do we know about final velocity? 00:04:51.870 --> 00:04:57.850 We know that the change in velocity is equal to 00:04:57.850 --> 00:05:01.620 acceleration times time, assuming that time 00:05:01.620 --> 00:05:03.900 starts a t equals 0. 00:05:03.900 --> 00:05:09.040 The change in velocity is the same thing is vf minus vi is 00:05:09.040 --> 00:05:11.860 equal to acceleration times time. 00:05:11.860 --> 00:05:14.250 We know that the final velocity is equal to the 00:05:14.250 --> 00:05:18.860 initial velocity plus acceleration times time. 00:05:18.860 --> 00:05:21.550 Let's substitute that back into what I was 00:05:21.550 --> 00:05:22.890 writing right here. 00:05:22.890 --> 00:05:28.170 We have distance is equal to the initial velocity plus the 00:05:28.170 --> 00:05:31.530 final velocity, so let's substitute this expression 00:05:31.530 --> 00:05:32.435 right here. 00:05:32.435 --> 00:05:36.835 The initial velocity, plus, now the final velocity is now 00:05:36.835 --> 00:05:41.910 the initial velocity, plus acceleration times time, and 00:05:41.910 --> 00:05:48.000 then we divide all of that by 2 times time. 00:05:48.000 --> 00:05:53.610 We get d is equal to-- we have 2 in the numerator, we have 2 00:05:53.610 --> 00:06:02.510 initial velocity, 2vi's plus at over 2, and all 00:06:02.510 --> 00:06:05.600 of that times t. 00:06:05.600 --> 00:06:07.730 Then we can simplify this. 00:06:07.730 --> 00:06:12.380 This equals d is equal to-- this 2 cancels out this 2, and 00:06:12.380 --> 00:06:16.260 then we distribute this t across both terms-- so d is 00:06:16.260 --> 00:06:24.820 equal to vit plus-- this term is at over 2, but then you 00:06:24.820 --> 00:06:27.300 multiply the t times here, too-- so it's at squared over 00:06:27.300 --> 00:06:34.190 2 plus at squared over 2. 00:06:34.190 --> 00:06:36.480 We could use this formula if we know the change in 00:06:36.480 --> 00:06:40.490 distance, or the distance-- this actually should be the 00:06:40.490 --> 00:06:43.190 change in distance, and the change in time-- is equal to 00:06:43.190 --> 00:06:47.030 the initial velocity times time plus acceleration times 00:06:47.030 --> 00:06:48.860 squared divided by 2. 00:06:48.860 --> 00:06:51.860 Let me summarize all of the equations we have, because we 00:06:51.860 --> 00:06:55.630 really now have in our arsenal every equation that you really 00:06:55.630 --> 00:07:00.400 need to solve one dimensional projectile problems-- things 00:07:00.400 --> 00:07:03.080 going either just left, right, east, west, or north, south, 00:07:03.080 --> 00:07:04.000 but not both. 00:07:04.000 --> 00:07:06.340 I will do that in the next video. 00:07:06.340 --> 00:07:09.060 Let's summarize everything we know. 00:07:13.820 --> 00:07:19.970 We know the change in distance divided by the change in time 00:07:19.970 --> 00:07:23.730 is equal to velocity-- average velocity, and it would equal 00:07:23.730 --> 00:07:26.830 velocity if velocity's not changing, but average when 00:07:26.830 --> 00:07:29.597 velocity does change-- and we have constant acceleration, 00:07:29.597 --> 00:07:31.700 which is an important assumption. 00:07:31.700 --> 00:07:35.870 We know that the change in velocity divided by the change 00:07:35.870 --> 00:07:40.500 in time is equal to acceleration. 00:07:40.500 --> 00:07:47.300 We know the average velocity is equal to the final velocity 00:07:47.300 --> 00:07:51.650 plus the initial velocity over 2, and this assumes 00:07:51.650 --> 00:07:52.900 acceleration is constant. 00:08:01.870 --> 00:08:04.870 If we know the initial velocity, acceleration, and 00:08:04.870 --> 00:08:09.540 the distance, and we want to figure out the final velocity, 00:08:09.540 --> 00:08:20.300 we could use this formula: vf squared equals vi squared plus 00:08:20.300 --> 00:08:25.603 2a times-- really the change in distance, so I'm going to 00:08:25.603 --> 00:08:27.500 write the change in distance, because that sometimes matters 00:08:27.500 --> 00:08:30.890 when we're dealing with direction-- change in 00:08:30.890 --> 00:08:32.299 distance, but so you'll sometimes just 00:08:32.299 --> 00:08:34.919 write this as distance. 00:08:34.919 --> 00:08:36.970 Then we just did the equation-- I think I did this 00:08:36.970 --> 00:08:40.440 in the third video, as well, early on-- but we also learned 00:08:40.440 --> 00:08:47.870 that distance is equal to the initial velocity times time 00:08:47.870 --> 00:08:53.230 plus at squared over 2. 00:08:53.230 --> 00:08:56.230 In that example that I did a couple of videos ago, where we 00:08:56.230 --> 00:08:58.160 had a cliff-- actually, I only have a minute 00:08:58.160 --> 00:08:58.890 left in this video. 00:08:58.890 --> 00:09:01.880 I will do that in the next presentation. 00:09:01.880 --> 00:09:03.270 I'll see you soon.

Projectile motion (part 2)

https://www.youtube.com/watch?v=emdHj6WodLw

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WEBVTT Kind: captions Language: en 00:00:00.920 --> 00:00:05.090 In the last video, I dropped myself or a penny from the top 00:00:05.090 --> 00:00:06.630 of a cliff. 00:00:06.630 --> 00:00:09.770 We started off at 0 velocity, obviously because it was 00:00:09.770 --> 00:00:13.290 stationary, and at the bottom, it was 100 meters per second. 00:00:13.290 --> 00:00:17.660 We used that to figure out how high the cliff was, and we 00:00:17.660 --> 00:00:20.120 figured out that the cliff was 500 meters high. 00:00:20.120 --> 00:00:22.310 What I want to do now is let's do that same problem, but 00:00:22.310 --> 00:00:25.310 let's do in a general form, and see if we can figure out a 00:00:25.310 --> 00:00:29.270 general formula for a problem like that. 00:00:29.270 --> 00:00:34.920 Let's say that you have the same thing, and let's say the 00:00:34.920 --> 00:00:43.210 initial velocity-- you're given the initial velocity, 00:00:43.210 --> 00:00:46.900 you're given the final velocity, you're given the 00:00:46.900 --> 00:00:52.410 acceleration, and you want to figure out the distance. 00:00:52.410 --> 00:00:56.980 This is what you're given, and you want to 00:00:56.980 --> 00:00:58.230 figure out the distance. 00:01:03.730 --> 00:01:08.620 Doing it the exact same way we did in that last presentation, 00:01:08.620 --> 00:01:09.650 but now we're now [INAUDIBLE] 00:01:09.650 --> 00:01:15.740 formulas, we know that the change in distance is equal to 00:01:15.740 --> 00:01:20.950 the average velocity times-- we could actually say the 00:01:20.950 --> 00:01:23.300 change in time, but I'll just say it with time, because we 00:01:23.300 --> 00:01:24.470 always assume that we start with time 00:01:24.470 --> 00:01:27.810 equals 0-- times time. 00:01:27.810 --> 00:01:35.880 We know that the average velocity is the final velocity 00:01:35.880 --> 00:01:43.190 plus the initial velocity divided by 2, so that's the 00:01:43.190 --> 00:01:47.800 average velocity-- let me highlight-- this is the same 00:01:47.800 --> 00:01:52.890 thing as this, and then that times time. 00:01:52.890 --> 00:01:55.510 What's the time? 00:01:55.510 --> 00:01:58.120 You could figure out the time by saying, we know how fast 00:01:58.120 --> 00:02:00.430 we're accelerating, and we know the initial and final 00:02:00.430 --> 00:02:05.040 velocity, so we can figure out how long we had to accelerate 00:02:05.040 --> 00:02:07.580 that fast to get that change in velocity. 00:02:07.580 --> 00:02:09.280 Another way of saying that, or probably a simpler way of 00:02:09.280 --> 00:02:15.730 saying that, is change in velocity, which is the same 00:02:15.730 --> 00:02:19.220 thing as the final velocity minus the initial velocity is 00:02:19.220 --> 00:02:22.230 equal to acceleration times time. 00:02:25.560 --> 00:02:29.460 If you want to solve for time, you could say the time-- if I 00:02:29.460 --> 00:02:34.240 just divide both sides of this equation by a-- is equal to vf 00:02:34.240 --> 00:02:39.920 minus vi divided by a. 00:02:39.920 --> 00:02:42.210 We could take that and substitute that into this 00:02:42.210 --> 00:02:45.910 equation, and remember-- this is all change in distance. 00:02:45.910 --> 00:02:50.190 We say change in distance is equal to-- let me write this 00:02:50.190 --> 00:02:58.740 term in yellow-- vf plus vi over 2. 00:02:58.740 --> 00:03:03.650 Let me write this term in green. 00:03:03.650 --> 00:03:10.520 That's times vf minus vi over a. 00:03:15.130 --> 00:03:18.800 Then if we do a little multiplying of expressions on 00:03:18.800 --> 00:03:21.690 the top-- you might have recognized this-- this would 00:03:21.690 --> 00:03:27.760 be vf squared minus vi squared, and then we multiply 00:03:27.760 --> 00:03:30.040 the denominators over 2a. 00:03:30.040 --> 00:03:33.570 So the change in distance is equal to vf squared minus vi 00:03:33.570 --> 00:03:35.630 squared over 2a. 00:03:35.630 --> 00:03:37.710 That's exciting-- let me write that over again. 00:03:45.000 --> 00:03:53.940 The change in distance is equal to vf squared minus vi 00:03:53.940 --> 00:03:59.300 squared divided by 2 times acceleration. 00:03:59.300 --> 00:04:01.710 We could play around with this a little bit, and if we assume 00:04:01.710 --> 00:04:04.290 that we started distance is equal to 0, we could write d 00:04:04.290 --> 00:04:06.150 here, and that might simplify things. 00:04:06.150 --> 00:04:10.600 If we multiply both sides by 2a, we get-- and I'm just 00:04:10.600 --> 00:04:12.710 going to switch this to distance, if we assume that we 00:04:12.710 --> 00:04:15.370 always start at distances equal to 0. 00:04:15.370 --> 00:04:19.230 di, or initial distance, is always at point 0. 00:04:19.230 --> 00:04:24.360 We could right 2ad-- I'm just multiplying both sides by 2a-- 00:04:24.360 --> 00:04:31.860 is equal to vf squared minus vi squared, or you could write 00:04:31.860 --> 00:04:41.030 it as vf squared is equal to vi squared plus 2ad. 00:04:41.030 --> 00:04:43.330 I don't know what your physics teacher might show you or 00:04:43.330 --> 00:04:46.400 written in your physics book, but of these variations will 00:04:46.400 --> 00:04:48.010 show up in your physics book. 00:04:48.010 --> 00:04:49.870 The reason why I wanted to show you that previous problem 00:04:49.870 --> 00:04:52.470 first is that I wanted to show you that you could actually 00:04:52.470 --> 00:04:55.900 figure out these problems without having to always 00:04:55.900 --> 00:04:58.780 memorize formulas and resort to the formula. 00:04:58.780 --> 00:05:00.930 With that said, it's probably not bad idea to memorize some 00:05:00.930 --> 00:05:04.600 form of this formula, although you should understand how it 00:05:04.600 --> 00:05:06.892 was derived, and when to apply it. 00:05:09.670 --> 00:05:12.230 Now that you have memorized it, or I showed you that maybe 00:05:12.230 --> 00:05:15.920 you don't have to memorize it, let's use this. 00:05:15.920 --> 00:05:17.970 Let's say I have the same cliff, and it 00:05:17.970 --> 00:05:21.490 has now turned purple. 00:05:21.490 --> 00:05:27.070 It was 500 meters high-- it's a 500 meter high cliff. 00:05:27.070 --> 00:05:29.960 This time, with the penny, instead of just dropping it 00:05:29.960 --> 00:05:35.470 straight down, I'm going to throw it straight up at 00:05:35.470 --> 00:05:37.930 positive 30 meters per second. 00:05:37.930 --> 00:05:40.640 The positive matters, because remember, we said negative is 00:05:40.640 --> 00:05:43.450 down, positive is up-- that's just the convention we use. 00:05:43.450 --> 00:05:49.370 Let's use this formula, or any version of this formula, to 00:05:49.370 --> 00:05:56.940 figure out what our final velocity was when we hit the 00:05:56.940 --> 00:05:59.000 bottom of the ground. 00:05:59.000 --> 00:06:01.040 This is probably the easiest formula to use, because it 00:06:01.040 --> 00:06:03.240 actually solves for final velocity. 00:06:03.240 --> 00:06:08.430 We can say the final velocity vf squared is equal to the 00:06:08.430 --> 00:06:12.780 initial velocity squared-- so what's our initial velocity? 00:06:12.780 --> 00:06:20.500 It's plus 30 meters per second, so it's 30 meters per 00:06:20.500 --> 00:06:26.600 second squared plus 2ad. 00:06:26.600 --> 00:06:30.550 So, 2a is the acceleration of gravity, which is minus 10, 00:06:30.550 --> 00:06:36.840 because it's going down, so it's 2a times minus 10-- I'm 00:06:36.840 --> 00:06:39.060 going to give up the units for a second, just so I don't run 00:06:39.060 --> 00:06:43.300 out of space-- 2 times minus 10, and what's the height? 00:06:43.300 --> 00:06:45.320 What's the change in distance? 00:06:45.320 --> 00:06:47.950 Actually, I should be correct about using change in 00:06:47.950 --> 00:06:51.490 distance, because it matters for this problem. 00:06:51.490 --> 00:06:55.520 In this case, the final distance is equal to minus 00:06:55.520 --> 00:07:00.550 500, and the initial distance is equal to 0. 00:07:00.550 --> 00:07:07.170 The change in distance is minus 500. 00:07:07.170 --> 00:07:08.450 So what does this get us? 00:07:08.450 --> 00:07:15.710 We get vf squared is equal to 900, and the negatives cancel 00:07:15.710 --> 00:07:26.570 out-- 10 times 500 is 5,000, and 5,000 times 2 is 10,000. 00:07:26.570 --> 00:07:37.220 So vf squared is equal to 10,900. 00:07:37.220 --> 00:07:45.460 So the final velocity is equal to the square root of 10,900. 00:07:45.460 --> 00:07:46.000 What is that? 00:07:46.000 --> 00:07:53.310 Let me bring over my trusty Windows-provided default 00:07:53.310 --> 00:07:54.550 calculator. 00:07:54.550 --> 00:08:01.830 It's 10,900, and the square root. 00:08:01.830 --> 00:08:15.110 It's about 104 meters per second, so my final velocity 00:08:15.110 --> 00:08:17.530 is approximately-- that squiggly equals is 00:08:17.530 --> 00:08:22.060 approximately-- 104 meters per second. 00:08:22.060 --> 00:08:23.090 That's interesting. 00:08:23.090 --> 00:08:26.710 If I just dropped something-- if I just drop it straight 00:08:26.710 --> 00:08:28.820 from the top-- we figured out in the last problem that at 00:08:28.820 --> 00:08:31.810 the end, I'm at 100 meters per second. 00:08:31.810 --> 00:08:35.000 But this time, if I throw it straight up at 30 meters per 00:08:35.000 --> 00:08:38.559 second, when the penny hits the ground, it's actually 00:08:38.559 --> 00:08:40.480 going even faster. 00:08:40.480 --> 00:08:43.799 You might want to think about why that is, and you might 00:08:43.799 --> 00:08:44.380 realize it. 00:08:44.380 --> 00:08:46.960 When I throw it up, the highest point of the penny-- 00:08:46.960 --> 00:08:50.230 if I throw it up at 30 meters per second, the highest point 00:08:50.230 --> 00:08:53.320 of the penny is going to be higher than 500 meters-- is 00:08:53.320 --> 00:08:56.250 going to make some positive distance first, and then it's 00:08:56.250 --> 00:08:59.030 going to come down, so it's going to have even more time 00:08:59.030 --> 00:09:00.910 to accelerate. 00:09:00.910 --> 00:09:05.660 I think that makes some intuitive sense to you. 00:09:05.660 --> 00:09:07.890 That's all the time I have now, and in the next 00:09:07.890 --> 00:09:10.800 presentation, maybe I'll use this formula to solve a couple 00:09:10.800 --> 00:09:13.420 of other types of problems.

Projectile motion (part 1)

https://www.youtube.com/watch?v=15zliAL4llE

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https://www.youtube.com/api/timedtext?v=15zliAL4llE&ei=dmeUZY34G4mzvdIP0LK0wAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=CDD97AFEFB30966BCE6FA5EB21DB5A843092AB5B.1EEACF06DF51CBB7C646C2B7A86E7BCBE709FD8E&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.730 --> 00:00:01.740 Welcome back. 00:00:01.740 --> 00:00:05.500 I'm not going to do a bunch of projectile motion problems, 00:00:05.500 --> 00:00:07.800 and this is because I think you learn more just seeing 00:00:07.800 --> 00:00:09.610 someone do it, and thinking out loud, 00:00:09.610 --> 00:00:11.040 than all the formulas. 00:00:11.040 --> 00:00:14.570 I have a strange notion that I might have done more harm than 00:00:14.570 --> 00:00:16.700 good by confusing you with a lot of what I did in the last 00:00:16.700 --> 00:00:19.480 couple of videos, so hopefully I can undo any damage if I 00:00:19.480 --> 00:00:21.430 have done any, or even better-- hopefully, you did 00:00:21.430 --> 00:00:24.530 learn from those, and we'll just add to the learning. 00:00:24.530 --> 00:00:28.350 Let's start with a general problem. 00:00:28.350 --> 00:00:32.150 Let's say that I'm at the top of a cliff, and I jump-- 00:00:32.150 --> 00:00:35.750 instead of throwing something, I just jump off the cliff. 00:00:35.750 --> 00:00:38.770 We won't worry about my motion from side to side, but just 00:00:38.770 --> 00:00:40.060 assume that I go straight down. 00:00:40.060 --> 00:00:42.740 We could even think that someone just dropped me off of 00:00:42.740 --> 00:00:43.750 the top of the cliff. 00:00:43.750 --> 00:00:46.670 I know these are getting kind of morbid, but let's just 00:00:46.670 --> 00:00:49.120 assume that nothing bad happens to me. 00:00:49.120 --> 00:00:54.560 Let's say that at the top of the cliff, my initial 00:00:54.560 --> 00:01:01.160 velocity-- velocity initial-- is going to be 0, because I'm 00:01:01.160 --> 00:01:06.160 stationary before the person drops me or before I jump. 00:01:06.160 --> 00:01:16.740 At the bottom of the cliff my velocity is 00:01:16.740 --> 00:01:23.730 100 meters per second. 00:01:23.730 --> 00:01:26.750 My question is, what is the height of this cliff? 00:01:29.590 --> 00:01:32.970 I think this is a good time to actually introduce the 00:01:32.970 --> 00:01:35.440 direction notion of velocity, to show 00:01:35.440 --> 00:01:37.200 you this scalar quantity. 00:01:37.200 --> 00:01:41.010 Let's assume up is positive, and down is negative. 00:01:41.010 --> 00:01:44.160 My velocity is actually 100 meters per second down-- I 00:01:44.160 --> 00:01:46.720 could have assumed the opposite. 00:01:46.720 --> 00:01:49.480 The final velocity is 100 meters per second down, and 00:01:49.480 --> 00:01:51.860 since we're saying that down is negative, and gravity is 00:01:51.860 --> 00:01:54.210 always pulling you down, we're going to say that our 00:01:54.210 --> 00:01:59.030 acceleration is equal to gravity, which is equal to 00:01:59.030 --> 00:02:02.935 minus 10 meters per second squared. 00:02:06.000 --> 00:02:08.120 I just wrote that ahead of times, because when we're 00:02:08.120 --> 00:02:09.780 dealing with anything of throwing or jumping or 00:02:09.780 --> 00:02:13.340 anything on this planet, we could just use this constant-- 00:02:13.340 --> 00:02:15.950 the actual number is 9.81, but I want to be able to do this 00:02:15.950 --> 00:02:19.110 without a calculator, so I'll just stick with minus 10 00:02:19.110 --> 00:02:20.060 meters per second squared. 00:02:20.060 --> 00:02:23.440 It's pulling me down, so that's why the minus is there. 00:02:23.440 --> 00:02:26.400 My question is: I know my initial velocity, I know my 00:02:26.400 --> 00:02:30.700 final velocity, right before I hit the ground or right when I 00:02:30.700 --> 00:02:32.420 hit the ground, what's the distance? 00:02:35.770 --> 00:02:38.010 In this circumstance, what does distance represent? 00:02:38.010 --> 00:02:41.060 Distance would be the height of the cliff, and so how do we 00:02:41.060 --> 00:02:42.310 figure this out? 00:02:45.400 --> 00:02:48.470 What's the only formula that we know for distance, or 00:02:48.470 --> 00:02:52.090 actually the change in distance, but in this case, 00:02:52.090 --> 00:02:52.990 it's the same thing. 00:02:52.990 --> 00:02:58.790 Change in distance is equal to the average velocity. 00:02:58.790 --> 00:03:01.650 When you learned this in middle school, or probably 00:03:01.650 --> 00:03:03.580 even elementary school, you didn't say average velocity, 00:03:03.580 --> 00:03:05.720 because you always assumed velocity was constant-- the 00:03:05.720 --> 00:03:07.990 average and the instantaneous velocity was 00:03:07.990 --> 00:03:09.560 kind of the same thing. 00:03:09.560 --> 00:03:12.100 Now, since the velocity is changing, we're going to say 00:03:12.100 --> 00:03:13.790 the average velocity. 00:03:13.790 --> 00:03:16.350 So, the change in distance is equal to the average velocity 00:03:16.350 --> 00:03:18.210 times time. 00:03:18.210 --> 00:03:20.020 This should be intuitive to you at this point. 00:03:20.020 --> 00:03:23.530 Velocity really is just distance divided by time, or 00:03:23.530 --> 00:03:26.290 actually, change in distance divided by times change in 00:03:26.290 --> 00:03:28.430 time-- or, change in distance divided by 00:03:28.430 --> 00:03:30.270 change in times is velocity. 00:03:30.270 --> 00:03:32.406 Let me actually change this to change in time. 00:03:35.030 --> 00:03:38.145 Since we always assume-- or we normally assume-- that we 00:03:38.145 --> 00:03:42.870 start at distance is equal to 0, and we assume that start at 00:03:42.870 --> 00:03:46.380 time is equal to 0, we can write distance is equal to 00:03:46.380 --> 00:03:49.970 velocity average times time. 00:03:49.970 --> 00:03:52.110 Maybe later on we'll do situations where we're not 00:03:52.110 --> 00:03:55.620 starting at time 0, or distance 0, and in that case, 00:03:55.620 --> 00:03:57.680 we will have to be a little more formal and say change in 00:03:57.680 --> 00:04:02.915 distance is equal to average velocity the change in time. 00:04:06.430 --> 00:04:10.250 This is a formula we know, and let's see what 00:04:10.250 --> 00:04:11.580 we can figure out. 00:04:11.580 --> 00:04:16.959 Can we figure out the average velocity? 00:04:16.959 --> 00:04:19.459 The average velocity is just the average of the initial 00:04:19.459 --> 00:04:21.940 velocity and the final velocity. 00:04:21.940 --> 00:04:26.500 The average velocity is just equal to the average of these 00:04:26.500 --> 00:04:32.520 two numbers: so, minus 100 plus 0 over 2-- and I'm just 00:04:32.520 --> 00:04:39.230 averaging the numbers-- equals minus 50 meters per second. 00:04:39.230 --> 00:04:40.790 We were able to figure that out, so can 00:04:40.790 --> 00:04:42.040 we figure out time? 00:04:44.250 --> 00:04:48.230 We know also that velocity, or let's say the change in 00:04:48.230 --> 00:04:57.020 velocity, is equal to the final velocity minus the 00:04:57.020 --> 00:04:58.400 initial velocity. 00:04:58.400 --> 00:05:01.640 This is nothing fancy-- you don't have to memorize this. 00:05:01.640 --> 00:05:04.390 This hopefully is intuitive to you, that the change is just 00:05:04.390 --> 00:05:06.755 the final velocity minus the initial velocity, and that 00:05:06.755 --> 00:05:12.380 that equals acceleration times time. 00:05:12.380 --> 00:05:15.920 So what's the change in velocity in this situation? 00:05:15.920 --> 00:05:22.815 Final velocity is minus 100 meters per second, and then 00:05:22.815 --> 00:05:27.260 the initial velocity is 0, so the change in velocity is 00:05:27.260 --> 00:05:31.290 equal to minus 100 meters per second. 00:05:31.290 --> 00:05:34.030 I'm kind of jumping in and out of the units, but I think you 00:05:34.030 --> 00:05:35.780 get what I'm doing. 00:05:35.780 --> 00:05:39.230 That equals acceleration times time-- what's the 00:05:39.230 --> 00:05:39.560 acceleration? 00:05:39.560 --> 00:05:42.150 It's minus 10 meters per second squared, because I'm 00:05:42.150 --> 00:05:46.920 going straight down-- minus 10 meters per second squared 00:05:46.920 --> 00:05:48.550 times time. 00:05:48.550 --> 00:05:51.365 This is a pretty straightforward equation. 00:05:51.365 --> 00:05:54.400 Let's divide both sides by the acceleration, by the minus 10 00:05:54.400 --> 00:05:59.160 meters per second squared, and you'll get time is equal to-- 00:05:59.160 --> 00:06:03.870 the negatives cancel out, as they should, because negative 00:06:03.870 --> 00:06:07.290 time is difficult, we're assuming positive time, and 00:06:07.290 --> 00:06:09.630 it's good we got a positive time answer-- but the 00:06:09.630 --> 00:06:11.410 negatives cancel out and we get time 00:06:11.410 --> 00:06:14.620 is equal to 10 seconds. 00:06:14.620 --> 00:06:17.560 There we have it: we figured out time, we figured out the 00:06:17.560 --> 00:06:20.030 average velocity, and so now we can figure out the height 00:06:20.030 --> 00:06:21.110 of the cliff. 00:06:21.110 --> 00:06:27.200 The distance is equal to the average velocity minus 50 00:06:27.200 --> 00:06:32.690 meters per second times 10 seconds. 00:06:32.690 --> 00:06:36.210 The distance-- this is going to be an interesting notion to 00:06:36.210 --> 00:06:45.240 you-- the distance it's going to be minus 500 meters. 00:06:45.240 --> 00:06:47.930 This might not make a lot of sense to you-- what does minus 00:06:47.930 --> 00:06:50.510 500 meters mean? 00:06:50.510 --> 00:06:53.660 This is actually right, because this formula is 00:06:53.660 --> 00:06:54.910 actually the change in distance. 00:06:58.780 --> 00:07:00.680 We said if we did it formally, it would be 00:07:00.680 --> 00:07:01.900 the change in distance. 00:07:01.900 --> 00:07:09.470 So if we have a cliff-- let me change colors with it-- and if 00:07:09.470 --> 00:07:13.780 we assume that we start at this point right here, and 00:07:13.780 --> 00:07:21.390 this distance is equal to 0, then the ground, if this cliff 00:07:21.390 --> 00:07:24.855 is 500 hundred meters high, your final distance-- this is 00:07:24.855 --> 00:07:29.140 the initial distance-- your final distance df is actually 00:07:29.140 --> 00:07:33.760 going to be at minus 500 hundred meters. 00:07:33.760 --> 00:07:35.136 We could have done it the other way around: we could 00:07:35.136 --> 00:07:37.930 have said this is plus 500 meters, and then this is 0, 00:07:37.930 --> 00:07:40.010 but all that matters is really the change in distance. 00:07:40.010 --> 00:07:42.240 We're saying from the top of the cliff to the ground, the 00:07:42.240 --> 00:07:45.260 change in distance is minus 500 meters. 00:07:45.260 --> 00:07:48.250 And minus, based on our convention, we said minus is 00:07:48.250 --> 00:07:52.990 down, so the change is 500 meters down, and that's height 00:07:52.990 --> 00:07:53.380 of the cliff. 00:07:53.380 --> 00:07:54.370 That's pretty interesting. 00:07:54.370 --> 00:08:04.160 If you go to a 500 meter cliff-- 500 is about 1,500 00:08:04.160 --> 00:08:08.960 feet-- so that's roughly the size of maybe a very tall 00:08:08.960 --> 00:08:10.480 skyscraper, like the World Trade Center 00:08:10.480 --> 00:08:12.440 or the Sears Tower. 00:08:12.440 --> 00:08:15.860 If you jump off of something like that, assuming no air 00:08:15.860 --> 00:08:18.930 resistance, which is a big assumption, or if you were to 00:08:18.930 --> 00:08:21.110 drop a penny-- because a penny has very little air 00:08:21.110 --> 00:08:24.330 resistance-- if you were to drop a penny off of the top of 00:08:24.330 --> 00:08:28.800 Sears Tower or a building like that, at the bottom it will be 00:08:28.800 --> 00:08:31.600 going 100 meters per second. 00:08:31.600 --> 00:08:35.130 That's extremely fast, and that's why you shouldn't be 00:08:35.130 --> 00:08:38.520 doing it, because that is fast enough to kill somebody, and I 00:08:38.520 --> 00:08:41.830 don't want to give you any bad ideas if you're a bad person. 00:08:41.830 --> 00:08:44.710 It's just interesting that physics allows you to solve 00:08:44.710 --> 00:08:45.950 these types of problems. 00:08:45.950 --> 00:08:47.760 In the next presentation, I'm just going to keep doing 00:08:47.760 --> 00:08:51.090 problems, and hopefully you'll realize that everything really 00:08:51.090 --> 00:08:54.880 just boils down to average velocity-- change in velocity 00:08:54.880 --> 00:08:57.830 is acceleration times time, and change in distance is 00:08:57.830 --> 00:09:03.380 equal to change in time times average velocity, which we all 00:09:03.380 --> 00:09:05.180 did just now. 00:09:05.180 --> 00:09:06.940 I'll see you in the next presentation.

Introduction to motion (part 3)

https://www.youtube.com/watch?v=enmHaVxLfAE

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https://www.youtube.com/api/timedtext?v=enmHaVxLfAE&ei=dmeUZb-LJODjxN8PkrCz0Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=111A52FDCF191498699DE3A68DC93FCD0889DAEF.E0C2BACCA1C54A0AD688E1BDC15720F5B5ACD2E9&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.950 --> 00:00:02.640 I am back. 00:00:02.640 --> 00:00:03.510 Where were we? 00:00:03.510 --> 00:00:08.750 We were saying that we know that velocity, or kind of a 00:00:08.750 --> 00:00:11.440 change in velocity, is acceleration times time. 00:00:11.440 --> 00:00:14.660 I just wrote that a little bit more formally, really kind of 00:00:14.660 --> 00:00:15.990 incorporating the change in velocity. 00:00:15.990 --> 00:00:16.580 Right? 00:00:16.580 --> 00:00:19.460 The final velocity is equal to the initial velocity plus 00:00:19.460 --> 00:00:20.535 acceleration times time. 00:00:20.535 --> 00:00:22.180 I actually could have written it like this: I could have 00:00:22.180 --> 00:00:27.640 written vf minus vi is equal to acceleration times time, 00:00:27.640 --> 00:00:30.010 and this is the change in velocity. 00:00:30.010 --> 00:00:31.190 Actually, that's the way I should be doing it. 00:00:31.190 --> 00:00:34.280 As you can tell, I kind of do some of this stuff on the fly, 00:00:34.280 --> 00:00:37.070 but I do that for a reason-- it's because I want you to get 00:00:37.070 --> 00:00:40.460 the same intuition that I hopefully have, instead of 00:00:40.460 --> 00:00:43.480 just kind of doing it in a very formal way in a book, and 00:00:43.480 --> 00:00:46.400 sometimes the book doesn't necessarily make the 00:00:46.400 --> 00:00:49.110 connections in the most natural way. 00:00:49.110 --> 00:00:52.130 This is going straight from my brain to this video, and 00:00:52.130 --> 00:00:54.410 hopefully into your brain. 00:00:54.410 --> 00:00:56.560 These are all ways of saying the same thing, and I actually 00:00:56.560 --> 00:01:01.960 should write this as change in velocity-- that triangle, or 00:01:01.960 --> 00:01:03.910 delta, just means change. 00:01:03.910 --> 00:01:06.220 The final velocity, my initial velocity, is equal to 00:01:06.220 --> 00:01:08.590 acceleration times time. 00:01:08.590 --> 00:01:11.200 The average velocity, you could just figure-- you take 00:01:11.200 --> 00:01:12.830 the final, and you take the initial, and you average the 00:01:12.830 --> 00:01:16.220 two, and it's equal to this. 00:01:16.220 --> 00:01:21.280 Then I said, we know what the final velocity is-- this is 00:01:21.280 --> 00:01:24.920 the final velocity, and the average velocity is this. 00:01:24.920 --> 00:01:27.470 We substitute it for the final velocity, and then we came to 00:01:27.470 --> 00:01:31.860 this equation for average velocity. 00:01:31.860 --> 00:01:35.650 Then before I almost ran out of time, I said I'm going to 00:01:35.650 --> 00:01:38.720 take this formula for the average velocity-- and I 00:01:38.720 --> 00:01:40.820 really encourage you to just play around with these 00:01:40.820 --> 00:01:43.100 formulas yourself and derive it yourself, because it's 00:01:43.100 --> 00:01:45.930 going to pay huge rewards later on when you forget the 00:01:45.930 --> 00:01:50.650 formulas on your exam, but you can work it out anyway. 00:01:50.650 --> 00:01:52.430 We have this formula for average velocity, and let's 00:01:52.430 --> 00:01:57.600 substitute it back into this, so we can say that distance is 00:01:57.600 --> 00:02:07.290 equal to the average velocity, and that's this: vi plus at 00:02:07.290 --> 00:02:12.680 over 2 times time. 00:02:12.680 --> 00:02:16.160 If we just distributed that t, we have the initial velocity 00:02:16.160 --> 00:02:26.440 times time, plus acceleration times time squared over 2. 00:02:26.440 --> 00:02:29.670 So, distance is equal to the initial velocity-- let me draw 00:02:29.670 --> 00:02:32.200 a line here, so we don't confuse things-- distance is 00:02:32.200 --> 00:02:36.400 equal to the initial velocity times time plus acceleration 00:02:36.400 --> 00:02:40.510 times time squared divided by 2. 00:02:40.510 --> 00:02:43.250 Sometimes the physics teacher might just teach at squared 00:02:43.250 --> 00:02:45.100 2-- that's sometimes what people memorize, and that's 00:02:45.100 --> 00:02:48.490 because in a lot of these projectile motion problems, 00:02:48.490 --> 00:02:51.460 your initial velocity is 0, especially when you're 00:02:51.460 --> 00:02:52.440 dropping a rock. 00:02:52.440 --> 00:02:56.610 If your initial velocity is 0, this term would cancel out. 00:02:56.610 --> 00:02:59.640 If you do that last problem that we just did using this 00:02:59.640 --> 00:03:02.680 example, you'll get the same answer. 00:03:02.680 --> 00:03:06.230 I said we're accelerating with gravity, so a is equal to 10 00:03:06.230 --> 00:03:11.490 meters per second squared, then time is equal to 2 00:03:11.490 --> 00:03:17.600 seconds, and then initial velocity is equal to 0, and 00:03:17.600 --> 00:03:19.110 let me make some space here. 00:03:22.360 --> 00:03:25.420 The initial velocity is 0-- so this term just cancels out-- 00:03:25.420 --> 00:03:29.570 plus acceleration, 10 meters per second 00:03:29.570 --> 00:03:32.170 squared times time squared. 00:03:46.830 --> 00:03:53.620 You have 10 meters per second squared times time squared, 00:03:53.620 --> 00:03:56.750 time is two seconds, so it's 4. 00:03:56.750 --> 00:04:00.210 Since we've squared the number, we should also square 00:04:00.210 --> 00:04:04.780 the units, so it's 4 seconds squared, and all of 00:04:04.780 --> 00:04:07.180 that is over 2. 00:04:07.180 --> 00:04:11.620 Like we learned before-- 10 times 4 divided 2 is 20, and 00:04:11.620 --> 00:04:13.800 we have seconds squared the denominator and we have 00:04:13.800 --> 00:04:15.430 seconds squared in the numerator here. 00:04:15.430 --> 00:04:18.353 They cancel out, and we're just left with meters. 00:04:21.329 --> 00:04:23.690 Actually, I'm going to leave it there for now, and in the 00:04:23.690 --> 00:04:27.630 next presentation, I'll explore some of these 00:04:27.630 --> 00:04:29.150 mechanics even further. 00:04:29.150 --> 00:04:30.840 I'll see you soon.

Introduction to motion (part 2)

https://www.youtube.com/watch?v=OKXyKt40WFE

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https://www.youtube.com/api/timedtext?v=OKXyKt40WFE&ei=dmeUZZyUH_e1vdIPot2CsA0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=E8C779FDA777A4D7E3C5D0EA2646358A20A4CA0D.7B0814BF8C3554F10B236297A1E3C43BFB11664C&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.980 --> 00:00:01.800 All right. 00:00:01.800 --> 00:00:05.090 Where I left off in the last presentation I was dropping a 00:00:05.090 --> 00:00:08.990 penny from the top of a building-- once again, you 00:00:08.990 --> 00:00:10.850 should not do, because you can kill somebody. 00:00:14.060 --> 00:00:20.380 Here's the building, and here's the bad person who's 00:00:20.380 --> 00:00:22.560 going to drop something. 00:00:22.560 --> 00:00:27.480 Let's say they just hold it out, and the penny drops. 00:00:27.480 --> 00:00:29.420 The penny is going to accelerate at the rate of 00:00:29.420 --> 00:00:32.030 gravity, so it's going to accelerate downwards at 10 00:00:32.030 --> 00:00:34.070 meters per second squared. 00:00:34.070 --> 00:00:36.180 Let's start with an interesting question. 00:00:36.180 --> 00:00:41.730 After after two seconds-- and lets say they drop it right at 00:00:41.730 --> 00:00:53.800 t equals 0-- so after two seconds how fast is it going? 00:00:53.800 --> 00:00:57.435 Time is equal to two seconds-- we could even say this change 00:00:57.435 --> 00:00:59.450 in time, but we're assuming that we're starting at time 00:00:59.450 --> 00:01:02.970 equals 0, so and change in time the same thing. 00:01:02.970 --> 00:01:05.193 If time is equal to two seconds, how fast is it going 00:01:05.193 --> 00:01:10.030 to be going after two seconds? 00:01:10.030 --> 00:01:11.240 Let's use that formula. 00:01:11.240 --> 00:01:16.610 Velocity is equal to acceleration-- acceleration is 00:01:16.610 --> 00:01:19.060 the acceleration of gravity, and that's 10 meters per 00:01:19.060 --> 00:01:24.910 second squared-- so velocity will be 10 meters per second 00:01:24.910 --> 00:01:29.940 squared times time, which is times two seconds. 00:01:37.710 --> 00:01:41.310 We can multiply the numbers, and you get 20. 00:01:41.310 --> 00:01:44.520 Just like the numbers, you can treat the units almost like 00:01:44.520 --> 00:01:48.230 variables-- the seconds is the same thing as this s, so this 00:01:48.230 --> 00:01:50.120 s is going to the numerator, and then you have an s squared 00:01:50.120 --> 00:01:53.120 in the denominator. 00:01:53.120 --> 00:01:56.190 This s will cancel out with one of the two s's that are 00:01:56.190 --> 00:01:59.600 multiplied down here, so we'll end up-- actually, let me 00:01:59.600 --> 00:02:04.520 write it down-- it'll be 10 meter seconds per second 00:02:04.520 --> 00:02:07.900 squared, and that's the same thing as 20. 00:02:07.900 --> 00:02:12.430 That cancels out, this makes that 1, and so that equals 20 00:02:12.430 --> 00:02:14.520 meters per second. 00:02:14.520 --> 00:02:16.610 Hopefully, you're starting to get a little intuition of why 00:02:16.610 --> 00:02:20.290 acceleration's units are meters per second squared. 00:02:20.290 --> 00:02:25.820 After two seconds, we're going 20 meters per second. 00:02:25.820 --> 00:02:29.090 Let me ask you a slightly more difficult problem that might 00:02:29.090 --> 00:02:30.860 have not been obvious to you. 00:02:30.860 --> 00:02:35.540 After two seconds, how far has the penny gone? 00:02:35.540 --> 00:02:38.700 This is interesting. 00:02:38.700 --> 00:02:42.250 We have this formula here: distance is equal to velocity 00:02:42.250 --> 00:02:48.410 times time, but the velocity is changing the entire time. 00:02:48.410 --> 00:02:52.640 We know after two seconds that the velocity is 20 meters per 00:02:52.640 --> 00:02:55.530 second, so we could call this the final velocity-- we'll 00:02:55.530 --> 00:02:57.340 called v sub f. 00:02:57.340 --> 00:02:59.870 That's just a fancy way of saying final velocity. 00:02:59.870 --> 00:03:01.880 Right when we start at t equals 0, 00:03:01.880 --> 00:03:04.480 what was the velocity? 00:03:04.480 --> 00:03:08.870 Right when it started, the initial velocity-- v sub i, 00:03:08.870 --> 00:03:14.230 for initial-- is equal to 0 meters per second. 00:03:17.610 --> 00:03:20.290 Can we use this formula? 00:03:20.290 --> 00:03:23.390 You might think of a way to already do it. 00:03:23.390 --> 00:03:26.990 Since the acceleration is constant, and you can only do 00:03:26.990 --> 00:03:30.130 this when the acceleration is constant-- most of what you'll 00:03:30.130 --> 00:03:32.152 encounter in a first year physics course, the 00:03:32.152 --> 00:03:34.480 acceleration will be constant, and especially when you're 00:03:34.480 --> 00:03:36.160 dealing with gravity, the acceleration will be 00:03:36.160 --> 00:03:40.770 constant-- you can actually take the average velocity to 00:03:40.770 --> 00:03:43.270 figure out the distance. 00:03:43.270 --> 00:03:46.370 So what was the average velocity over the two seconds? 00:03:50.400 --> 00:03:53.550 My final velocity was 20 meters per second, and my 00:03:53.550 --> 00:03:55.650 initial velocity was 0 meters per second. 00:03:55.650 --> 00:03:59.560 Obviously, I went continuously over those two seconds from 0 00:03:59.560 --> 00:04:03.940 to 20, so my average velocity-- actually, I've 00:04:03.940 --> 00:04:06.620 never seen it done this way before, but let's just call it 00:04:06.620 --> 00:04:10.910 average velocity-- is equal to the final velocity plus the 00:04:10.910 --> 00:04:13.300 initial velocity divided by 2. 00:04:13.300 --> 00:04:16.490 I just took the average of the initial and the final, which 00:04:16.490 --> 00:04:23.830 is 20 plus 0-- which is 20-- divided by 2, which is equal 00:04:23.830 --> 00:04:30.820 to 10 meters per second. 00:04:30.820 --> 00:04:33.850 Right when I let go of the rock, the ball, or the penny, 00:04:33.850 --> 00:04:36.780 whatever I'm dropping, the thing is stationary, and so 00:04:36.780 --> 00:04:38.400 it's 0 meters per second. 00:04:38.400 --> 00:04:42.110 After two seconds-- we used this acceleration formula-- 00:04:42.110 --> 00:04:43.840 after two seconds, it accelerated to 00:04:43.840 --> 00:04:46.290 20 meters per second. 00:04:46.290 --> 00:04:49.780 Over the course of those two seconds, its average velocity 00:04:49.780 --> 00:04:52.340 was 10 meters per second. 00:04:52.340 --> 00:04:56.600 We can now use that average velocity in this 00:04:56.600 --> 00:04:58.680 formula right here. 00:04:58.680 --> 00:05:02.620 The average velocity, distance equals average velocity times 00:05:02.620 --> 00:05:05.530 time-- you can make a mental footnote, so it's average 00:05:05.530 --> 00:05:08.370 velocity times time when the velocity is changing and 00:05:08.370 --> 00:05:10.570 acceleration is constant, which is most of what you'll 00:05:10.570 --> 00:05:13.650 see in most projectile motion problems. 00:05:13.650 --> 00:05:17.180 Now we could say distance is equal to the average velocity 00:05:17.180 --> 00:05:24.590 times time, which equals 10 meters per second times two 00:05:24.590 --> 00:05:27.650 seconds-- once again, the s's cancel out-- 00:05:27.650 --> 00:05:30.070 so we're at 20 meters. 00:05:30.070 --> 00:05:33.030 After two seconds, not only is my velocity 20 meters per 00:05:33.030 --> 00:05:36.110 second down-- once again, if I said speed, it would just be 00:05:36.110 --> 00:05:43.930 20 meters per second-- but my distance is the ball, or the 00:05:43.930 --> 00:05:46.270 rock, assuming no air resistance, 00:05:46.270 --> 00:05:51.900 has dropped 20 meters. 00:05:51.900 --> 00:05:55.020 Hopefully, that makes a little bit of intuition for you. 00:05:57.550 --> 00:06:00.760 If you are taking physics-- which you don't have to view 00:06:00.760 --> 00:06:03.110 these videos, that's the idea-- I wanted to show you 00:06:03.110 --> 00:06:07.300 that this is actually exactly like one of the formulas that 00:06:07.300 --> 00:06:09.860 you'll see in your physics class. 00:06:09.860 --> 00:06:11.605 It's kind of a shame, but people tend to just memorize 00:06:11.605 --> 00:06:15.040 it in physics without-- when they're learning projectile 00:06:15.040 --> 00:06:19.330 motion without really appreciating that it just 00:06:19.330 --> 00:06:21.900 comes from distance is equal to velocity times time. 00:06:26.400 --> 00:06:27.900 Before, I said velocity is equal to 00:06:27.900 --> 00:06:29.340 acceleration times to time. 00:06:29.340 --> 00:06:31.440 Let me just expand that a little bit, because I assume 00:06:31.440 --> 00:06:33.420 that my initial velocity is 0. 00:06:33.420 --> 00:06:37.270 Let me just say that the final velocity is equal to the 00:06:37.270 --> 00:06:40.870 initial velocity, because you could already be going 10 00:06:40.870 --> 00:06:43.360 meters per second, and then you're going to accelerate. 00:06:43.360 --> 00:06:46.150 Final velocity is equal to the initial velocity-- this is an 00:06:46.150 --> 00:06:50.110 i-- plus acceleration times time. 00:06:52.840 --> 00:06:57.650 We said that the distance-- we could rewrite this as the 00:06:57.650 --> 00:07:01.360 distance is equal to the average velocity times time. 00:07:03.920 --> 00:07:05.915 I just realized how funny that character looks. 00:07:09.350 --> 00:07:11.840 So, the final velocity is equal to the initial velocity 00:07:11.840 --> 00:07:15.090 plus acceleration times time, and the distance is equal to 00:07:15.090 --> 00:07:18.910 the average velocity times time. 00:07:18.910 --> 00:07:20.820 Let's see if we can use these two formulas, which we 00:07:20.820 --> 00:07:22.810 essentially just applied in the previous example-- we 00:07:22.810 --> 00:07:27.300 didn't do it exactly so formally-- to come up with a 00:07:27.300 --> 00:07:32.130 formula for distance, given acceleration and time. 00:07:32.130 --> 00:07:37.230 We know that the average velocity-- oh, I switched 00:07:37.230 --> 00:07:43.210 colors-- the average velocity is equal to the final velocity 00:07:43.210 --> 00:07:46.180 plus the initial velocity divided by 2. 00:07:51.720 --> 00:07:53.880 What is the final velocity? 00:07:53.880 --> 00:08:00.650 The final velocity is equal to this: substitute, and so the 00:08:00.650 --> 00:08:09.200 initial velocity plus acceleration times time plus 00:08:09.200 --> 00:08:13.470 the initial velocity-- my i's are getting blurred, they're 00:08:13.470 --> 00:08:15.590 not showing up-- these are all i's for initial velocity. 00:08:15.590 --> 00:08:17.730 They look like a 2, but I think you get the idea-- 00:08:17.730 --> 00:08:19.000 that's all initial velocity. 00:08:19.000 --> 00:08:22.300 All of that is over 2. 00:08:22.300 --> 00:08:24.910 The average velocity is equal to the initial velocity plus 00:08:24.910 --> 00:08:29.115 acceleration times time, plus the initial velocity, all of 00:08:29.115 --> 00:08:30.710 that divided by 2. 00:08:30.710 --> 00:08:37.429 That just equals 2 times initial velocity-- that looks 00:08:37.429 --> 00:08:41.929 like an i now-- plus acceleration times time 00:08:41.929 --> 00:08:47.740 divided by 2, and that equals the initial velocity plus 00:08:47.740 --> 00:08:51.410 acceleration times time divided by 2. 00:08:51.410 --> 00:08:53.720 This might be intuitive for you, as well, that the average 00:08:53.720 --> 00:08:58.590 velocity is equal to your initial velocity plus-- this 00:08:58.590 --> 00:09:04.240 is essentially the difference between how much you're 00:09:04.240 --> 00:09:07.430 accelerating over that time and speed is going to be that 00:09:07.430 --> 00:09:09.780 divided by 2, because we're taking the average. 00:09:09.780 --> 00:09:11.570 If what I just said confused you, don't worry about it; you 00:09:11.570 --> 00:09:14.400 could just backtrack into what we said before. 00:09:14.400 --> 00:09:17.940 Think about a lot of these formulas yourself, and plug in 00:09:17.940 --> 00:09:20.900 numbers, and I think it'll start to make more sense. 00:09:20.900 --> 00:09:23.520 We figured out that the average velocity is equal to 00:09:23.520 --> 00:09:26.670 the initial velocity plus acceleration times time. 00:09:26.670 --> 00:09:29.430 We can just substitute that back into 00:09:29.430 --> 00:09:32.280 this original equation. 00:09:32.280 --> 00:09:34.620 Once again, I realized I'm running out of time, so I'll 00:09:34.620 --> 00:09:36.420 see you shortly.

Introduction to motion

https://www.youtube.com/watch?v=8wZugqi_uCg

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https://www.youtube.com/api/timedtext?v=8wZugqi_uCg&ei=dmeUZcuKFMSQxN8PpKGVkAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=24D666B5BF2709BE0B4B7AB224BBDDE469CE9815.C406ED9B74B975E99E4387B6B241466C5953A383&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.460 --> 00:00:03.730 Hello, and welcome. 00:00:03.730 --> 00:00:07.530 This will be the first in the series of lectures on physics. 00:00:07.530 --> 00:00:11.120 My goal is really to give you an intuitive feeling of what 00:00:11.120 --> 00:00:13.340 physics is all about, because especially on the mechanics 00:00:13.340 --> 00:00:16.530 side of things-- projectile motion, force, and momentum-- 00:00:16.530 --> 00:00:19.170 it's actually pretty intuitive. 00:00:19.170 --> 00:00:21.980 Let me know if I'm not giving you an intuitive sense. 00:00:21.980 --> 00:00:25.170 Let's just start with probably the most basic formula in 00:00:25.170 --> 00:00:27.890 physics, and a lot of you have already seen this: it's 00:00:27.890 --> 00:00:34.690 distance is equal to velocity times time. 00:00:34.690 --> 00:00:36.090 You might have seen it in different forms-- you might 00:00:36.090 --> 00:00:38.560 have seen it written as the distance is equal to rate 00:00:38.560 --> 00:00:42.850 times time, distance is equal to speed times time, or 00:00:42.850 --> 00:00:44.910 instead of a d, if you're doing it in math class, 00:00:44.910 --> 00:00:46.830 sometimes you'll write an s for distance, because they use 00:00:46.830 --> 00:00:49.190 d for derivatives-- but it's all the same thing. 00:00:49.190 --> 00:00:52.140 It just says the distance you travel is equal to the speed 00:00:52.140 --> 00:00:54.570 or the velocity you travel times time. 00:00:54.570 --> 00:00:57.490 Before I do a couple of quick example problems just to hit 00:00:57.490 --> 00:00:59.780 this point home, I just want to make a slight distinction 00:00:59.780 --> 00:01:01.100 between velocity and speed. 00:01:03.920 --> 00:01:07.300 When people use it in everyday language, it tends to be used 00:01:07.300 --> 00:01:10.330 interchangeably-- it's how fast are you going, velocity 00:01:10.330 --> 00:01:11.810 versus speed. 00:01:11.810 --> 00:01:14.590 Technically, there's a difference: velocity is a 00:01:14.590 --> 00:01:21.700 vector of measurement, and speed is a scalar. 00:01:21.700 --> 00:01:24.740 I probably have already confused you, but all you have 00:01:24.740 --> 00:01:30.550 to know is a vector has a magnitude and a direction. 00:01:30.550 --> 00:01:33.580 If I were to give a velocity, I really shouldn't just say 00:01:33.580 --> 00:01:34.740 five miles per hour. 00:01:34.740 --> 00:01:37.930 I should say five miles per hour north. 00:01:37.930 --> 00:01:40.630 Or, instead of saying five miles per hour north, I could 00:01:40.630 --> 00:01:43.140 say negative five miles per hour south. 00:01:43.140 --> 00:01:43.390 Right? 00:01:43.390 --> 00:01:45.740 Those would be the same thing. 00:01:45.740 --> 00:01:50.320 Speed, or a scalar, only has magnitude, so speed would say 00:01:50.320 --> 00:01:52.710 five miles per hour, but I don't know what 00:01:52.710 --> 00:01:53.770 direction I'm going in. 00:01:53.770 --> 00:01:55.990 I could be going forward, backward, left, right, north, 00:01:55.990 --> 00:01:58.030 south, up, down, who knows? 00:01:58.030 --> 00:02:02.130 That's the technical difference between a vector 00:02:02.130 --> 00:02:05.180 and a scalar and between velocity and speed. 00:02:05.180 --> 00:02:08.210 It might not seem so obvious, and probably on these few 00:02:08.210 --> 00:02:11.150 example problems we're doing right now, the distinction 00:02:11.150 --> 00:02:13.720 probably could be used interchangeably. 00:02:13.720 --> 00:02:17.420 Later on, as we progress to fancier problems, I think 00:02:17.420 --> 00:02:21.020 you'll see that velocity is a more useful notion, because 00:02:21.020 --> 00:02:23.010 there's the idea of a negative velocity. 00:02:23.010 --> 00:02:25.440 You can go in one direction, which is positive, and then 00:02:25.440 --> 00:02:27.800 you can turn around and go negative. 00:02:27.800 --> 00:02:30.710 With speed, there's no direction involved. 00:02:30.710 --> 00:02:32.860 With that said-- I don't want to dwell on that too much, 00:02:32.860 --> 00:02:34.380 because I don't want to make you think that this is 00:02:34.380 --> 00:02:38.580 difficult-- let's do a couple of really quick problems. 00:02:38.580 --> 00:02:44.830 Let's say I went 50 meters, so distance is equal to 50 00:02:44.830 --> 00:02:47.230 meters-- I go 50 meters. 00:02:47.230 --> 00:02:52.640 Time is equal to-- this is should really be distance, 00:02:52.640 --> 00:02:55.270 because it actually changes in distance, but for the problems 00:02:55.270 --> 00:02:57.706 we'll do, it doesn't make a difference-- let's say the 00:02:57.706 --> 00:03:01.630 time is-- and this could be change in time, as well, the 00:03:01.630 --> 00:03:06.990 time is 10 seconds. 00:03:06.990 --> 00:03:12.480 So if we use that formula, we have 50 meters is equal to 00:03:12.480 --> 00:03:17.240 velocity times 10 seconds. 00:03:17.240 --> 00:03:19.560 This is a pretty simple algebra equation. 00:03:19.560 --> 00:03:23.700 If we want to solve for velocity, we just divide both 00:03:23.700 --> 00:03:26.740 sides of this situation by 10 seconds. 00:03:26.740 --> 00:03:28.570 I'm doing this because I actually want to show you that 00:03:28.570 --> 00:03:30.550 when you divide the numbers, you should also divide the 00:03:30.550 --> 00:03:32.620 units with them, and then you always end up 00:03:32.620 --> 00:03:33.920 with the right units. 00:03:33.920 --> 00:03:36.850 Let's multiply both sides of this equation 00:03:36.850 --> 00:03:38.960 by 1 over 10 seconds. 00:03:38.960 --> 00:03:47.750 I get 1 over 10 seconds times 50 meters is equal to velocity 00:03:47.750 --> 00:03:51.470 times-- I'll write it all out, later on, I'll skip some 00:03:51.470 --> 00:03:55.800 steps, and this obviously cancels out, because that's 00:03:55.800 --> 00:03:59.965 why we did in the first place-- and then the 50 and 00:03:59.965 --> 00:04:03.230 the 10 cancel and this becomes 5. 00:04:03.230 --> 00:04:09.240 We're left with 5, meter in the numerator, and the seconds 00:04:09.240 --> 00:04:11.380 in the denominator. 00:04:11.380 --> 00:04:14.910 I could write it as sec, or just an s-- so 5 meters per 00:04:14.910 --> 00:04:17.540 second is our velocity, and you could have done that. 00:04:17.540 --> 00:04:21.350 The one thing I just wanted to highlight is that the units, 00:04:21.350 --> 00:04:24.020 you can manipulate with the numbers, and then you always 00:04:24.020 --> 00:04:25.380 get the right answer. 00:04:25.380 --> 00:04:27.700 It might have been obvious in this case-- you didn't have to 00:04:27.700 --> 00:04:29.930 do it this way-- but once again, later on, when we start 00:04:29.930 --> 00:04:35.480 doing power and work and energy, which is actually the 00:04:35.480 --> 00:04:37.640 same thing as work, but once we start doing those things, 00:04:37.640 --> 00:04:40.670 then the units might not seem so obvious. 00:04:40.670 --> 00:04:45.590 It's good to be able to deal with the units this way. 00:04:45.590 --> 00:04:50.020 We could solve-- if I said that the velocity is equal to 00:04:50.020 --> 00:04:59.460 7 meters per second, and that time is equal to 5 seconds-- 00:04:59.460 --> 00:05:00.620 how far did I go? 00:05:00.620 --> 00:05:03.770 I could use that formula again: distance is equal to 00:05:03.770 --> 00:05:09.070 velocity, which is 7 meters per second, times time, which 00:05:09.070 --> 00:05:11.820 is 5 seconds. 00:05:11.820 --> 00:05:15.550 Once again, not only can we multiply the numbers-- 7 times 00:05:15.550 --> 00:05:19.600 5 is 35-- but we can multiply the units, so we have meters 00:05:19.600 --> 00:05:22.360 over second times second. 00:05:22.360 --> 00:05:24.200 You can almost treat them like variables, but they're not-- 00:05:24.200 --> 00:05:25.570 they're units. 00:05:25.570 --> 00:05:29.180 Meters over seconds times seconds-- the second in the 00:05:29.180 --> 00:05:31.300 numerator and the second in the denominator cancel out, 00:05:31.300 --> 00:05:33.670 and so you're left with 35 meters. 00:05:33.670 --> 00:05:36.550 There-- not only do you have the right number, you have the 00:05:36.550 --> 00:05:37.780 right units. 00:05:37.780 --> 00:05:41.020 Actually, this is going to be super useful when you have to 00:05:41.020 --> 00:05:46.390 convert from centimeters to meters, and hours to seconds, 00:05:46.390 --> 00:05:47.910 and all of that-- maybe we'll do a couple of examples. 00:05:47.910 --> 00:05:51.060 Actually, a while ago, I actually made a separate video 00:05:51.060 --> 00:05:55.350 on unit conversion, and that's going to come and really handy 00:05:55.350 --> 00:05:57.540 when we do the physics. 00:05:57.540 --> 00:06:00.100 With that out of the way, let's make things a little bit 00:06:00.100 --> 00:06:01.350 more complicated. 00:06:05.680 --> 00:06:09.550 Most of what you've probably experienced so far-- distance 00:06:09.550 --> 00:06:14.420 is equal to velocity times time, or rate times time-- is 00:06:14.420 --> 00:06:16.590 where velocity is constant. 00:06:16.590 --> 00:06:19.080 You're going 30 meters per second, you're always going 00:06:19.080 --> 00:06:21.480 the 30 meters per second, and you'll stay going 30 meters 00:06:21.480 --> 00:06:22.200 per second. 00:06:22.200 --> 00:06:28.500 But we know from moving, generally, that your velocity 00:06:28.500 --> 00:06:30.700 isn't-- sometimes, you're stationary, then sometimes 00:06:30.700 --> 00:06:33.460 you're moving, and in order to start stationary, and then get 00:06:33.460 --> 00:06:37.760 moving, your velocity has to change. 00:06:37.760 --> 00:06:40.860 How could we describe a change in velocity? 00:06:40.860 --> 00:06:43.120 Once again, I don't think I'm teaching you anything 00:06:43.120 --> 00:06:44.580 fundamentally new. 00:06:44.580 --> 00:06:47.090 You know what it is-- it's acceleration. 00:06:47.090 --> 00:06:52.300 So, velocity is acceleration times time. 00:06:52.300 --> 00:06:57.690 There's a pretty good analogy here: just as distance is 00:06:57.690 --> 00:06:59.340 velocity times time, and velocity is 00:06:59.340 --> 00:07:00.650 acceleration times time. 00:07:00.650 --> 00:07:05.920 Or, you view it as the change in distance over the change in 00:07:05.920 --> 00:07:12.020 time is velocity, while the change in velocity versus the 00:07:12.020 --> 00:07:16.940 change in time is equal to acceleration. 00:07:16.940 --> 00:07:19.220 My phone's ringing-- let me answer that a little bit 00:07:19.220 --> 00:07:23.440 later, because once again, you are more important. 00:07:23.440 --> 00:07:25.000 So where was I? 00:07:25.000 --> 00:07:28.060 I was saying how they're very similar-- at least, there's 00:07:28.060 --> 00:07:30.550 kind of an analogy here. 00:07:30.550 --> 00:07:32.500 What can we do with this notion? 00:07:32.500 --> 00:07:36.460 I'm going to do a bunch of what I think you'll find is 00:07:36.460 --> 00:07:39.120 pretty useful-- they're called projectile problems, and 00:07:39.120 --> 00:07:42.120 projectile problems involve the acceleration of gravity. 00:07:42.120 --> 00:07:47.620 We could do other acceleration problems involving the 00:07:47.620 --> 00:07:49.470 acceleration of cars, and actually, I 00:07:49.470 --> 00:07:51.040 probably will do that. 00:07:51.040 --> 00:07:58.050 The acceleration of gravity is actually 9.8 meters per second 00:07:58.050 --> 00:08:02.400 downwards-- once again, acceleration is actually not 00:08:02.400 --> 00:08:04.530 downwards, but towards the center of the earth. 00:08:04.530 --> 00:08:08.060 Acceleration is also a vector quantity, but for the sake of 00:08:08.060 --> 00:08:13.260 our computations, we'll just say 10 meters per second. 00:08:13.260 --> 00:08:17.850 Acceleration is equal to g-- g is normally the variable used 00:08:17.850 --> 00:08:21.160 when people talk about the acceleration of gravity, and 00:08:21.160 --> 00:08:27.290 let's say that equals 10 meters per second squared. 00:08:27.290 --> 00:08:31.410 I know what you're thinking, that this is kind of a strange 00:08:31.410 --> 00:08:34.270 set of units-- meters per second squared, and it's hard 00:08:34.270 --> 00:08:35.059 to visualize. 00:08:35.059 --> 00:08:38.610 That's acceleration, and I think once you see it used in 00:08:38.610 --> 00:08:41.070 some of these formulas, it'll start to make a little sense 00:08:41.070 --> 00:08:46.770 in terms of how these units work out. 00:08:46.770 --> 00:08:49.080 Let's start with a fairly simple problem. 00:08:49.080 --> 00:08:53.640 Let's say I drop a rock from-- I don't recommend you do this, 00:08:53.640 --> 00:08:57.170 you could kill somebody-- not a rock, a penny, from the 00:08:57.170 --> 00:09:01.500 Empire State Building, and I'm assuming no air resistance. 00:09:01.500 --> 00:09:03.920 Actually, I just looked at the clock, and I realize that I'm 00:09:03.920 --> 00:09:05.030 running out of time. 00:09:05.030 --> 00:09:06.830 I'm going to actually start this problem in the next 00:09:06.830 --> 00:09:07.760 presentation. 00:09:07.760 --> 00:09:09.370 I'll see you soon.

Trig identities part 3 (part 5 if you watch the proofs)

https://www.youtube.com/watch?v=JXCiFbEMTZ4

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en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:03.400 Welcome back, because I was hitting against the 00:00:03.400 --> 00:00:04.760 YouTube 10-minute limit. 00:00:04.760 --> 00:00:06.110 But all I was saying is, we said, you know, 00:00:06.110 --> 00:00:07.260 cosine of minus a. 00:00:07.260 --> 00:00:12.500 So I drew a right triangle with a, and then I showed you minus 00:00:12.500 --> 00:00:15.160 a, and I said, well, all of the lengths are going to be the 00:00:15.160 --> 00:00:18.520 same, but now the direction of-- and we're kind of assuming 00:00:18.520 --> 00:00:19.670 this is all on the unit circle. 00:00:19.670 --> 00:00:23.410 If you don't remember the unit circle, maybe you'll want to 00:00:23.410 --> 00:00:26.000 rewatch the videos we have on that. 00:00:26.000 --> 00:00:28.890 But I'm just showing you that the cosine of minus a is equal 00:00:28.890 --> 00:00:32.140 to this side over the hypotenuse, and this hypotenuse 00:00:32.140 --> 00:00:34.350 is the same as this hypotenuse, right? 00:00:34.350 --> 00:00:38.250 So cosine of minus a is adjacent over this hypotenuse, 00:00:38.250 --> 00:00:42.290 while cosine of a is adjacent over this hypotenuse. 00:00:42.290 --> 00:00:44.660 But it's the same thing, so we know that cosine of minus 00:00:44.660 --> 00:00:47.640 a is equal to cosine of a. 00:00:47.640 --> 00:00:50.430 Actually, by definition, that makes it a-- I don't want to 00:00:50.430 --> 00:00:52.600 confuse you too much, but that makes cosine an even function, 00:00:52.600 --> 00:00:53.510 and I'll show you more. 00:00:53.510 --> 00:00:55.160 Actually, I should do a whole presentation on 00:00:55.160 --> 00:00:57.030 even and odd functions. 00:00:57.030 --> 00:00:59.490 Now, let's see what sine of minus a is. 00:00:59.490 --> 00:01:06.030 Sine of minus a is equal to-- so this is minus a. 00:01:06.030 --> 00:01:11.100 So it's this side, so it's the minus length of-- let's call 00:01:11.100 --> 00:01:15.480 this x, let's call this y, and let's call this, well, 00:01:15.480 --> 00:01:17.030 let's leave that h, right? 00:01:17.030 --> 00:01:20.880 If that is x, this is y, this length is y, then this length 00:01:20.880 --> 00:01:22.880 right here is minus x, right? 00:01:22.880 --> 00:01:25.890 So the sine of minus a is minus x/h. 00:01:30.670 --> 00:01:33.390 What's the sine of a? 00:01:33.390 --> 00:01:38.740 Sine of a is equal to-- this is a-- opposite or hypotenuse, 00:01:38.740 --> 00:01:43.540 x over h, right? 00:01:43.540 --> 00:01:50.030 So sine of minus a is equal to minus 1 times x over h, right? 00:01:50.030 --> 00:01:52.960 Or this is just the same thing as-- I mean, we could multiply 00:01:52.960 --> 00:01:56.350 both sides of this by minus 1, minus x over h, right? 00:01:56.350 --> 00:02:00.670 So sine of minus a is equal to minus sine of a. 00:02:04.100 --> 00:02:07.700 So let me clear this out and rewrite this identity. 00:02:07.700 --> 00:02:10.030 And as you can see, all I'm doing is I'm just playing 00:02:10.030 --> 00:02:14.010 around with triangles and showing you that, you know, 00:02:14.010 --> 00:02:17.430 just using the basic SOHCAHTOA, you can actually discover a 00:02:17.430 --> 00:02:19.590 whole set of trigonometric identities. 00:02:22.300 --> 00:02:23.300 So let's clear that. 00:02:23.300 --> 00:02:26.100 And, you know, it is useful to memorize. 00:02:26.100 --> 00:02:29.680 I normally don't advocate memorizing, but it's helpful 00:02:29.680 --> 00:02:30.890 just to do things quickly. 00:02:30.890 --> 00:02:33.940 But I'd also advocate being able to prove it to yourself, 00:02:33.940 --> 00:02:37.150 so if you ever forget it, and you don't have a cheat sheet 00:02:37.150 --> 00:02:40.170 available, you can prove it, and if you ever have to teach 00:02:40.170 --> 00:02:45.130 it, then you'll be able to explain the underlying 00:02:45.130 --> 00:02:46.020 themes a little bit better. 00:02:46.020 --> 00:02:47.490 So let's clear this. 00:02:47.490 --> 00:02:49.895 Let's see if we can discover some more trig identities. 00:02:52.710 --> 00:02:58.890 So we know that-- so let's see, if we have sine-- 00:02:58.890 --> 00:03:01.370 what's sine of a plus pi/2? 00:03:04.960 --> 00:03:05.980 a plus pi/2. 00:03:05.980 --> 00:03:09.600 Well, we could use our handy sine of a plus b identity, 00:03:09.600 --> 00:03:12.250 which we've already proved, so we can use it now. 00:03:12.250 --> 00:03:15.580 So that tells us that it's the sine of a-- that equals the 00:03:15.580 --> 00:03:34.810 sine of a times the cosine of pi/2 plus the sine of pi/2. 00:03:34.810 --> 00:03:36.270 And we're in radians, of course. 00:03:36.270 --> 00:03:38.570 This could have been 90 degrees instead, if we 00:03:38.570 --> 00:03:40.140 wanted to be in degrees. 00:03:40.140 --> 00:03:47.310 sine of pi/2 times the cosine of a, right? 00:03:47.310 --> 00:03:52.890 Well, this equals the sine of -- what's cosine sign of pi/2? 00:03:52.890 --> 00:03:55.040 Or cosine of 90 degrees? 00:03:55.040 --> 00:03:57.160 Well, that's when we're on the unit circle, we're 00:03:57.160 --> 00:03:58.350 pointing straight up. 00:03:58.350 --> 00:04:00.450 And so the x-coordinate is 0. 00:04:00.450 --> 00:04:02.490 I could draw it out, but I think-- you might want to draw 00:04:02.490 --> 00:04:04.310 the unit circle and figure it out for yourself, or if you 00:04:04.310 --> 00:04:06.840 don't, do it on a calculator, but you will learn 00:04:06.840 --> 00:04:07.620 that it is 0. 00:04:07.620 --> 00:04:09.620 The cosine of pi/2 is 0. 00:04:09.620 --> 00:04:12.690 Plus sine of pi/2, for the same reason, we're pointing straight 00:04:12.690 --> 00:04:15.470 up on the unit circle, so the y-coordinate, or the sine 00:04:15.470 --> 00:04:19.710 coordinate, is 1, right on the unit-- is essentially at the 00:04:19.710 --> 00:04:22.400 point 0, 1 on the unit circle. 00:04:22.400 --> 00:04:29.300 So sine of pi/2 is 1, and then times cosine of a. 00:04:29.300 --> 00:04:31.820 So sine of a times 0 is 0. 00:04:31.820 --> 00:04:34.250 1 times cosine of a is just cosine of a. 00:04:34.250 --> 00:04:37.010 So we have a new, useful trig identity. 00:04:40.060 --> 00:04:48.100 Sine a plus pi/2 is equal to cosine of a. 00:04:48.100 --> 00:04:49.240 Fascinating! 00:04:49.240 --> 00:04:53.880 So really, this is just telling us that cosine of a is the same 00:04:53.880 --> 00:04:57.100 thing as sine of a shifted. 00:04:57.100 --> 00:04:59.030 So if we were to think of this graphically, if we were to 00:04:59.030 --> 00:05:03.070 think of, you know, if we were to draw the graph, if you shift 00:05:03.070 --> 00:05:09.150 the sine graph to the left by pi/2, you get the cosine graph. 00:05:09.150 --> 00:05:10.960 And if you haven't learned about shifting yet, 00:05:10.960 --> 00:05:11.670 don't worry about that. 00:05:11.670 --> 00:05:13.930 Or you might want to actually graph the two, and I 00:05:13.930 --> 00:05:16.860 think you'll get a sense of what I'm saying. 00:05:16.860 --> 00:05:19.900 So let's do-- I don't know. 00:05:19.900 --> 00:05:23.550 And another way to rewrite this exact same thing is the sine of 00:05:23.550 --> 00:05:36.090 a is equal to the cosine of a minus pi/2, right? 00:05:36.090 --> 00:05:40.640 Let's say I said that b is a plus pi/2, right? 00:05:40.640 --> 00:05:45.570 Let's say I said that b is equal to a plus pi/2, then we 00:05:45.570 --> 00:05:50.770 can say that this is b, and then this would b minus pi/2. 00:05:50.770 --> 00:05:52.770 I'm just switching around variables. 00:05:52.770 --> 00:05:55.520 I'm doing this in a much more loosey-goosey fashion than I 00:05:55.520 --> 00:05:57.840 normally do a lot of videos, but I want to show you that a 00:05:57.840 --> 00:06:00.560 lot of this trigonometry can just be-- you know, it's 00:06:00.560 --> 00:06:02.960 just kind of discovery. 00:06:02.960 --> 00:06:09.490 What's sine of a minus b? 00:06:09.490 --> 00:06:11.400 Well, that looks like a new one, doesn't it? 00:06:11.400 --> 00:06:13.930 Well, let's try to figure it out. 00:06:13.930 --> 00:06:26.080 Well, that equals sine of a cosine of minus b plus sine 00:06:26.080 --> 00:06:31.940 of minus b times the cosine of a, right? 00:06:31.940 --> 00:06:34.850 Well, what do we know about the cosine of minus b? 00:06:34.850 --> 00:06:36.970 Before I cleared the screen, we just figured out that the 00:06:36.970 --> 00:06:40.660 cosine of minus b, since it's an even function, is the same 00:06:40.660 --> 00:06:42.410 thing as the cosine of b. 00:06:42.410 --> 00:06:44.810 So we can rewrite that as that equals the 00:06:44.810 --> 00:06:49.900 sine of a cosine of b. 00:06:49.900 --> 00:06:52.600 And then what's the sine of minus b? 00:06:52.600 --> 00:06:56.380 Well, that's the same thing as the minus sine of b. 00:06:56.380 --> 00:07:00.120 That's what we just proved, that the sine of minus 00:07:00.120 --> 00:07:03.070 b, that this is equal to minus sine of b. 00:07:03.070 --> 00:07:05.340 You could draw the triangle and the unit circle, if you don't 00:07:05.340 --> 00:07:07.060 believe me, but we just did that. 00:07:07.060 --> 00:07:11.860 So we can say that that is equal to minus sine 00:07:11.860 --> 00:07:14.270 of b cosine of a. 00:07:14.270 --> 00:07:15.010 Interesting! 00:07:15.010 --> 00:07:16.580 I encourage you do the same thing with the 00:07:16.580 --> 00:07:18.120 cosine of a minus b. 00:07:18.120 --> 00:07:21.650 These are all just, you know, we're using one or two or three 00:07:21.650 --> 00:07:25.480 trig identities together and trying to come up 00:07:25.480 --> 00:07:26.290 with new things. 00:07:26.290 --> 00:07:29.250 And I think at this point, we've literally gone over 00:07:29.250 --> 00:07:32.260 everything, that almost every trig identity you've seen in 00:07:32.260 --> 00:07:36.100 your book, you should be able to get there somehow, 00:07:36.100 --> 00:07:37.270 just by keep on playing. 00:07:37.270 --> 00:07:39.990 And obviously, all of these identities, you can invert the 00:07:39.990 --> 00:07:44.130 sines and the cosines and the tangents, and you can get 00:07:44.130 --> 00:07:47.840 identities for secant and cotangent and cosecant 00:07:47.840 --> 00:07:50.750 and keep playing around. 00:07:50.750 --> 00:07:51.620 And I encourage you to do so. 00:07:51.620 --> 00:07:53.120 And do it graphically. 00:07:53.120 --> 00:07:54.860 Draw the triangles. 00:07:54.860 --> 00:07:56.490 It's also interesting to sometimes actually draw the 00:07:56.490 --> 00:08:00.420 graph on the x-y plane of, say, you know, cosine of x plus 00:08:00.420 --> 00:08:04.680 pi/2, or sine of x plus pi/2, or sine of x. 00:08:04.680 --> 00:08:07.410 And I think in the future, I'll do a video where I really 00:08:07.410 --> 00:08:08.930 do explore all of that. 00:08:08.930 --> 00:08:11.370 Well, I hope I haven't thoroughly confused you. 00:08:11.370 --> 00:08:16.170 I wanted to just kind of show you that a lot of trig-- it 00:08:16.170 --> 00:08:18.990 all comes from SOHCAHTOA and playing around with SOHCAHTOA 00:08:18.990 --> 00:08:22.390 and triangles, and you can pretty much get-- you can 00:08:22.390 --> 00:08:24.740 pretty much solve for everything you learn 00:08:24.740 --> 00:08:26.240 in trigonometry. 00:08:26.240 --> 00:08:27.510 And if you don't have SOHCAHTOA, at least the unit 00:08:27.510 --> 00:08:29.795 circle definition, which is actually better, because 00:08:29.795 --> 00:08:30.580 it's more extensive. 00:08:30.580 --> 00:08:32.230 But anyway, that's all for now. 00:08:32.230 --> 00:08:33.732 See you soon.

Trig identities part 2 (part 4 if you watch the proofs)

https://www.youtube.com/watch?v=ZWSoyUxAQW0

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en

WEBVTT Kind: captions Language: en 00:00:00.600 --> 00:00:01.680 Welcome back. 00:00:01.680 --> 00:00:04.210 I'm now going to do a bit of a review of everything we've 00:00:04.210 --> 00:00:06.080 learned so far about maybe even trigonometry and 00:00:06.080 --> 00:00:07.350 trig identities. 00:00:07.350 --> 00:00:10.450 And then we'll see if we can come up with-- maybe use what 00:00:10.450 --> 00:00:13.050 we already know to come up with a couple more trig identities. 00:00:13.050 --> 00:00:18.940 So we know that from SOH-CAH-TOA we know that sine 00:00:18.940 --> 00:00:26.070 of theta is equal to the opposite over the hypotenuse. 00:00:26.070 --> 00:00:28.270 Let me draw a triangle here. 00:00:28.270 --> 00:00:31.560 If I were to draw a triangle here-- whoops. 00:00:31.560 --> 00:00:33.380 Oh, there you go. 00:00:33.380 --> 00:00:35.780 OK, so this is theta. 00:00:35.780 --> 00:00:37.180 This is the opposite. 00:00:37.180 --> 00:00:38.410 This is the adjacent. 00:00:38.410 --> 00:00:40.400 This is hypotenuse. 00:00:40.400 --> 00:00:42.460 Then sine of theta is equal to opposite over hypotenuse. 00:00:42.460 --> 00:00:46.040 Cosine of theta-- this is basic review, hopefully at this 00:00:46.040 --> 00:00:49.170 point-- is the adjacent over the hypotenuse. 00:00:49.170 --> 00:00:54.470 The tangent of theta is equal to the opposite over the 00:00:54.470 --> 00:00:59.770 adjacent, which is also equal to the sine of theta over 00:00:59.770 --> 00:01:00.590 the cosine of theta. 00:01:00.590 --> 00:01:03.380 And we showed this in a couple of videos ago. 00:01:03.380 --> 00:01:08.160 And then, these are kind of almost definitional, but the 00:01:08.160 --> 00:01:17.840 cosecant of theta is equal to the hypotenuse over the 00:01:17.840 --> 00:01:21.260 opposite, which is the same thing as 1 over sine of theta. 00:01:21.260 --> 00:01:23.900 You can just memorize this. 00:01:23.900 --> 00:01:26.120 I mean, I kind of find is silly that there is such 00:01:26.120 --> 00:01:27.100 a thing as cosecant. 00:01:27.100 --> 00:01:29.330 I guess it's just for convenience because you know 00:01:29.330 --> 00:01:31.410 everyone knows it's just 1 over sine of theta. 00:01:31.410 --> 00:01:34.830 And same thing for secant. 00:01:34.830 --> 00:01:37.140 Secant of theta-- it's really for convenience. 00:01:37.140 --> 00:01:39.990 Instead of having to say, in the case of secant-- oh, that's 00:01:39.990 --> 00:01:42.910 1 you know-- if you end up with the equation 1 over cosine of 00:01:42.910 --> 00:01:44.200 theta you can just say, oh, that's just the 00:01:44.200 --> 00:01:45.550 secant of theta. 00:01:45.550 --> 00:01:48.573 I think it actually has some obvious properties and if you 00:01:48.573 --> 00:01:50.450 were to draw unit circle and all of that too. 00:01:50.450 --> 00:01:54.760 But anyway, so that's equal to the hypotenuse over the 00:01:54.760 --> 00:01:59.400 adjacent, which is equal to 1 over cosine of theta. 00:01:59.400 --> 00:02:05.160 And then, of course, cotangent of theta is equal to the 00:02:05.160 --> 00:02:08.430 adjacent over the opposite, which is equal to 00:02:08.430 --> 00:02:12.260 1 over tan theta. 00:02:12.260 --> 00:02:16.010 And of course, that's also equal to cosine of 00:02:16.010 --> 00:02:19.430 theta over sine theta. 00:02:19.430 --> 00:02:22.310 It's just the opposite of the tangent of theta. 00:02:22.310 --> 00:02:27.470 Or that's the same thing as what? 00:02:27.470 --> 00:02:32.230 That's the same thing as the secant-- no, no, no. 00:02:32.230 --> 00:02:36.632 It's the same thing as the cosecant-- no, no no. 00:02:36.632 --> 00:02:38.600 Let me make sure I get this right. 00:02:38.600 --> 00:02:45.700 It's the same thing as the-- I just want to get the inverses. 00:02:45.700 --> 00:02:47.050 Well, let's prove what it is actually. 00:02:47.050 --> 00:02:48.110 I always confuse myself. 00:02:48.110 --> 00:02:59.060 So this is the same thing as 1 over the secant of theta over 00:02:59.060 --> 00:03:01.730 1 over the cosecant of theta. 00:03:04.770 --> 00:03:06.370 Secant of theta, cosecant theta. 00:03:06.370 --> 00:03:12.700 And then that equals the cosecant of theta over 00:03:12.700 --> 00:03:14.190 the secant of theta. 00:03:14.190 --> 00:03:16.110 I wouldn't waste your time memorizing. 00:03:16.110 --> 00:03:17.930 So we know that a cotangent of theta is equal to 00:03:17.930 --> 00:03:19.240 1 over tangent theta. 00:03:19.240 --> 00:03:21.300 Is equal to the cosine over the sine. 00:03:21.300 --> 00:03:24.040 And it also equals the cosecant over the secant. 00:03:24.040 --> 00:03:26.690 And I wouldn't worry about really memorizing this. 00:03:26.690 --> 00:03:27.840 You could derive it if you had to. 00:03:27.840 --> 00:03:29.460 As you could tell, I really didn't have 00:03:29.460 --> 00:03:30.850 this memorized either. 00:03:30.850 --> 00:03:38.380 And we also learned in previous videos that the sine squared 00:03:38.380 --> 00:03:43.440 theta plus the cosine squared of theta is equal to 1. 00:03:43.440 --> 00:03:45.530 And that just comes from the pythagorean theorem. 00:03:45.530 --> 00:03:48.170 And if you play around with this a little bit you'd also 00:03:48.170 --> 00:03:55.950 get that the tangent squared theta plus 1 is equal to 00:03:55.950 --> 00:04:00.380 the secant squared theta. 00:04:00.380 --> 00:04:02.110 You actually go from here to here if you just divide both 00:04:02.110 --> 00:04:05.010 sides of this equation by cosine squared. 00:04:05.010 --> 00:04:05.860 So we know that. 00:04:05.860 --> 00:04:11.020 And then if you've watched the last two proof videos I made, 00:04:11.020 --> 00:04:20.180 we also know that the sine of-- let's say a plus b-- is equal 00:04:20.180 --> 00:04:31.670 to the sine of a times the cosine of b. 00:04:31.670 --> 00:04:34.890 Plus-- let me erase some of this because I don't 00:04:34.890 --> 00:04:38.590 think that that is an important trig identity. 00:04:38.590 --> 00:04:39.730 You can derive it on your own. 00:04:39.730 --> 00:04:43.570 I just wanted to show you that you could figure it out. 00:04:43.570 --> 00:04:45.780 I'm using too much space. 00:04:45.780 --> 00:04:48.320 OK, now I have space. 00:04:48.320 --> 00:04:50.330 Let me find that blue color I was using and make 00:04:50.330 --> 00:04:51.650 sure my pen is small. 00:04:51.650 --> 00:04:52.015 OK. 00:04:52.015 --> 00:04:56.040 So it's the sine of a times the cosine of b plus the sine 00:04:56.040 --> 00:05:02.430 of b times the cosine of a. 00:05:02.430 --> 00:05:03.690 And you might want to just memorize it. 00:05:03.690 --> 00:05:05.370 This actually becomes really useful when you actually start 00:05:05.370 --> 00:05:08.090 doing calculus because you have to solve derivatives and 00:05:08.090 --> 00:05:09.720 integrals that you might have to know the identity. 00:05:09.720 --> 00:05:11.870 And it's not that hard to memorize. 00:05:11.870 --> 00:05:15.080 It's the sine of one of them times the cosine of one of them 00:05:15.080 --> 00:05:16.370 plus the other way around. 00:05:16.370 --> 00:05:17.130 That's all this is. 00:05:17.130 --> 00:05:23.200 And then we also learned that the cosine of a plus b-- it's 00:05:23.200 --> 00:05:27.960 the cosine of both of them minus the sine of both them. 00:05:27.960 --> 00:05:36.890 So that is equal to the cosine of a times the cosine of b. 00:05:36.890 --> 00:05:39.400 And I proved this in another video, hopefully did it 00:05:39.400 --> 00:05:41.180 to your satisfaction. 00:05:41.180 --> 00:05:48.160 Minus the sine of a times the sine of b. 00:05:51.990 --> 00:05:54.110 These are pretty useful because from these can we can come up 00:05:54.110 --> 00:05:57.040 with a bunch of other trig identities. 00:05:57.040 --> 00:05:59.280 For example, what is the sine of 2a? 00:06:05.900 --> 00:06:08.360 Well, that's just the same thing as the sine of a plus a. 00:06:11.670 --> 00:06:15.010 And if we use this trig identity up here, that is equal 00:06:15.010 --> 00:06:27.850 to sine of a cosine of a plus the sine of a, cosine of a. 00:06:27.850 --> 00:06:31.560 I just used this sine of a plus b identity up here and well, 00:06:31.560 --> 00:06:32.982 want a and b are both a. 00:06:32.982 --> 00:06:34.110 Now what does this equal? 00:06:34.110 --> 00:06:35.990 Well, this is two terms that are just both sine 00:06:35.990 --> 00:06:36.810 of a, cosine of a. 00:06:36.810 --> 00:06:43.300 So that just equals 2 sine of a, cosine of a. 00:06:43.300 --> 00:06:47.030 So we now have derived another trigonometric identity that 00:06:47.030 --> 00:06:51.650 might be in the inside cover of your trig, or actually, 00:06:51.650 --> 00:06:54.270 your calculus book. 00:06:54.270 --> 00:06:55.940 All of these actually, I could draw a square 00:06:55.940 --> 00:06:56.540 around all of them. 00:06:56.540 --> 00:07:00.110 Let's do another one. 00:07:00.110 --> 00:07:03.070 Once you have a bit of a library of trig identities you 00:07:03.070 --> 00:07:05.680 can really just keep playing around and seeing what else you 00:07:05.680 --> 00:07:07.930 can-- and I encourage you to do so. 00:07:07.930 --> 00:07:10.890 And you'd be amazed how many other trig identities 00:07:10.890 --> 00:07:12.610 you could come up with. 00:07:12.610 --> 00:07:15.490 For example, let's do cosine of 2a. 00:07:15.490 --> 00:07:25.410 Cosine of 2a is equal to cosine of a plus a. 00:07:25.410 --> 00:07:27.530 And cosine of a plus a, what did we say? 00:07:27.530 --> 00:07:33.220 It's the cosine of both of the terms times each other minus 00:07:33.220 --> 00:07:34.160 the sine of both of the terms. 00:07:34.160 --> 00:07:39.750 So that equals cosine of a, cosine of a, right? 00:07:39.750 --> 00:07:46.020 Cosine of a times cosine of a minus sine of a, sine of a. 00:07:46.020 --> 00:07:49.080 This identity was the cosine of a plus b identity. 00:07:49.080 --> 00:07:50.260 Minus sine of a. 00:07:50.260 --> 00:07:51.030 So what is this? 00:07:51.030 --> 00:08:03.280 This is equal to cosine squared a minus sine squared a. 00:08:03.280 --> 00:08:04.720 That's interesting. 00:08:04.720 --> 00:08:05.310 We could play around. 00:08:05.310 --> 00:08:07.830 This is interesting because this is the form a 00:08:07.830 --> 00:08:09.880 squared minus b squared. 00:08:09.880 --> 00:08:13.770 So that's also the same thing as a plus b times a minus b. 00:08:13.770 --> 00:08:22.120 So that's the same thing as cosine of a plus sine of a 00:08:22.120 --> 00:08:26.950 times cosine of a minus sine of a. 00:08:26.950 --> 00:08:27.360 I don't know. 00:08:27.360 --> 00:08:29.250 This isn't really a trig identity, but I'm just showing 00:08:29.250 --> 00:08:30.310 you could play with things. 00:08:30.310 --> 00:08:34.650 Cosine of 2a is equal to cosine of a plus sine of a times 00:08:34.650 --> 00:08:37.290 cosine of a minus sine of a. 00:08:37.290 --> 00:08:41.400 So the sum of the cosine and sine of a then 00:08:41.400 --> 00:08:42.000 times the difference. 00:08:42.000 --> 00:08:42.760 That's just interesting. 00:08:42.760 --> 00:08:44.870 I'm just showing you that what's fun about trigonometry 00:08:44.870 --> 00:08:46.740 is you can kind of keep playing around with it. 00:08:46.740 --> 00:08:49.010 And actually, that's probably-- that is how all of the trig 00:08:49.010 --> 00:08:51.680 identities were discovered. 00:08:51.680 --> 00:08:57.830 So let's say that we have-- we want to figure out what cosine 00:08:57.830 --> 00:09:01.940 of let's say, negative a is. 00:09:01.940 --> 00:09:02.945 Well, let me draw a right triangle. 00:09:06.260 --> 00:09:06.645 Whoops. 00:09:06.645 --> 00:09:08.265 That's almost a right triangle. 00:09:08.265 --> 00:09:10.850 Now let's say this angle is a. 00:09:10.850 --> 00:09:13.450 So negative a, unit circle would look something 00:09:13.450 --> 00:09:16.240 like this, right? 00:09:16.240 --> 00:09:17.250 Negative a. 00:09:17.250 --> 00:09:22.750 So cosine of a, if we say that this side is the adjacent 00:09:22.750 --> 00:09:24.260 side, this is the hypotenuse. 00:09:24.260 --> 00:09:26.280 This would still be the hypotenuse, right? 00:09:26.280 --> 00:09:27.710 And this is the opposite. 00:09:27.710 --> 00:09:30.350 This is the negative opposite. 00:09:30.350 --> 00:09:34.500 So cosine of minus a is equal to what? 00:09:34.500 --> 00:09:38.260 This is minus a, so it's the adjacent over the hypotenuse. 00:09:38.260 --> 00:09:41.860 So it equals the adjacent over the hypotenuse, 00:09:41.860 --> 00:09:43.470 which we just say is h. 00:09:43.470 --> 00:09:47.070 But that's the same thing as cosine of a, right? 00:09:47.070 --> 00:09:51.130 Because cosine of a is also the adjacent over the hypotenuse. 00:09:53.870 --> 00:09:54.960 Oh, I'm almost out of time. 00:09:54.960 --> 00:09:57.050 Let me switch to a new video.

Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)

https://www.youtube.com/watch?v=V3-xCPDzQ1Q

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https://www.youtube.com/api/timedtext?v=V3-xCPDzQ1Q&ei=dmeUZe2sM8TzmLAP7pCvoA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5B0A2DCF021040C45BA9333A5BE076E3523E3AB6.0F21C250A8C9D4FE119899DF5D0BE2097F23D5BD&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.940 --> 00:00:01.970 Welcome back. 00:00:01.970 --> 00:00:05.000 We'll now try to see what trigonometric identity we can 00:00:05.000 --> 00:00:15.490 come up with if we start off with cosine of alpha plus beta. 00:00:18.880 --> 00:00:21.520 Let's see if we can rewrite this as another combination 00:00:21.520 --> 00:00:24.170 of cosines and sines of alpha and beta. 00:00:24.170 --> 00:00:25.310 So let's get started. 00:00:25.310 --> 00:00:30.660 And if you've already watched the sine equivalent of this, 00:00:30.660 --> 00:00:33.320 this proof will be pretty similar in how we operate. 00:00:33.320 --> 00:00:35.670 And we get a kind of similar answer. 00:00:35.670 --> 00:00:37.870 And something interesting is to kind of compare the difference 00:00:37.870 --> 00:00:40.385 between sine of alpha plus beta and cosine of alpha plus 00:00:40.385 --> 00:00:41.900 beta after we're done. 00:00:41.900 --> 00:00:45.460 So just like that last proof, let's say that this angle-- no, 00:00:45.460 --> 00:00:47.182 that color isn't bright enough. 00:00:47.182 --> 00:00:48.580 Let's do yellow. 00:00:48.580 --> 00:00:53.450 Let's say that this angle right here is alpha, and this 00:00:53.450 --> 00:00:59.400 angle right here is beta. 00:00:59.400 --> 00:01:00.160 Right? 00:01:00.160 --> 00:01:03.340 We want to know cosine of alpha plus beta. 00:01:03.340 --> 00:01:07.560 So alpha plus beta is this large angle right here. 00:01:07.560 --> 00:01:08.140 Right? 00:01:08.140 --> 00:01:09.550 So what's the cosine of that? 00:01:09.550 --> 00:01:11.890 SOH, CAH, TOA. 00:01:11.890 --> 00:01:14.930 So cosine is adjacent over hypotenuse. 00:01:14.930 --> 00:01:15.160 Right? 00:01:15.160 --> 00:01:16.390 SOH, CAH, TOA. 00:01:16.390 --> 00:01:16.830 CAH. 00:01:21.240 --> 00:01:28.980 So cosine is equal to adjacent over hypotenuse. 00:01:28.980 --> 00:01:31.730 So for this large angle, what's the adjacent? 00:01:31.730 --> 00:01:33.570 It's line AC. 00:01:33.570 --> 00:01:38.350 So that equals-- line AC, that's the adjacent. 00:01:41.120 --> 00:01:45.890 The length of line AC over the length of line-- 00:01:45.890 --> 00:01:47.790 what's the hypotenuse? 00:01:47.790 --> 00:01:50.270 AB, right? 00:01:50.270 --> 00:01:50.810 All right. 00:01:50.810 --> 00:01:56.400 AC over AB. 00:01:56.400 --> 00:01:58.670 Now let's see what we can do with this. 00:01:58.670 --> 00:02:02.640 AC-- adjacent over hypotenuse. 00:02:02.640 --> 00:02:06.100 Now can we write AC in any other interesting way-- a 00:02:06.100 --> 00:02:12.290 combination of some of the other lines on this very 00:02:12.290 --> 00:02:14.070 fortunately designed graph? 00:02:14.070 --> 00:02:15.230 Let's see. 00:02:15.230 --> 00:02:20.640 Well isn't AC the same thing as AF, this big line, 00:02:20.640 --> 00:02:22.660 minus-- what is this? 00:02:22.660 --> 00:02:23.610 This is a D, right? 00:02:23.610 --> 00:02:23.805 OK. 00:02:23.805 --> 00:02:25.020 That's a D. 00:02:25.020 --> 00:02:26.920 Let me rewrite that as a D. 00:02:26.920 --> 00:02:29.770 D as in dog. 00:02:29.770 --> 00:02:30.440 There you go. 00:02:30.440 --> 00:02:32.570 So AF minus DE. 00:02:32.570 --> 00:02:33.210 Right? 00:02:33.210 --> 00:02:35.090 Oh, I forgot to draw some things. 00:02:35.090 --> 00:02:37.330 We assume that this line is perpendicular to that line. 00:02:37.330 --> 00:02:39.790 We assume this line is perpendicular to that line. 00:02:39.790 --> 00:02:43.060 We assume that line is perpendicular to that line. 00:02:43.060 --> 00:02:43.450 Right? 00:02:43.450 --> 00:02:46.470 And then by definition, that is because we drew it that way. 00:02:46.470 --> 00:02:47.610 But anyway. 00:02:47.610 --> 00:02:51.720 So now you know that this line is parallel to this line and 00:02:51.720 --> 00:02:52.730 this line is perpendicular. 00:02:52.730 --> 00:02:59.770 So we know that AF, this long line, minus DE is equal to AC. 00:02:59.770 --> 00:03:00.560 Does that make sense? 00:03:00.560 --> 00:03:08.800 AF, this big line, minus the shorter line is 00:03:08.800 --> 00:03:10.070 the same thing as AC. 00:03:10.070 --> 00:03:10.600 Right? 00:03:10.600 --> 00:03:11.570 So let me write that down. 00:03:11.570 --> 00:03:25.850 That equals AF minus DE, all of that over AB. 00:03:29.720 --> 00:03:32.935 And then, of course, we can rewrite that as-- and I'm going 00:03:32.935 --> 00:03:43.080 to switch to some different colors-- as AF over AB. 00:03:45.940 --> 00:03:48.650 Let me switch to maybe green. 00:03:48.650 --> 00:03:58.100 Minus DE over AB. 00:03:58.100 --> 00:04:07.600 So we have now AF over AB minus DE over AB. 00:04:07.600 --> 00:04:10.320 And those are kind of nonsensical ratios to me. 00:04:10.320 --> 00:04:12.300 Wouldn't it be great if we could express it 00:04:12.300 --> 00:04:15.640 somehow as AF over AE? 00:04:15.640 --> 00:04:19.300 Because then we could say well that's cosine of alpha, and 00:04:19.300 --> 00:04:20.050 do something from there. 00:04:20.050 --> 00:04:20.850 Well let's try. 00:04:20.850 --> 00:04:23.960 So let's try to rewrite this first expression. 00:04:23.960 --> 00:04:25.720 So I'll switch back to the purple just so you 00:04:25.720 --> 00:04:27.890 know where this first expression is coming from. 00:04:27.890 --> 00:04:31.690 Let's see if we can break this down as AF over AE 00:04:31.690 --> 00:04:32.690 times something else. 00:04:32.690 --> 00:04:34.710 Well, we could just algebraically do it. 00:04:34.710 --> 00:04:41.760 That's equivalent to AF over-- I might run out of space-- 00:04:41.760 --> 00:04:52.170 over AE times AE over AB. 00:04:52.170 --> 00:04:54.370 And you're saying, Sal where did you get that from? 00:04:54.370 --> 00:04:57.870 Well, you can kind of say my motivation was to have AF 00:04:57.870 --> 00:04:59.540 as a ratio of over AE. 00:04:59.540 --> 00:05:04.330 And I just set it up so that the multiplication would cancel 00:05:04.330 --> 00:05:06.310 out, because the AE's would cancel out and you'd be 00:05:06.310 --> 00:05:07.550 left with AF over AB. 00:05:07.550 --> 00:05:07.810 Right? 00:05:07.810 --> 00:05:09.490 So this is a reasonable thing to do. 00:05:09.490 --> 00:05:11.200 I hope you see. 00:05:11.200 --> 00:05:15.520 And let me switch to the green and do something similar. 00:05:15.520 --> 00:05:19.310 DE over AB doesn't make much sense. 00:05:19.310 --> 00:05:24.680 But if I could maybe do DE over BE, then if this angle I can 00:05:24.680 --> 00:05:27.670 see is similar to alpha or beta then maybe I can 00:05:27.670 --> 00:05:29.060 make some progress. 00:05:29.060 --> 00:05:45.420 So let's say DE over BE times-- and we'll do the same thing. 00:05:45.420 --> 00:05:50.245 You just have to multiply times BE over AB. 00:05:53.390 --> 00:05:56.790 And just like in that sine proof, and we'll do the same 00:05:56.790 --> 00:05:59.760 thing here, let's figure out what this angle up here is. 00:05:59.760 --> 00:06:00.040 Right? 00:06:00.040 --> 00:06:03.950 Because if we know that then these ratios become useful. 00:06:03.950 --> 00:06:06.730 So if this angle is an alpha, then we know that this 00:06:06.730 --> 00:06:08.860 angle right here is alpha. 00:06:08.860 --> 00:06:09.140 Right? 00:06:09.140 --> 00:06:12.800 Because DE-- because this line-- is parallel to AF. 00:06:12.800 --> 00:06:14.100 You learned that in geometry. 00:06:14.100 --> 00:06:17.395 And if this angle is alpha we know that this angle right here 00:06:17.395 --> 00:06:21.850 is 90 minus alpha, because it's complementary. 00:06:21.850 --> 00:06:22.110 Right? 00:06:22.110 --> 00:06:24.020 Because this whole angle is 90 degrees, so this 00:06:24.020 --> 00:06:25.730 is 90 minus alpha. 00:06:25.730 --> 00:06:29.670 And since this angle 90 minus alpha, this angle 90, and this 00:06:29.670 --> 00:06:34.580 angle add up to 180, we could figure out that this is alpha. 00:06:34.580 --> 00:06:39.030 And if you don't believe me add up alpha plus 90 plus 90 minus 00:06:39.030 --> 00:06:42.190 alpha, and you will get 180 degrees. 00:06:42.190 --> 00:06:46.022 So this angle up here, angle DBE, is alpha. 00:06:46.022 --> 00:06:47.860 So that's very interesting. 00:06:47.860 --> 00:06:53.450 So can we rewrite these ratios as the sines or cosines 00:06:53.450 --> 00:06:54.360 of alpha's or beta's? 00:06:54.360 --> 00:06:56.230 Well, let's try. 00:06:56.230 --> 00:06:58.970 Let me switch back to purple. 00:06:58.970 --> 00:07:02.330 So that equals-- what is AF over AE? 00:07:06.340 --> 00:07:09.880 Well if we look at this right triangle, that's the adjacent 00:07:09.880 --> 00:07:11.820 over the hypotenuse for alpha. 00:07:11.820 --> 00:07:12.760 Right? 00:07:12.760 --> 00:07:15.000 Adjacent over hypotenuse, that's cosine. 00:07:15.000 --> 00:07:16.200 So it's cosine of alpha. 00:07:22.290 --> 00:07:24.030 And what's AE over AB? 00:07:28.300 --> 00:07:29.920 Well, they're similar. 00:07:29.920 --> 00:07:33.840 If we look at this big right triangle right here that is the 00:07:33.840 --> 00:07:35.950 adjacent over the hypotenuse for beta, so it's 00:07:35.950 --> 00:07:36.930 cosine of beta. 00:07:42.790 --> 00:07:45.970 Switch my colors. 00:07:45.970 --> 00:07:49.515 Minus DE over BE. 00:07:52.930 --> 00:07:54.570 Well this is alpha, right? 00:07:54.570 --> 00:07:55.440 Now there's a little smudge. 00:07:55.440 --> 00:07:56.420 You probably can't read it. 00:07:56.420 --> 00:07:56.980 But that was alpha. 00:07:56.980 --> 00:07:58.420 We showed that that was alpha. 00:07:58.420 --> 00:08:01.310 So DE is the opposite and BE is the hypotenuse. 00:08:01.310 --> 00:08:03.540 Opposite over hypotenuse is sine, right? 00:08:03.540 --> 00:08:04.880 So that's sine of alpha. 00:08:09.780 --> 00:08:11.780 And what is BE over AB? 00:08:15.710 --> 00:08:16.900 Look at this triangle again. 00:08:16.900 --> 00:08:20.000 Well, for beta that is BE is the opposite and 00:08:20.000 --> 00:08:21.250 AB is the hypotenuse. 00:08:21.250 --> 00:08:23.280 So opposite over hypotenuse for beta. 00:08:23.280 --> 00:08:25.170 So it's the sine of beta. 00:08:25.170 --> 00:08:26.240 Times the sine of beta. 00:08:26.240 --> 00:08:26.860 I'm running out of space. 00:08:26.860 --> 00:08:27.660 I have to go to another line. 00:08:32.330 --> 00:08:34.390 Pretty neat. 00:08:34.390 --> 00:08:39.255 I'll rewrite everything in a new and exciting color. 00:08:44.230 --> 00:08:44.620 OK. 00:08:44.620 --> 00:08:46.860 Let me do it in light blue. 00:08:46.860 --> 00:08:55.340 So we now know that the cosine of alpha plus beta is equal to 00:08:55.340 --> 00:09:00.970 the cosine of both of them multiplied-- so cosine of 00:09:00.970 --> 00:09:10.070 alpha, cosine of beta-- minus the sine of both of 00:09:10.070 --> 00:09:10.880 them multiplied. 00:09:10.880 --> 00:09:17.140 Minus sine of alpha times sine of beta. 00:09:17.140 --> 00:09:19.850 I hope you found that as satisfying as I do. 00:09:19.850 --> 00:09:22.080 See you in the next presentation.

Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b)

https://www.youtube.com/watch?v=zw0waJCEc-w

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https://www.youtube.com/api/timedtext?v=zw0waJCEc-w&ei=dmeUZeGyJIWhhcIPx-eKsAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=C2A45DF06A812D69BE9C566FF4EEB7A2612178E4.EE887ED6D12D4BAB3A85A14C4A951827A4B12271&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.810 --> 00:00:01.700 Welcome back. 00:00:01.700 --> 00:00:05.380 I'm now going to do a proof of a trig identity, which 00:00:05.380 --> 00:00:07.520 I think is pretty amazing. 00:00:07.520 --> 00:00:09.660 Although, I think, the proof isn't that obvious. 00:00:09.660 --> 00:00:12.250 And I'll have to admit ahead of time, this isn't something that 00:00:12.250 --> 00:00:14.070 would have occurred to me naturally. 00:00:14.070 --> 00:00:16.802 I wouldn't have naturally drawn this figure just 00:00:16.802 --> 00:00:20.010 to start off with. 00:00:20.010 --> 00:00:22.740 Let's just say we want to figure out some other way to 00:00:22.740 --> 00:00:33.440 write the sine of alpha plus beta, where alpha and beta are 00:00:33.440 --> 00:00:34.580 let's say, two separate angles. 00:00:34.580 --> 00:00:42.180 So if I had the sine of 40 and 50 degrees, I'd want to know-- 00:00:42.180 --> 00:00:44.140 this would obviously be the sine of 90, which is easy. 00:00:44.140 --> 00:00:47.040 But could I rewrite that as some combination of the sine 00:00:47.040 --> 00:00:49.280 of 40 and the sine of 50 or whatever? 00:00:49.280 --> 00:00:51.040 I think you'll see where this is going. 00:00:51.040 --> 00:00:52.790 So let's go back to this diagram and let's say 00:00:52.790 --> 00:00:56.040 that this-- let me pick a better color. 00:00:56.040 --> 00:01:01.710 Let's say that this is angle alpha and this is angle beta. 00:01:05.240 --> 00:01:12.020 Than this whole angle right here is angle alpha plus beta. 00:01:12.020 --> 00:01:14.970 So we want to figure out the sine of alpha plus beta. 00:01:14.970 --> 00:01:16.810 Well, the sine of alpha plus beta, the sine of this 00:01:16.810 --> 00:01:19.120 whole angle, opposite over hypotenuse. 00:01:19.120 --> 00:01:24.440 Opposite this whole angle is if we use this right angle-- or 00:01:24.440 --> 00:01:26.830 this right triangle, triangle BAC. 00:01:26.830 --> 00:01:33.262 Opposite is BC, so that equals BC. 00:01:33.262 --> 00:01:35.170 I'll draw a little line over it. 00:01:35.170 --> 00:01:37.790 BC over the hypotenuse, AB. 00:01:43.520 --> 00:01:48.650 BC over AB is the sine of alpha plus beta. 00:01:48.650 --> 00:01:53.770 Well, can be write BC over AB differently? 00:01:53.770 --> 00:01:54.900 Let's see if we can. 00:01:54.900 --> 00:01:56.920 And probably, the person who first figured out this proof 00:01:56.920 --> 00:01:58.440 was just playing around. 00:01:58.440 --> 00:02:01.090 They drew this diagram, they said, can I write 00:02:01.090 --> 00:02:02.480 BC any differently? 00:02:02.480 --> 00:02:08.730 Well BC-- this whole length-- is the sum of BD and EF. 00:02:08.730 --> 00:02:11.070 And we know that because this is a horizontal line right now 00:02:11.070 --> 00:02:12.620 and you can figure that out just by looking at all 00:02:12.620 --> 00:02:13.750 the right angles. 00:02:13.750 --> 00:02:14.800 But this is a horizontal line. 00:02:14.800 --> 00:02:19.970 So BC is the same thing is BD plus EF. 00:02:19.970 --> 00:02:20.880 Let's write that one down. 00:02:20.880 --> 00:02:30.750 BC is the same thing as BD plus EF. 00:02:34.050 --> 00:02:38.290 And then still, all of that, over AB. 00:02:38.290 --> 00:02:42.730 All I did is I rewrote BC as a sum of this segment and this 00:02:42.730 --> 00:02:44.700 segment, which should make sense to you, hopefully. 00:02:44.700 --> 00:02:52.970 And then we can of course, rewrite that as equal to BD 00:02:52.970 --> 00:03:05.460 over AB plus EF over AB. 00:03:05.460 --> 00:03:13.060 So BD over AB plus EF over AB. 00:03:13.060 --> 00:03:15.020 And these are kind of nonsensical ratios, right? 00:03:15.020 --> 00:03:16.920 BD over AB, what can I do with that? 00:03:16.920 --> 00:03:19.420 And EF over AB, what can I do with that? 00:03:19.420 --> 00:03:23.190 Wouldn't it be more interesting if I could do like BD over BE. 00:03:23.190 --> 00:03:25.520 That'd be an interesting ratio because that would be a 00:03:25.520 --> 00:03:27.500 segment over its hypotenuse. 00:03:27.500 --> 00:03:30.400 So let's see if we can rewrite it somehow like that. 00:03:30.400 --> 00:03:33.610 Well, we could just do it mathematically. 00:03:33.610 --> 00:03:52.900 We could say this is equal to BD over BE times BE over AB. 00:03:52.900 --> 00:03:54.560 So this might seem non-intuitive to you, but 00:03:54.560 --> 00:03:55.820 it kind of makes sense. 00:03:55.820 --> 00:03:57.505 We didn't pick BE completely arbitrarily. 00:03:57.505 --> 00:04:01.750 We said we know what BD is, so let me pick another side that I 00:04:01.750 --> 00:04:05.130 can do something maybe with real trig ratios. 00:04:05.130 --> 00:04:10.330 And so I said BD over BE times BE over AB is 00:04:10.330 --> 00:04:12.380 equal to BD over AB. 00:04:12.380 --> 00:04:14.190 I hope I don't confuse you with all these letters. 00:04:14.190 --> 00:04:15.210 But that makes sense, right? 00:04:15.210 --> 00:04:16.850 Because these two terms would just cancel out. 00:04:16.850 --> 00:04:18.430 If we're just multiplying these fractions then you would 00:04:18.430 --> 00:04:21.770 get back to this top term. 00:04:21.770 --> 00:04:24.090 Let me actually make sure that you understand 00:04:24.090 --> 00:04:25.240 that this-- whoops. 00:04:25.240 --> 00:04:29.630 That this term and this term are the same thing. 00:04:29.630 --> 00:04:31.810 And now let's do that second term. 00:04:31.810 --> 00:04:34.960 We know EF, wouldn't it be good if we could relate EF to 00:04:34.960 --> 00:04:37.570 something, like it's the hypotenuse of this 00:04:37.570 --> 00:04:38.280 right triangle? 00:04:38.280 --> 00:04:39.220 Like AE. 00:04:39.220 --> 00:04:40.075 So let's do that. 00:04:42.810 --> 00:04:44.750 So let's put the plus sign there. 00:04:44.750 --> 00:05:01.450 EF over AB is the same thing as EF over AE times AE over AB. 00:05:01.450 --> 00:05:03.480 Once again, we're just multiplying fractions. 00:05:03.480 --> 00:05:06.170 These would cancel out and you would get this again. 00:05:06.170 --> 00:05:10.760 Let me make sure you understand that this term is the 00:05:10.760 --> 00:05:11.660 same thing as this term. 00:05:11.660 --> 00:05:13.560 And you can just multiple out the fractions and that's 00:05:13.560 --> 00:05:15.570 what you would get. 00:05:15.570 --> 00:05:19.920 Now before we progress with this whole line of 00:05:19.920 --> 00:05:20.900 thought that we're doing. 00:05:20.900 --> 00:05:22.860 Let's see if we could figure out something else interesting 00:05:22.860 --> 00:05:26.990 about this strange set of triangles and shapes 00:05:26.990 --> 00:05:27.850 that I've drawn. 00:05:27.850 --> 00:05:30.060 It's actually pretty neat. 00:05:30.060 --> 00:05:36.390 IF this angle is alpha-- we have line AF. 00:05:36.390 --> 00:05:38.800 EF is perpendicular to it, right? 00:05:38.800 --> 00:05:41.450 And DE is perpendicular to EF, right? 00:05:41.450 --> 00:05:45.260 So DE, this line, and AF are parallel. 00:05:45.260 --> 00:05:51.110 Since AF is parallel to DE and then, AE intersects both, 00:05:51.110 --> 00:05:52.350 we know that, what is that? 00:05:52.350 --> 00:05:53.460 The inner angles? 00:05:53.460 --> 00:05:55.780 Yeah, I think that's called inner angles 00:05:55.780 --> 00:05:56.730 with parallel lines. 00:05:56.730 --> 00:06:01.080 That this is also equal to alpha. 00:06:01.080 --> 00:06:04.330 You can imagine long parallel line here, long parallel here, 00:06:04.330 --> 00:06:05.890 and then this line intersects both. 00:06:05.890 --> 00:06:08.500 So if this is a little confusing maybe you want to 00:06:08.500 --> 00:06:11.880 review a little bit of the parallel line geometry, but I 00:06:11.880 --> 00:06:13.150 think this might make sense. 00:06:13.150 --> 00:06:17.380 So if this angle is alpha, then this angle right here 00:06:17.380 --> 00:06:19.300 is complementary to it. 00:06:19.300 --> 00:06:20.740 So it's 90 minus alpha. 00:06:23.470 --> 00:06:26.600 And if this angle is 90 minus alpha, this 00:06:26.600 --> 00:06:28.480 angle is obviously 90. 00:06:28.480 --> 00:06:31.210 Then we know that this angle plus this angle plus this 00:06:31.210 --> 00:06:32.460 angle has to equal 180. 00:06:32.460 --> 00:06:35.620 So we know that this is equal to alpha. 00:06:35.620 --> 00:06:38.980 If that doesn't make sense to you, think about this: alpha 00:06:38.980 --> 00:06:44.070 plus 90 minus alpha plus 90-- that's a minus. 00:06:44.070 --> 00:06:45.370 Minus alpha. 00:06:45.370 --> 00:06:47.120 Plus 90 is what? 00:06:47.120 --> 00:06:48.760 Alpha plus 90 minus alpha. 00:06:48.760 --> 00:06:51.920 So this minus alpha and alpha cancel out and you just have 90 00:06:51.920 --> 00:06:54.080 plus 90 and that equals 180. 00:06:54.080 --> 00:06:55.775 So we know that this angle right here, I know it's 00:06:55.775 --> 00:06:57.980 getting really small and probably hard to read. 00:06:57.980 --> 00:07:01.940 We know that this angle here is alpha. 00:07:01.940 --> 00:07:03.730 So let's get back to what we were progressing, 00:07:03.730 --> 00:07:05.060 what we were doing here. 00:07:05.060 --> 00:07:09.190 So what is BD over BE? 00:07:09.190 --> 00:07:12.850 BD over BE. 00:07:12.850 --> 00:07:15.990 Well, that's the adjacent to this alpha, which is 00:07:15.990 --> 00:07:17.690 the same angle really. 00:07:17.690 --> 00:07:23.530 BD over BE, so it's adjacent over hypotenuse. 00:07:23.530 --> 00:07:24.630 Cosine. 00:07:24.630 --> 00:07:28.780 So that is equal to the cosine of alpha. 00:07:32.880 --> 00:07:35.360 And what's BE over AB? 00:07:39.900 --> 00:07:44.770 Well, if we look at this larger right triangle, that is the 00:07:44.770 --> 00:07:50.230 opposite of beta times its hypotenuse. 00:07:50.230 --> 00:07:52.620 So what's opposite over hypotenuse? 00:07:52.620 --> 00:07:53.740 SOH. 00:07:53.740 --> 00:07:54.710 S O H. 00:07:54.710 --> 00:07:55.480 Sine. 00:07:55.480 --> 00:07:59.000 So sine of beta is BE over AB. 00:07:59.000 --> 00:08:00.310 So this is sine of beta. 00:08:06.940 --> 00:08:10.780 And now let me switch to magenta. 00:08:10.780 --> 00:08:13.610 What's EF over AE? 00:08:17.210 --> 00:08:19.960 If we look at this right triangle right here, 00:08:19.960 --> 00:08:24.220 is opposite over hypotenuse for alpha. 00:08:24.220 --> 00:08:26.050 So it's sine of alpha. 00:08:26.050 --> 00:08:27.030 Opposite over hypotenuse. 00:08:30.000 --> 00:08:32.660 And what's AE over AB? 00:08:37.240 --> 00:08:40.330 So now we're looking at this large right triangle here. 00:08:40.330 --> 00:08:42.780 AE over AB. 00:08:42.780 --> 00:08:46.710 Well, that's the adjacent of beta over the hypotenuse. 00:08:46.710 --> 00:08:48.580 Well, what's adjacent over hypotenuse? 00:08:48.580 --> 00:08:51.430 That's the cosine. 00:08:51.430 --> 00:08:53.230 CAH. 00:08:53.230 --> 00:08:57.140 Cosine of beta, of this beta right here. 00:08:57.140 --> 00:08:58.650 I think we're done. 00:08:58.650 --> 00:09:02.140 This is to me, fairly mind blowing. 00:09:02.140 --> 00:09:14.010 That the sine of alpha plus beta is equal to the cosine of 00:09:14.010 --> 00:09:15.920 alpha times the sine of beta. 00:09:15.920 --> 00:09:20.310 Plus the sine of alpha times the cosine of beta. 00:09:20.310 --> 00:09:22.590 What's neat about this is that it kind of came out of this 00:09:22.590 --> 00:09:24.970 nice symmetric formula. 00:09:24.970 --> 00:09:27.370 It's not this big, hairy thing. 00:09:27.370 --> 00:09:28.530 You might have even guessed it. 00:09:28.530 --> 00:09:29.940 I don't know. 00:09:29.940 --> 00:09:30.950 I just find it very neat. 00:09:30.950 --> 00:09:33.430 We went through this big convoluted proof with this big 00:09:33.430 --> 00:09:37.780 convoluted shape, but we got this nice symmetric trig 00:09:37.780 --> 00:09:38.980 identity out of it. 00:09:38.980 --> 00:09:41.780 So hopefully you found that amazing as well and in the next 00:09:41.780 --> 00:09:45.900 presentation I'll do a proof for cosine of alpha plus beta. 00:09:45.900 --> 00:09:47.460 See you soon.

Trigonometric Identities

https://www.youtube.com/watch?v=OLzXqIqZZz0

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.880 --> 00:00:01.780 Welcome back. 00:00:01.780 --> 00:00:03.590 I'm now going to do a series of videos on the 00:00:03.590 --> 00:00:05.430 trigonometric identities. 00:00:05.430 --> 00:00:08.250 So let's just review what we already know about the 00:00:08.250 --> 00:00:09.920 trig function, so let me just write SOHCAHTOA. 00:00:15.960 --> 00:00:18.320 That tells us, and we've actually extended this with the 00:00:18.320 --> 00:00:20.670 unit circle definition, but if you watch those videos, you'll 00:00:20.670 --> 00:00:23.040 realize that the unit circle definition directly 00:00:23.040 --> 00:00:23.910 uses SOHCAHTOA. 00:00:23.910 --> 00:00:25.690 So we'll just stick with SOHCAHTOA because I think it'll 00:00:25.690 --> 00:00:28.690 help make some of what we're about to do seem a little bit 00:00:28.690 --> 00:00:31.380 more straightforward and will kind of verge on the unit 00:00:31.380 --> 00:00:32.760 circle definition anyway. 00:00:32.760 --> 00:00:40.150 So we know that sine of theta is equal to opposite 00:00:40.150 --> 00:00:41.390 over hypotenuse, right? 00:00:41.390 --> 00:00:49.290 So cosine of theta is equal to adjacent over hypotenuse, and 00:00:49.290 --> 00:00:55.930 then the tangent of theta is equal to opposite 00:00:55.930 --> 00:00:58.310 over adjacent. 00:00:58.310 --> 00:01:01.420 So let's draw that out on a right triangle. 00:01:01.420 --> 00:01:03.230 We could do this with the unit circle as well, and 00:01:03.230 --> 00:01:05.590 it would make sense. 00:01:05.590 --> 00:01:07.960 Let's see if we can find a relationship between sine, 00:01:07.960 --> 00:01:10.760 cosine and tangent. 00:01:10.760 --> 00:01:11.915 There's my right triangle. 00:01:14.490 --> 00:01:16.690 Let's call this theta. 00:01:16.690 --> 00:01:18.910 This is the hypotenuse h. 00:01:18.910 --> 00:01:21.820 This is the opposite side, right, opposite of theta. 00:01:21.820 --> 00:01:23.480 This is theta right here. 00:01:23.480 --> 00:01:27.090 This is the adjacent side, right? 00:01:27.090 --> 00:01:28.940 Well, what do we know about the relationship between 00:01:28.940 --> 00:01:31.860 the opposite adjacent side and then the hypotenuse? 00:01:31.860 --> 00:01:34.030 What does the Pythagorean theorem tell us? 00:01:34.030 --> 00:01:38.350 Oh, yeah, this side squared plus this side squared is equal 00:01:38.350 --> 00:01:41.290 to the hypotenuse squared, so we could write that down. 00:01:41.290 --> 00:01:48.100 a squared plus o squared is equal to the hypotenuse 00:01:48.100 --> 00:01:49.900 squared, right? 00:01:49.900 --> 00:01:52.980 And then this is just an equation, so if we want to, we 00:01:52.980 --> 00:01:56.620 could divide both sides of this equation by h squared, 00:01:56.620 --> 00:01:57.790 and so what do we get? 00:01:57.790 --> 00:02:07.880 We get a squared over h squared plus o squared over h squared 00:02:07.880 --> 00:02:10.590 is equal to 1, right? 00:02:10.590 --> 00:02:22.070 And then I could rewrite that as a over h squared plus o 00:02:22.070 --> 00:02:26.710 over h squared is equal to 1. 00:02:26.710 --> 00:02:30.930 Now, do these look at all familiar? 00:02:30.930 --> 00:02:32.080 Well, we have them here, right? 00:02:32.080 --> 00:02:35.570 This is a over h, this is o over h, so we could 00:02:35.570 --> 00:02:36.750 just substitute. 00:02:36.750 --> 00:02:41.280 So this is just cosine of theta squared. 00:02:41.280 --> 00:02:42.520 And this is how you write cosine squared. 00:02:42.520 --> 00:02:44.280 You could put a parentheses around the whole thing and then 00:02:44.280 --> 00:02:47.760 square it, but this is just the notation people use. 00:02:47.760 --> 00:02:53.510 Plus opposite over adjacent squared, so that's sine theta 00:02:53.510 --> 00:02:55.280 squared is equal to 1. 00:02:55.280 --> 00:02:56.530 So that's our first trig identity. 00:02:56.530 --> 00:03:00.440 So if you know the sine of theta, it's very easy to figure 00:03:00.440 --> 00:03:01.680 out the cosine of theta, right? 00:03:01.680 --> 00:03:04.550 You could just solve this equation. 00:03:04.550 --> 00:03:06.430 If I know that the-- I don't know. 00:03:06.430 --> 00:03:14.050 Let's say I know that the sine of theta is 1/2, right? 00:03:14.050 --> 00:03:16.351 Then what is the cosine of theta? 00:03:16.351 --> 00:03:18.630 The cosine of theta is what? 00:03:18.630 --> 00:03:20.950 Well, I know the sine of theta is 1/2, right? 00:03:20.950 --> 00:03:28.160 So I would say cosine squared of theta plus sine of theta 00:03:28.160 --> 00:03:34.770 is 1/2, so 1/2 squared is equal to 1, right? 00:03:34.770 --> 00:03:41.480 So cosine squared theta plus 1/4 is equal to 1. 00:03:41.480 --> 00:03:47.870 So we have cosine squared theta is equal to 3/4, or cosine of 00:03:47.870 --> 00:03:50.830 theta would be the square root of this, right? 00:03:50.830 --> 00:03:52.010 We just take the square root of both sides. 00:03:52.010 --> 00:03:54.510 It would be the square root of 3/2. 00:03:54.510 --> 00:03:56.770 And you probably remember that from our whole presentation 00:03:56.770 --> 00:03:57.900 on the 30-60-90 triangle. 00:03:57.900 --> 00:04:03.400 So I just wanted to show you a use of this trig identity 00:04:03.400 --> 00:04:05.520 that's usually written sine squared plus cosine 00:04:05.520 --> 00:04:07.860 squared is equal to 1. 00:04:07.860 --> 00:04:09.740 So let's extend that one a little bit. 00:04:09.740 --> 00:04:12.750 Let's just play with the ratios and see what else we can-- 00:04:12.750 --> 00:04:15.950 other identities we can discover. 00:04:15.950 --> 00:04:17.180 Whoops! 00:04:17.180 --> 00:04:20.370 Clear image, invert colors. 00:04:20.370 --> 00:04:28.310 So we know that sine squared theta plus cosine squared 00:04:28.310 --> 00:04:31.390 theta is equal to 1. 00:04:31.390 --> 00:04:35.220 The one thing we could do is we could divide both sides of this 00:04:35.220 --> 00:04:38.880 equation by cosine squared of theta, and let's just see what 00:04:38.880 --> 00:04:40.260 happens when we do that. 00:04:40.260 --> 00:04:44.020 So if we say cosine squared theta, right? 00:04:44.020 --> 00:04:46.530 You have to distribute across both terms. 00:04:46.530 --> 00:04:57.710 Cosine squared of theta, and then cosine squared of theta. 00:04:57.710 --> 00:05:00.360 Well, what's sine squared theta over cosine squared theta? 00:05:00.360 --> 00:05:07.720 That's the same thing as sine of theta over cosine of theta 00:05:07.720 --> 00:05:16.550 squared plus this is 1 over cosine theta squared, right? 00:05:16.550 --> 00:05:19.240 I mean, 1 squared is 1, so I just rewrote it. 00:05:19.240 --> 00:05:21.980 So sine over cosine theta, I think we learned that already. 00:05:21.980 --> 00:05:23.410 That's just the tangent of theta. 00:05:26.000 --> 00:05:28.280 And in case you actually haven't learned that already, 00:05:28.280 --> 00:05:30.190 think about it this way. 00:05:30.190 --> 00:05:34.030 Sine is opposite over the hypotenuse, right? 00:05:34.030 --> 00:05:36.350 So that's opposite over hypotenuse. 00:05:36.350 --> 00:05:40.410 And then cosine is adjacent over hypotenuse. 00:05:40.410 --> 00:05:42.290 So adjacent over hypotenuse. 00:05:42.290 --> 00:05:46.350 So then that equals opposite over hypotenuse times 00:05:46.350 --> 00:05:49.030 hypotenuse over adjacent, right? 00:05:49.030 --> 00:05:51.720 Just dividing by a fraction is the same thing as multiplying 00:05:51.720 --> 00:05:53.160 by its reciprocal. 00:05:53.160 --> 00:05:54.000 That's all I did. 00:05:54.000 --> 00:05:56.780 And that equals opposite over adjacent, right? 00:05:59.290 --> 00:06:01.930 So that just says sine of theta over cosine of theta is 00:06:01.930 --> 00:06:03.720 equal to tangent of theta. 00:06:03.720 --> 00:06:08.270 So sine squared theta over cosine squared theta is tan 00:06:08.270 --> 00:06:15.310 squared theta, then plus 1 is 1 over cosine theta squared. 00:06:15.310 --> 00:06:19.300 And now I'm going to introduce a new trig ratio, it's really 00:06:19.300 --> 00:06:21.800 just 1 over cosine theta. 00:06:21.800 --> 00:06:24.945 So 1 over cosine theta-- and I'm going to summarize this at 00:06:24.945 --> 00:06:28.050 the end, just so it's not too confusing-- is actually 00:06:28.050 --> 00:06:30.340 the secant of theta. 00:06:30.340 --> 00:06:31.780 And this is just another ratio, right? 00:06:31.780 --> 00:06:35.170 The secant of theta, instead of being the adjacent over the 00:06:35.170 --> 00:06:38.580 hypotenuse, would be the hypotenuse over the 00:06:38.580 --> 00:06:39.180 adjacent, right? 00:06:39.180 --> 00:06:40.370 It's just 1 over cosine theta. 00:06:40.370 --> 00:06:42.710 Nothing fancy here. 00:06:42.710 --> 00:06:44.000 So secant of theta. 00:06:44.000 --> 00:06:47.580 So that equals secant squared of theta. 00:06:47.580 --> 00:06:49.280 I know it can be a little overwhelming initially, just 00:06:49.280 --> 00:06:51.750 because I'm, you know, throwing out all these new terms, secant 00:06:51.750 --> 00:06:54.670 is 1 over cosine theta, but once you just play around with 00:06:54.670 --> 00:06:56.790 these enough and get familiar with the terms, it'll make 00:06:56.790 --> 00:06:58.130 sense, and it'll be a little more natural to you. 00:06:58.130 --> 00:06:59.760 So this could be-- you could view this as 00:06:59.760 --> 00:07:01.970 another trig identity. 00:07:01.970 --> 00:07:04.160 And actually, I don't even remember if I've 00:07:04.160 --> 00:07:05.690 taught it already. 00:07:05.690 --> 00:07:07.460 I mean, you could view this as a trig identity, although 00:07:07.460 --> 00:07:09.110 that's almost definitional. 00:07:09.110 --> 00:07:12.730 And then, of course, you can-- in case I haven't done it 00:07:12.730 --> 00:07:18.340 already, you now know that sine of theta over cosine of theta 00:07:18.340 --> 00:07:22.570 is equal to tangent of theta. 00:07:22.570 --> 00:07:25.210 And that's right here with, I guess you could 00:07:25.210 --> 00:07:27.770 say, the proof of it. 00:07:27.770 --> 00:07:30.750 So let me keep introducing you to more things, and if this is 00:07:30.750 --> 00:07:33.540 really daunting, maybe you just can rewatch it, and 00:07:33.540 --> 00:07:34.890 hopefully, it'll make sense. 00:07:34.890 --> 00:07:37.810 Let me see, clear image. 00:07:37.810 --> 00:07:39.150 So what have we learned so far? 00:07:39.150 --> 00:07:45.540 We learned that sine squared theta plus cosine squared 00:07:45.540 --> 00:07:47.540 theta is equal to 1. 00:07:47.540 --> 00:07:53.880 We learned that sine of theta over cosine of theta is 00:07:53.880 --> 00:07:56.700 equal to tangent of theta. 00:07:56.700 --> 00:08:04.750 We learned that the tangent squared of theta plus 1 is 00:08:04.750 --> 00:08:08.650 equal to the secant of theta. 00:08:08.650 --> 00:08:11.030 And here, let me actually write this definition down. 00:08:11.030 --> 00:08:13.675 The secant of theta-- oops, is equal to the secant 00:08:13.675 --> 00:08:15.790 squared of theta, sorry. 00:08:15.790 --> 00:08:19.800 And the secant of theta is just 1 over cosine of theta. 00:08:19.800 --> 00:08:21.670 This is something you really should just memorize, that 00:08:21.670 --> 00:08:22.950 secant is 1 over cosine. 00:08:22.950 --> 00:08:27.430 And if you're wondering what 1 over sine is, 1 over sine of 00:08:27.430 --> 00:08:33.350 theta, it's the cosecant-- the abbreviation is csc-- of theta. 00:08:33.350 --> 00:08:37.690 And if you're wondering what 1 over the tangent is, 00:08:37.690 --> 00:08:39.730 it's the cotangent. 00:08:39.730 --> 00:08:41.660 And you just might want to memorize these. 00:08:41.660 --> 00:08:46.600 And this often confuses me, that 1 over the cosine is the 00:08:46.600 --> 00:08:50.340 secant, but 1 over the sine is the cosecant, so it's kind of 00:08:50.340 --> 00:08:52.250 almost the opposite, right? 00:08:52.250 --> 00:08:55.230 1 over the sine has a co in it, while 1 over the cosine 00:08:55.230 --> 00:08:56.060 doesn't have the co in it. 00:08:56.060 --> 00:09:00.150 So that might help you remember things. 00:09:00.150 --> 00:09:02.760 So I think that's all I have time for now. 00:09:02.760 --> 00:09:06.120 In the next presentation, I'm going to introduce you to a 00:09:06.120 --> 00:09:07.575 couple more trig identities. 00:09:07.575 --> 00:09:09.380 See you soon.

Rates-of-change (part 2)

https://www.youtube.com/watch?v=xmgk8_l3lig

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en

WEBVTT Kind: captions Language: en 00:00:00.900 --> 00:00:01.790 Welcome back. 00:00:01.790 --> 00:00:04.320 I'm now going to actually do a couple more rates of change 00:00:04.320 --> 00:00:06.480 problems because I think the first one I did was probably a 00:00:06.480 --> 00:00:09.420 little bit more complicated than I wanted, and the hard 00:00:09.420 --> 00:00:11.460 part was actually the geometry and not the calculus. 00:00:11.460 --> 00:00:14.660 So let's just say I drop a rock into a pond 00:00:14.660 --> 00:00:16.140 and it has a ripple. 00:00:16.140 --> 00:00:18.250 And the ripple spreads out-- it'll probably have multiple 00:00:18.250 --> 00:00:20.760 ripples, but let's just say we focus on one ripple. 00:00:20.760 --> 00:00:24.450 And the ripple is spreading out. 00:00:24.450 --> 00:00:25.550 So let me draw. 00:00:25.550 --> 00:00:28.560 This is where I dropped the rock, and then this is 00:00:28.560 --> 00:00:31.000 a ripple in the pond. 00:00:31.000 --> 00:00:33.310 It would be an actual circle, which isn't what 00:00:33.310 --> 00:00:34.780 I drew, but close enough. 00:00:38.620 --> 00:00:41.310 So let me say that-- let's call r. 00:00:41.310 --> 00:00:42.800 r is not for ripple. 00:00:42.800 --> 00:00:45.330 r is the distance that the ripple is from the center 00:00:45.330 --> 00:00:48.440 from where it started from the get go. 00:00:48.440 --> 00:00:52.320 And it's not by chance that that's also for radius. 00:00:52.320 --> 00:00:55.700 So let's say that the ripple is moving out at 00:00:55.700 --> 00:00:57.720 2 meters per second. 00:00:57.720 --> 00:01:03.400 So the rate at which r is changing with respect to t 00:01:03.400 --> 00:01:07.660 is 2 meters per second. 00:01:07.660 --> 00:01:14.190 And what I want to know is what is the rate at which the area 00:01:14.190 --> 00:01:20.250 that's kind of included by this ripple-- this entire area-- how 00:01:20.250 --> 00:01:24.430 fast is that changing with respect to time. 00:01:24.430 --> 00:01:31.840 When the ripple is let's say 3 meters from the center. 00:01:31.840 --> 00:01:35.430 When the ripple is equal to 3 meters. 00:01:35.430 --> 00:01:42.530 Well do we know any relation between the area of the circle 00:01:42.530 --> 00:01:46.690 essentially, and the distance of the ripple, or the radius? 00:01:46.690 --> 00:01:49.250 Well yes we do. 00:01:49.250 --> 00:01:57.770 Area is equal to pi r squared. 00:01:57.770 --> 00:02:00.290 So we want to figure out the rate at which area changes 00:02:00.290 --> 00:02:01.220 with respect to time. 00:02:01.220 --> 00:02:02.630 So what does a chain rule tell us? 00:02:02.630 --> 00:02:07.270 The chain rule tells us that the rate at which a changes 00:02:07.270 --> 00:02:11.890 with respect to t is equal to the rate at which a changes 00:02:11.890 --> 00:02:17.490 with respect to r times the rate at which r changes 00:02:17.490 --> 00:02:19.910 with respect to t. 00:02:19.910 --> 00:02:22.120 Well we already know this piece, right? 00:02:22.120 --> 00:02:23.990 We already know the rate at which r is changing 00:02:23.990 --> 00:02:25.270 with respect to time. 00:02:25.270 --> 00:02:28.010 That's right here. 00:02:28.010 --> 00:02:31.270 All we have to do to figure out the rate at which a is changing 00:02:31.270 --> 00:02:34.320 with respect to time is we have to figure out the rate at which 00:02:34.320 --> 00:02:37.320 a changes with respect to r. 00:02:37.320 --> 00:02:38.410 Well that's just the derivative. 00:02:38.410 --> 00:02:42.230 The derivative with respect to r. 00:02:42.230 --> 00:02:48.260 The derivative with respect to r-- do it of both sides-- of a 00:02:48.260 --> 00:02:58.076 equals the derivative with respect to r of pi r squared. 00:02:58.076 --> 00:03:04.160 So we get da/dr is equal to-- what's the derivative 00:03:04.160 --> 00:03:05.250 with respect to r here? 00:03:05.250 --> 00:03:05.850 Well that's easy. 00:03:05.850 --> 00:03:13.390 Just 2 times pi r to the 1. 00:03:13.390 --> 00:03:18.660 That's pretty interesting by itself that the rate at which 00:03:18.660 --> 00:03:23.650 the derivative of the area-- formula really-- is what? 00:03:23.650 --> 00:03:27.100 This formula by itself looks interesting, right? 00:03:27.100 --> 00:03:29.350 That's a formula for the circumference of a circle. 00:03:29.350 --> 00:03:32.590 So the derivative of an area of a circle with respect to the 00:03:32.590 --> 00:03:34.690 radius is a circumference. 00:03:34.690 --> 00:03:37.510 That is mildly fascinating. 00:03:37.510 --> 00:03:44.070 Something even more interesting is to figure out what the 00:03:44.070 --> 00:03:47.270 antiderivative is of the area and compare that to the 00:03:47.270 --> 00:03:48.740 volume of a sphere. 00:03:48.740 --> 00:03:51.700 You can look it up on Wikipedia or something. 00:03:51.700 --> 00:03:54.465 Or even compare that to the surface area of the sphere, and 00:03:54.465 --> 00:03:57.570 just keep picking derivatives and integrals and they'll be 00:03:57.570 --> 00:03:58.960 some pretty interesting relationships. 00:03:58.960 --> 00:03:59.350 But anyway. 00:03:59.350 --> 00:04:00.030 Back to the problem. 00:04:00.030 --> 00:04:03.060 Not to go on too far of a tangent. 00:04:03.060 --> 00:04:04.420 So we figured out the rate at which 8 changes 00:04:04.420 --> 00:04:05.460 with respect to r. 00:04:05.460 --> 00:04:09.380 So going back to our original thing, we now know that the 00:04:09.380 --> 00:04:15.230 rate at which a changes with respect to t is equal to the 00:04:15.230 --> 00:04:17.790 rate at which a changes with respect to r. 00:04:17.790 --> 00:04:20.116 Well that's just right here, right? 00:04:20.116 --> 00:04:22.280 The rate at which a changes with respect to r. 00:04:22.280 --> 00:04:28.760 Well that's just 2 pi r times the rate at which r is 00:04:28.760 --> 00:04:31.700 changing with respect to t. 00:04:31.700 --> 00:04:32.720 Well the rate at which r changes with respect to 00:04:32.720 --> 00:04:34.880 t is just right here. 00:04:34.880 --> 00:04:37.400 2 meters per second, we figured that out. 00:04:37.400 --> 00:04:45.970 So 2 pi r times 2-- well I won't include the units 00:04:45.970 --> 00:04:47.195 because that might confuse you right now. 00:04:47.195 --> 00:04:49.135 So the rate at at which a is changing with respect 00:04:49.135 --> 00:04:50.560 to t is 2 pi r times 2. 00:04:50.560 --> 00:04:51.840 Well are we done yet? 00:04:51.840 --> 00:04:53.880 Well no, because we haven't substituted yet for r. 00:04:53.880 --> 00:04:54.930 So what is r? 00:04:54.930 --> 00:05:03.300 Well we know. r is three meters, so da/dt is equal to 00:05:03.300 --> 00:05:06.230 2 times pi times 3 times 2. 00:05:06.230 --> 00:05:07.090 So what is that? 00:05:07.090 --> 00:05:09.610 2 times 3 is 6 times 2 is 12. 00:05:09.610 --> 00:05:14.850 12 pi and it's meters squared per second, because it's the 00:05:14.850 --> 00:05:17.280 rate at which area-- square meters-- is changing 00:05:17.280 --> 00:05:18.110 with respect to time. 00:05:18.110 --> 00:05:20.320 And if we multiply out the units we would have gotten 00:05:20.320 --> 00:05:22.490 the same thing right here. 00:05:22.490 --> 00:05:24.990 So in the next video i'm going to do a slightly harder rate of 00:05:24.990 --> 00:05:28.700 change problem just so that you see that it doesn't only apply 00:05:28.700 --> 00:05:30.730 to kind of purely geometric things. 00:05:30.730 --> 00:05:32.350 It applies to pretty much anything that you can 00:05:32.350 --> 00:05:35.990 find a relationship between two values. 00:05:35.990 --> 00:05:36.360 See you in the next module.

Introduction to rate-of-change problems

https://www.youtube.com/watch?v=Zyq6TmQVBxk

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en

WEBVTT Kind: captions Language: en 00:00:00.880 --> 00:00:03.300 We've learned a lot about derivatives, and now we will 00:00:03.300 --> 00:00:07.840 use them to solve something that is hopefully may 00:00:07.840 --> 00:00:08.970 be kind of useful. 00:00:08.970 --> 00:00:11.980 So let's just start with a review of the chain rule, 00:00:11.980 --> 00:00:13.760 and I'm going to write in a different way. 00:00:13.760 --> 00:00:19.920 So let's say I had the function f of g of x. 00:00:19.920 --> 00:00:22.530 So what I'm going to do is I'm going actually write this in a 00:00:22.530 --> 00:00:25.430 way that might be a bit foreign to you, but I think with a 00:00:25.430 --> 00:00:27.370 little bit of explanation you'll realize that this is the 00:00:27.370 --> 00:00:32.500 same thing as the chain rule that we all know and love. 00:00:32.500 --> 00:00:34.830 I changed colors really for no reason. 00:00:34.830 --> 00:00:36.220 Just to change colors. 00:00:36.220 --> 00:00:37.230 Sometimes I do that. 00:00:37.230 --> 00:00:38.570 I'll change colors again. 00:00:38.570 --> 00:00:45.500 The derivative of f of g of x is equal to the rate at which 00:00:45.500 --> 00:00:48.790 f changes with respect to g. 00:00:48.790 --> 00:01:01.870 So that's df to dg times the rate at which g changes 00:01:01.870 --> 00:01:04.290 with respect to x. 00:01:04.290 --> 00:01:06.900 And you're saying Sal, this looks completely foreign to me. 00:01:06.900 --> 00:01:09.550 And one, you could think about what I just said and 00:01:09.550 --> 00:01:10.490 I think it'll make sense. 00:01:10.490 --> 00:01:12.760 Or I could just rewrite this in the traditional 00:01:12.760 --> 00:01:14.540 chain rule format. 00:01:14.540 --> 00:01:16.130 And this isn't really the traditional. 00:01:16.130 --> 00:01:17.440 What I wrote is the traditional, but the 00:01:17.440 --> 00:01:19.100 way I've taught you. 00:01:19.100 --> 00:01:22.300 So the way I taught you, what's the change of 00:01:22.300 --> 00:01:24.450 f with respect to g? 00:01:24.450 --> 00:01:34.130 Well that's just f prime of g of x times-- and what's the 00:01:34.130 --> 00:01:37.025 rate at which g changes with respect to x? 00:01:37.025 --> 00:01:40.870 Well that's just g prime of x. 00:01:40.870 --> 00:01:43.270 So hopefully this make sense to you that these are just 00:01:43.270 --> 00:01:45.410 two different ways of writing the chain rule. 00:01:45.410 --> 00:01:47.670 I think what was this, this is either Lagrange or Leibniz's 00:01:47.670 --> 00:01:49.430 notation, and this the other guy. 00:01:49.430 --> 00:01:50.640 But I forget. 00:01:50.640 --> 00:01:53.040 And this actually makes sense just from a fractions point of 00:01:53.040 --> 00:01:55.500 view that this term and that term cancels out and you get, 00:01:55.500 --> 00:01:59.730 you know that this equals that rate at which f changes 00:01:59.730 --> 00:02:02.190 with respect x. 00:02:02.190 --> 00:02:04.960 So with that out of the way, let's use this to solve 00:02:04.960 --> 00:02:08.020 something kind of hopefully a little bit neat. 00:02:08.020 --> 00:02:11.635 Let's say I have a cone. 00:02:11.635 --> 00:02:13.830 Let's say it's one of those cups you have 00:02:13.830 --> 00:02:15.280 at the water cooler. 00:02:15.280 --> 00:02:16.350 That's the top of the cup. 00:02:20.510 --> 00:02:27.500 And let's say at any point in the clone the ratio of the 00:02:27.500 --> 00:02:31.090 radius of the cone to the height-- if this is the 00:02:31.090 --> 00:02:35.110 height, then the radius is 1/2 of height. 00:02:35.110 --> 00:02:36.340 At any point. 00:02:36.340 --> 00:02:40.750 As you know, it's a fixed cone, these are lines on both sides. 00:02:40.750 --> 00:02:42.215 So that ratio stays fixed. 00:02:46.170 --> 00:02:49.850 So just to start off with some-- you maybe never 00:02:49.850 --> 00:02:50.605 even learned this. 00:02:53.180 --> 00:02:54.900 This will help us with the problem later on. 00:02:54.900 --> 00:02:59.020 What's the volume of-- let's say I'm filling water 00:02:59.020 --> 00:03:00.500 up to height h. 00:03:00.500 --> 00:03:04.650 Let's say this is where the water line is. 00:03:04.650 --> 00:03:09.130 But that's where the water is in this cone. 00:03:09.130 --> 00:03:13.870 What's the volume assuming that the height is h. 00:03:13.870 --> 00:03:17.680 Assuming that I've let's say 8 centimeters of water. 00:03:17.680 --> 00:03:20.210 What is the volume of water that I put in the cone? 00:03:20.210 --> 00:03:23.450 Well, if you don't know it, and I sometimes-- well actually 00:03:23.450 --> 00:03:28.180 later when we do rotations of solid into integration modules 00:03:28.180 --> 00:03:33.030 I can actually prove this to you-- but the volume of a cone 00:03:33.030 --> 00:03:38.300 is equal to 1/3 base times height, where the base is 00:03:38.300 --> 00:03:40.120 actually, you can kind of view it as the surface 00:03:40.120 --> 00:03:42.550 area of the water. 00:03:42.550 --> 00:03:44.370 And what's the base? 00:03:44.370 --> 00:03:47.595 Well that's just equal to volume is equal to 1/3. 00:03:47.595 --> 00:03:52.230 The base is equal to pi r squared, where r is the radius. 00:03:52.230 --> 00:03:54.770 This is just kind of solid geometry review. 00:03:54.770 --> 00:03:57.000 And what's the radius in this case? 00:03:57.000 --> 00:04:02.220 It's 1/3 pi r squared times h. 00:04:02.220 --> 00:04:04.370 And in this example I said that the radius 00:04:04.370 --> 00:04:06.960 is 1/2 of the height. 00:04:06.960 --> 00:04:13.690 So this equals 1/3 pi 1/2 the height-- I just replaced it 00:04:13.690 --> 00:04:15.450 for r squared-- times h. 00:04:15.450 --> 00:04:16.760 We haven't done any calculus yet. 00:04:16.760 --> 00:04:21.100 This is just complicated-- not complicated really-- just a 00:04:21.100 --> 00:04:23.770 little bit hairy solid geometry. 00:04:23.770 --> 00:04:28.220 And if we simplify that we get volume is equal to-- let's see. 00:04:28.220 --> 00:04:37.850 So we get 1/2 squared is 1/4 times 1/3 is 1/12 pi, and 00:04:37.850 --> 00:04:42.990 then h squared times h, h to the third. 00:04:42.990 --> 00:04:48.200 Very interesting, Now let's start doing some calculus. 00:04:48.200 --> 00:04:50.640 I think this might blow your mind. 00:04:50.640 --> 00:05:03.030 Let's say I am pouring water into this cup at a rate of 1 00:05:03.030 --> 00:05:06.410 cubic centimeter per second. 00:05:06.410 --> 00:05:10.800 And for you metric jocks, you'll know sometimes doctors 00:05:10.800 --> 00:05:13.740 or nurses will say 1 cc, and that's also equal 00:05:13.740 --> 00:05:15.090 to 1 millimeter. 00:05:15.090 --> 00:05:17.013 Just for other frames of reference, but I like to stay 00:05:17.013 --> 00:05:19.070 in centimeters cubed per second because we're going 00:05:19.070 --> 00:05:22.380 to work with centimeters in multiple dimensions. 00:05:22.380 --> 00:05:25.960 But let's say we're pouring 1 centimeter cubed per 00:05:25.960 --> 00:05:28.270 second into this cup. 00:05:28.270 --> 00:05:34.460 I want to know-- this is an interesting question-- how fast 00:05:34.460 --> 00:05:36.380 is the water level rising. 00:05:36.380 --> 00:05:40.750 How fast is h changing, the height of the water, at the 00:05:40.750 --> 00:05:45.640 moment when h is equal to 2 centimeters. 00:05:45.640 --> 00:05:46.600 So how do we do that? 00:05:46.600 --> 00:05:49.080 This just give us a static picture of if we know the 00:05:49.080 --> 00:05:51.510 height we can figure out the volume. 00:05:51.510 --> 00:05:54.980 But if we figure it out, if we took the rate of change with 00:05:54.980 --> 00:05:57.700 respect to time off of both sides of this equation, 00:05:57.700 --> 00:05:59.390 something interesting might happen. 00:05:59.390 --> 00:06:01.310 So let's take the derivative with respect time 00:06:01.310 --> 00:06:01.940 on both sides. 00:06:01.940 --> 00:06:06.970 So the derivative with respect to time of the volume. 00:06:06.970 --> 00:06:08.860 Well that's just dv/dt. 00:06:11.930 --> 00:06:15.880 And what's the derivative with respect to time of 00:06:15.880 --> 00:06:17.730 this side of the equation? 00:06:17.730 --> 00:06:21.530 Well it's the rate which v changes with respect to h. 00:06:24.750 --> 00:06:27.970 This is v as a function of h, right? 00:06:27.970 --> 00:06:36.040 So it's going to be dv as a function-- how fast does v 00:06:36.040 --> 00:06:39.730 change with respect to h-- times how fast does h 00:06:39.730 --> 00:06:41.300 change with respect to t. 00:06:41.300 --> 00:06:43.175 This is just the chain rule up here. 00:06:43.175 --> 00:06:44.740 I want you to think about this a little bit. 00:06:44.740 --> 00:06:46.370 It might not seem obvious but all we're doing 00:06:46.370 --> 00:06:47.280 is the chain rule. 00:06:47.280 --> 00:06:49.820 It's a little confusing because I had no t's in this equation 00:06:49.820 --> 00:06:51.345 before, and all of a sudden I'm picking a derivative 00:06:51.345 --> 00:06:52.690 with respect to t. 00:06:52.690 --> 00:06:56.670 But let's just say that h actually is a function of t, 00:06:56.670 --> 00:06:59.020 which as you know it is. 00:06:59.020 --> 00:07:01.640 So let's then just solve that. 00:07:01.640 --> 00:07:06.020 So dv-- the derivative of v with respect to t-- is equal 00:07:06.020 --> 00:07:08.840 to the derivative of v with respect to h. 00:07:08.840 --> 00:07:09.740 Well that's easy. 00:07:09.740 --> 00:07:12.850 The derivative of v with respect to h is just simple. 00:07:12.850 --> 00:07:17.230 So 3 times 1/12, that's 3/12. 00:07:17.230 --> 00:07:25.730 That's 1/4 pi, so we can just write pi over 4 h squared. 00:07:25.730 --> 00:07:26.720 So that's this part. 00:07:29.550 --> 00:07:34.800 And in this part is just still-- see, I can rewrite 00:07:34.800 --> 00:07:38.570 in this new color dh/dt, times dh/dt. 00:07:42.200 --> 00:07:44.250 So you're saying Sal, what have you now done. 00:07:44.250 --> 00:07:48.236 I said the rate of which the volume is changing with respect 00:07:48.236 --> 00:07:52.810 to time is equal to pi over 4 times the height squared times 00:07:52.810 --> 00:07:56.560 the rate at which the height is changing with respect to time. 00:07:56.560 --> 00:07:58.630 So can this solve the problem for us? 00:08:04.590 --> 00:08:06.870 Well what do we know? 00:08:06.870 --> 00:08:08.860 We know the rate at which the volume is changing 00:08:08.860 --> 00:08:09.980 with respect to time. 00:08:09.980 --> 00:08:12.330 1 centimeter cubed per second squared. 00:08:12.330 --> 00:08:17.030 So we could say dv/dt is equal to 1. 00:08:17.030 --> 00:08:18.210 I'm going to get rid of the units. 00:08:18.210 --> 00:08:20.620 Most physics instructors would cringe. 00:08:20.620 --> 00:08:28.080 But 1 centimeter cubed per second is equal to pi over 4. 00:08:28.080 --> 00:08:32.390 This one is v the rate at which the volume is changing 00:08:32.390 --> 00:08:34.450 with respect to time. 00:08:34.450 --> 00:08:35.520 pi over 4 h squared. 00:08:35.520 --> 00:08:38.210 We know what h is right now. h is 2, so it's 00:08:38.210 --> 00:08:41.410 2 squared times 4. 00:08:41.410 --> 00:08:44.785 We said the height is 2 right then, so 2 squared 00:08:44.785 --> 00:08:46.330 is 4 times dh/dt. 00:08:49.720 --> 00:08:54.820 So this cancels out, and we get 1 is equal to pi times dh/dt. 00:08:54.820 --> 00:08:58.150 And we solve for dt/dt we get-- let me make sure 00:08:58.150 --> 00:08:59.100 not to confuse you. 00:08:59.100 --> 00:09:02.423 The rate at which the height of water is changing with respect 00:09:02.423 --> 00:09:04.860 to time is just 1 over pi. 00:09:04.860 --> 00:09:06.710 Fascinating. 00:09:06.710 --> 00:09:10.180 Or 1 over pi centimeters per second. 00:09:10.180 --> 00:09:11.900 So we can figure out what this number is, it's going to 00:09:11.900 --> 00:09:15.150 be like 0.3 something. 00:09:15.150 --> 00:09:18.660 So 0.3 something centimeters per second is the rate at which 00:09:18.660 --> 00:09:22.900 the height of the water level is going to change as I put 1 00:09:22.900 --> 00:09:28.300 centimeter cubed of volume into this cup per second. 00:09:28.300 --> 00:09:29.620 I have probably confused you. 00:09:29.620 --> 00:09:31.090 You may want to watch this again. 00:09:31.090 --> 00:09:33.110 And I'll do a couple more videos with these rates of 00:09:33.110 --> 00:09:36.800 change problems because these tend to confuse people, but 00:09:36.800 --> 00:09:39.400 once you get the hang of it, I think you'll see that 00:09:39.400 --> 00:09:40.890 they're not so bad. 00:09:40.890 --> 00:09:41.650 I'll see you the next presentation.

Definite Integrals (part 4)

https://www.youtube.com/watch?v=11Bt6OhIeqA

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WEBVTT Kind: captions Language: en 00:00:00.730 --> 00:00:01.970 Welcome back. 00:00:01.970 --> 00:00:05.390 I'm now going to use definite integrals to figure out the 00:00:05.390 --> 00:00:07.890 areas under a bunch of curves and, if we have time, maybe 00:00:07.890 --> 00:00:09.290 even between some curves. 00:00:09.290 --> 00:00:11.660 So let me right down the fundamental theorem 00:00:11.660 --> 00:00:12.350 of calculus. 00:00:12.350 --> 00:00:15.560 I know I covered it really fast in the last presentation. 00:00:15.560 --> 00:00:17.310 Just to make sure you understand this formula. 00:00:17.310 --> 00:00:19.020 The last couple presentations were really to give you an 00:00:19.020 --> 00:00:20.800 intuition for this exact formula. 00:00:20.800 --> 00:00:34.440 Let's say that big f prime of x is equal to f of x, right? 00:00:34.440 --> 00:00:39.200 That's also like saying that the -- that's equivalent to 00:00:39.200 --> 00:00:45.950 saying that f of x -- big f of x -- is equal to the 00:00:45.950 --> 00:00:56.000 antiderivative of f of x, right? 00:00:56.000 --> 00:01:01.880 Well, let's just that it's one of the possible antiderivatives 00:01:01.880 --> 00:01:02.760 of f of x, right? 00:01:02.760 --> 00:01:04.420 Because there's always a constant term here and you're 00:01:04.420 --> 00:01:06.090 not sure whether it is. 00:01:06.090 --> 00:01:09.760 And this is why people tend to use this standard because we 00:01:09.760 --> 00:01:15.590 know that f of x is the derivative of big f prime of x. 00:01:15.590 --> 00:01:18.850 Big f of x is just one of the antiderivatives of f of x. 00:01:18.850 --> 00:01:21.100 So this is a little bit not true, but I think 00:01:21.100 --> 00:01:21.800 you get the idea. 00:01:21.800 --> 00:01:26.430 But the fundamental theorem of calculus tells us if this top 00:01:26.430 --> 00:01:38.480 line is true, then the definite integral from a to b of f of x 00:01:38.480 --> 00:01:45.520 d x is equal to its antiderivative evaluated at b 00:01:45.520 --> 00:01:50.010 minus its antiderivative evaluated at a. 00:01:50.010 --> 00:01:53.300 And I know I said here that big f isn't the only 00:01:53.300 --> 00:01:54.310 antiderivative, right? 00:01:54.310 --> 00:01:56.760 Because you could at any constant to this and that would 00:01:56.760 --> 00:01:58.200 also be the antiderivative. 00:01:58.200 --> 00:02:01.110 But when you subtract here, the constants will cancel out. 00:02:01.110 --> 00:02:02.575 So it really doesn't matter which of the 00:02:02.575 --> 00:02:03.250 constants you pick. 00:02:03.250 --> 00:02:04.420 The constant actually doesn't matter. 00:02:04.420 --> 00:02:06.765 So that's why I actually said the antiderivative. 00:02:06.765 --> 00:02:08.150 But let's apply this. 00:02:08.150 --> 00:02:09.960 You might be confused right now. 00:02:09.960 --> 00:02:13.795 So let me draw a graph. 00:02:21.740 --> 00:02:22.160 There you go. 00:02:22.160 --> 00:02:25.630 Look how straight that is. 00:02:25.630 --> 00:02:28.220 Draw the x-axis. 00:02:28.220 --> 00:02:29.510 Not perfect but it'll do. 00:02:37.640 --> 00:02:47.570 Let's say that my f of x is equal to x squared plus 1. 00:02:47.570 --> 00:02:49.620 So f of x looks like this. 00:02:49.620 --> 00:02:50.630 This is 1. 00:02:53.810 --> 00:02:54.770 So it'll start at 1. 00:02:54.770 --> 00:02:56.880 It'll just be a parabola. 00:02:56.880 --> 00:03:00.600 Let me see how good I can draw this. 00:03:00.600 --> 00:03:01.900 I've done worse. 00:03:01.900 --> 00:03:02.550 OK. 00:03:02.550 --> 00:03:03.550 So that's f of x. 00:03:03.550 --> 00:03:05.260 It's a parabola. y-intercept at 1. 00:03:05.260 --> 00:03:08.660 And let's say I want to figure out the area under the curve -- 00:03:08.660 --> 00:03:11.570 between the curve and, really, the x-axis. 00:03:11.570 --> 00:03:14.180 Let's say I want to figure out the area between the curve and 00:03:14.180 --> 00:03:20.030 the x-axis from x equals negative 1 to, I don't 00:03:20.030 --> 00:03:23.190 know, x equals 3. 00:03:23.190 --> 00:03:25.220 So this is the area I want to figure out. 00:03:25.220 --> 00:03:26.220 I'm going to shade it in. 00:03:29.970 --> 00:03:33.470 So this is the area. 00:03:33.470 --> 00:03:34.980 All of this stuff. 00:03:34.980 --> 00:03:36.660 I want to figure out this area. 00:03:36.660 --> 00:03:39.620 And you could imagine, before you knew calculus, figuring out 00:03:39.620 --> 00:03:41.870 an area of something with a curve -- it's kind 00:03:41.870 --> 00:03:42.630 of top boundary. 00:03:42.630 --> 00:03:44.100 It would have been very difficult. 00:03:44.100 --> 00:03:46.000 But we will now use the fundamental theorem of calculus 00:03:46.000 --> 00:03:48.970 and hopefully you have an intuition of why this works and 00:03:48.970 --> 00:03:52.470 how the integral is really just a sum of a bunch of little, 00:03:52.470 --> 00:03:55.400 little, small squares with infinitely small bases. 00:03:55.400 --> 00:03:57.030 But if you watched the last videos, hopefully that 00:03:57.030 --> 00:03:57.540 hit the point home. 00:03:57.540 --> 00:04:00.310 But now we'll just mechanically compute because, actually, 00:04:00.310 --> 00:04:02.925 understanding it is a bit harder than just doing it. 00:04:02.925 --> 00:04:04.390 But let's just mechanically compute it. 00:04:04.390 --> 00:04:07.470 So we are essentially just going to figure out the 00:04:07.470 --> 00:04:16.780 integral from minus 1 to 3 of f of x, which is x 00:04:16.780 --> 00:04:23.970 squared plus 1 d x. 00:04:23.970 --> 00:04:27.850 What's the antiderivative of x squared plus 1? 00:04:27.850 --> 00:04:30.010 This just equals the antiderivative. 00:04:30.010 --> 00:04:34.630 So it's just x to the third -- we could say 1/3 x to the third 00:04:34.630 --> 00:04:39.860 or x to the third over 3 -- plus x, right? 00:04:39.860 --> 00:04:41.130 The derivative of x is 1. 00:04:41.130 --> 00:04:42.770 And then we don't have to worry about plus c because we're 00:04:42.770 --> 00:04:44.600 going to subtract out the c's. 00:04:44.600 --> 00:04:45.030 You'll see. 00:04:45.030 --> 00:04:45.850 I think you'll get the point. 00:04:45.850 --> 00:04:46.340 It doesn't matter. 00:04:46.340 --> 00:04:47.800 You could pick an arbitrary c right here and it'll 00:04:47.800 --> 00:04:48.900 just cancel out. 00:04:48.900 --> 00:04:53.960 And we're going to evaluate that at 3 and negative 1 and 00:04:53.960 --> 00:04:57.590 we're going to subtract out big f of negative 00:04:57.590 --> 00:04:59.650 1 from big f of 3. 00:04:59.650 --> 00:05:00.940 This is just the notation they use. 00:05:00.940 --> 00:05:02.070 You figure out the antiderivative and 00:05:02.070 --> 00:05:03.710 you say where you're going to evaluate it. 00:05:03.710 --> 00:05:08.430 And then this is equal to -- So if I evaluate 3. 00:05:08.430 --> 00:05:10.820 3 to the third power is what? 00:05:10.820 --> 00:05:12.370 That's 27. 00:05:12.370 --> 00:05:18.010 27 divided by 3 is 9. 00:05:18.010 --> 00:05:21.420 And then 9 plus 3 is 12. 00:05:21.420 --> 00:05:22.360 Right? 00:05:22.360 --> 00:05:27.150 This is just big f of 3, right? 00:05:27.150 --> 00:05:30.520 Because I figured out the end -- This is big f of x. 00:05:30.520 --> 00:05:33.150 You can kind of view this as big f of x. 00:05:33.150 --> 00:05:36.340 But not to be confused with small, cursive f of x. 00:05:36.340 --> 00:05:37.860 This is big f of x. 00:05:37.860 --> 00:05:39.430 So this big f of 3. 00:05:39.430 --> 00:05:41.480 And then, from that we'll subtract big f 00:05:41.480 --> 00:05:43.280 of negative 1, right? 00:05:43.280 --> 00:05:45.950 Minus big f of negative 1. 00:05:45.950 --> 00:05:47.640 And if we put minus 1 here. 00:05:47.640 --> 00:05:50.020 Let's see, minus 1 to the third power is minus 1. 00:05:50.020 --> 00:05:54.570 So it's minus 1/3 and then plus minus 1, right? 00:05:54.570 --> 00:05:57.610 So minus 1/3 plus minus 1. 00:05:57.610 --> 00:06:02.690 I think that equals minus 4/3, correct? 00:06:02.690 --> 00:06:03.470 I think so. 00:06:03.470 --> 00:06:06.420 Maybe I'm making a mistake with negative signs. 00:06:06.420 --> 00:06:08.790 Minus 1/3 minus 4/3. 00:06:08.790 --> 00:06:10.840 And I'm going to subtract that, right? 00:06:10.840 --> 00:06:13.010 So if I'm subtracting minus 4/3, it's the same thing 00:06:13.010 --> 00:06:16.850 as adding minus 4/3. 00:06:16.850 --> 00:06:20.040 And then we have our answer. 00:06:20.040 --> 00:06:25.460 Actually, it's 12 and 4/3 -- whatever -- units. 00:06:25.460 --> 00:06:26.290 Squared units. 00:06:26.290 --> 00:06:27.880 12 and 4/3 squared units. 00:06:27.880 --> 00:06:30.480 We could write this as a mixed number as well. 00:06:30.480 --> 00:06:31.520 Let's do another one. 00:06:31.520 --> 00:06:33.285 I'll do a slight variation. 00:06:40.020 --> 00:06:42.850 OK. 00:06:42.850 --> 00:06:45.110 Let me draw again. 00:06:45.110 --> 00:06:47.596 Some coordinates. 00:06:47.596 --> 00:06:49.210 I don't know if I'm going to have time to do it in 00:06:49.210 --> 00:06:51.040 this video but I'll try. 00:06:51.040 --> 00:06:52.000 I always try. 00:06:55.060 --> 00:07:07.440 Let's say I have f of x is equal to the square root of x. 00:07:07.440 --> 00:07:08.590 So it looks something like this. 00:07:14.750 --> 00:07:17.910 That's actually a pretty nice looking, kind of sideways 00:07:17.910 --> 00:07:19.790 parabola, I think. 00:07:19.790 --> 00:07:22.960 This is f of x. 00:07:22.960 --> 00:07:25.660 And let's say I have another function. 00:07:25.660 --> 00:07:32.250 g of x which equals x squared. 00:07:32.250 --> 00:07:34.790 So g of x is actually going to look something like this. 00:07:41.980 --> 00:07:42.840 Whoops! 00:07:42.840 --> 00:07:45.650 I was doing well and then something happened. 00:07:45.650 --> 00:07:47.750 And, of course, it'll continue on this side as well. 00:07:47.750 --> 00:07:50.810 Because it is defined for negative numbers. 00:07:50.810 --> 00:07:53.990 But anyway, my question to you, or my question to myself, 00:07:53.990 --> 00:07:58.170 really, is what is the area between the curves 00:07:58.170 --> 00:07:59.730 where they intersect? 00:07:59.730 --> 00:08:01.340 What is this? 00:08:01.340 --> 00:08:02.420 What is this area? 00:08:08.980 --> 00:08:10.830 Well, the first thing you have to figure out is just what 00:08:10.830 --> 00:08:11.690 are the boundary points? 00:08:11.690 --> 00:08:12.922 What is this point? 00:08:12.922 --> 00:08:15.350 And what is this point? 00:08:15.350 --> 00:08:16.800 Well this point, I think, is pretty clear. 00:08:16.800 --> 00:08:19.220 It's 0, 0, right? 00:08:19.220 --> 00:08:21.570 They both intersect at 0, 0. 00:08:21.570 --> 00:08:24.640 And even this point, you could probably do it from intuition. 00:08:24.640 --> 00:08:30.310 But if you don't, I guess, want to do it through intuition, you 00:08:30.310 --> 00:08:32.990 could just set these 2 equations equal to 00:08:32.990 --> 00:08:33.580 each other, right? 00:08:33.580 --> 00:08:38.690 You could say x squared is equal to the square 00:08:38.690 --> 00:08:41.790 root of x, right? 00:08:41.790 --> 00:08:44.330 And then you could do a bunch of things. 00:08:44.330 --> 00:08:50.450 You could square both sides or -- well, actually, this is the 00:08:50.450 --> 00:08:51.910 same thing as doing it by intuition. 00:08:51.910 --> 00:08:54.170 But I think it's pretty obvious that the only places where x 00:08:54.170 --> 00:08:57.390 squared is equal to the square root of x are the points x 00:08:57.390 --> 00:09:02.510 equals 0, which you already know, and x equals 1. 00:09:02.510 --> 00:09:06.060 So this is the point 1, 1. 00:09:06.060 --> 00:09:07.310 Which is true for both of them. 00:09:07.310 --> 00:09:09.005 And this is more algebra, so I won't go into that 00:09:09.005 --> 00:09:09.890 in too much detail. 00:09:09.890 --> 00:09:12.220 I'm kind of running out of time. 00:09:12.220 --> 00:09:15.400 So we want to figure out the area between these 2 curves. 00:09:15.400 --> 00:09:18.280 So what we can do is -- maybe you want to pause it and think 00:09:18.280 --> 00:09:22.320 about it yourself -- we can figure out the area 00:09:22.320 --> 00:09:26.840 under the grey curve. 00:09:26.840 --> 00:09:28.180 We could figure out this area. 00:09:30.940 --> 00:09:34.840 So we want to figure out -- this is a boundary, right? 00:09:34.840 --> 00:09:35.760 Between 0 and 1. 00:09:35.760 --> 00:09:40.970 We could figure out this area and then we could figure out 00:09:40.970 --> 00:09:44.640 the entire area under the green curve separately and then we 00:09:44.640 --> 00:09:45.700 could subtract the difference. 00:09:45.700 --> 00:09:47.930 Which is exactly how we're going to do it in the next 00:09:47.930 --> 00:09:50.630 video because I have run out of time.

Definite Integrals (part 5)

https://www.youtube.com/watch?v=CmXmRNFrtFw

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WEBVTT Kind: captions Language: en 00:00:00.700 --> 00:00:01.940 Welcome back. 00:00:01.940 --> 00:00:05.730 Where I had just left off we were trying to figure out this 00:00:05.730 --> 00:00:08.460 area between these two curves, and we figured out that it's 00:00:08.460 --> 00:00:12.440 really the area between the curves between the point 0, 00:00:12.440 --> 00:00:14.520 x equals 0 and x equals 1. 00:00:14.520 --> 00:00:16.840 And I was proposing of a way to do it. 00:00:16.840 --> 00:00:19.620 Let's figure out the entire area under the square root of 00:00:19.620 --> 00:00:26.570 x from 0 to 1, and we can subtract from that, this 00:00:26.570 --> 00:00:29.580 purple area, which is the area under x squared. 00:00:29.580 --> 00:00:32.220 So just based on the last example we did, we could just 00:00:32.220 --> 00:00:34.490 write the indefinite integral, and I'm not going to rewrite 00:00:34.490 --> 00:00:36.970 the fundamental theorem from calculus, because I think 00:00:36.970 --> 00:00:38.770 you know that by now. 00:00:38.770 --> 00:00:41.080 Let me do it in a loud color. 00:00:41.080 --> 00:00:42.120 Magenta. 00:00:42.120 --> 00:00:45.770 So I want to know the larger area, right. 00:00:45.770 --> 00:00:47.355 The area under just the square root of x. 00:00:47.355 --> 00:00:49.660 That's Kind of like the combined area. 00:00:49.660 --> 00:00:58.710 Well that's just from 0 to 1, the integral of square root of 00:00:58.710 --> 00:01:04.050 x, because square root of x is the green function, dx. 00:01:04.050 --> 00:01:13.020 And I want to subtract from that the area from 0 to 1 00:01:13.020 --> 00:01:15.450 what's under x squared. 00:01:15.450 --> 00:01:18.110 x squared dx. 00:01:18.110 --> 00:01:19.240 And I just want to make a point. 00:01:19.240 --> 00:01:21.420 This could have just been rewritten this way. 00:01:21.420 --> 00:01:23.330 You could just rewrite a new function, which is the 00:01:23.330 --> 00:01:25.410 difference of these two functions, and it would 00:01:25.410 --> 00:01:26.120 have been equivalent. 00:01:26.120 --> 00:01:27.070 You could have said this. 00:01:27.070 --> 00:01:28.820 This isn't kind of a step of the problem, but you could 00:01:28.820 --> 00:01:29.520 have done it this way. 00:01:29.520 --> 00:01:32.080 In fact, some people start this way. 00:01:32.080 --> 00:01:35.960 See those are the same thing as the integral from 0 to 1 00:01:35.960 --> 00:01:42.120 of square root of x minus x squared dx. 00:01:42.120 --> 00:01:44.720 So you could do this two separate problems, two separate 00:01:44.720 --> 00:01:47.450 indefinite integrals, or you could do it as one 00:01:47.450 --> 00:01:48.080 indefinite integral. 00:01:48.080 --> 00:01:49.800 Actually that might be even simpler because when you 00:01:49.800 --> 00:01:52.620 evaluate it from 1 to 0 it simplify things a little bit. 00:01:52.620 --> 00:01:54.150 So let's stick with the second one. 00:01:54.150 --> 00:01:55.710 So first of all we just have to figure out what the 00:01:55.710 --> 00:01:58.930 antiderivative of this inner expression is. 00:01:58.930 --> 00:02:02.720 So you haven't seen square root of x yet. 00:02:02.720 --> 00:02:04.100 Do you think you know how to do it? 00:02:04.100 --> 00:02:06.480 Well, I think you do. 00:02:06.480 --> 00:02:10.060 Let's say that equals square root of x is just x to 00:02:10.060 --> 00:02:11.940 the 1/2 power, right? 00:02:11.940 --> 00:02:14.020 So we just use the same antiderivative rules 00:02:14.020 --> 00:02:15.160 we've always used. 00:02:15.160 --> 00:02:21.110 We raise it one more power so it becomes x to the 3/2. 00:02:21.110 --> 00:02:23.230 Right, it was 1/2, we added 1 to it. 00:02:23.230 --> 00:02:25.860 And then we divide by this new exponent. 00:02:25.860 --> 00:02:28.370 So dividing by a fraction is like multiplying 00:02:28.370 --> 00:02:30.920 by its reciprocal. 00:02:30.920 --> 00:02:36.840 So it's 2/3 x to the 3/2 and then minus-- I think the second 00:02:36.840 --> 00:02:42.780 term is pretty easy for you-- minus 1/3 x to the third. 00:02:42.780 --> 00:02:46.250 That's the antiderivative of minus x squared, minus 00:02:46.250 --> 00:02:48.200 1/3 x to the third. 00:02:48.200 --> 00:02:56.120 And we're going to have to evaluate this thing at 0 and 1 00:02:56.120 --> 00:02:57.770 and subtract the difference. 00:02:57.770 --> 00:03:01.040 Subtract this expression evaluated at 0 from this 00:03:01.040 --> 00:03:04.030 expression evaluated at 1. 00:03:04.030 --> 00:03:05.080 I think I'm running out of space. 00:03:05.080 --> 00:03:07.070 What happens when x equals 1? 00:03:07.070 --> 00:03:08.810 1 to the 3/2 is 1. 00:03:08.810 --> 00:03:09.755 1 to the third is 1. 00:03:09.755 --> 00:03:11.130 So it's 2/3 minus 1/3. 00:03:11.130 --> 00:03:12.370 Well that's easy. 00:03:12.370 --> 00:03:15.180 It's 1/3. 00:03:15.180 --> 00:03:17.030 I just put 1 in for x. 00:03:17.030 --> 00:03:19.470 And then when x is equal to 0 what is this expression equal? 00:03:19.470 --> 00:03:20.700 Well that's easy too. 00:03:20.700 --> 00:03:22.530 That's 0. 00:03:22.530 --> 00:03:23.720 So there you go. 00:03:23.720 --> 00:03:26.520 1/3 minus 0 or 1/3. 00:03:26.520 --> 00:03:27.610 That's kind of neat. 00:03:31.030 --> 00:03:35.940 You know I find is this of exciting because if just my 00:03:35.940 --> 00:03:38.940 intuition I was like, oh I have these two curves, and I mean 00:03:38.940 --> 00:03:43.320 they do intersect at the nice integer number, but you know 00:03:43.320 --> 00:03:45.830 what it's probably going to be some really messy number 00:03:45.830 --> 00:03:49.560 of what the areas between these two curves, right. 00:03:49.560 --> 00:03:51.990 Who knows, maybe it'll involve some you know-- a circle 00:03:51.990 --> 00:03:54.750 involves pi, which is this really messy number, so maybe 00:03:54.750 --> 00:03:57.100 all curves have these kind of messy areas. 00:03:57.100 --> 00:04:00.830 But this one it's just one of those neat things about math. 00:04:00.830 --> 00:04:05.360 The area between the square root of x and x squared is 1/3, 00:04:05.360 --> 00:04:07.000 which is a pretty clean number. 00:04:10.820 --> 00:04:13.100 Actually let me do one more problem since I have time. 00:04:18.260 --> 00:04:19.250 It's a bit of a trick problem. 00:04:19.250 --> 00:04:24.970 I mean, you might actually find this easy, but let's figure out 00:04:24.970 --> 00:04:38.310 the area between f of x is equal to-- I don't 00:04:38.310 --> 00:04:41.130 know, x to the fifth. 00:04:41.130 --> 00:04:42.120 I'm going to do something simple. 00:04:42.120 --> 00:04:43.645 Let me draw it actually. 00:04:56.930 --> 00:04:58.730 OK, I'm going to draw the x-axis. 00:05:04.660 --> 00:05:06.990 x to the fifth is going to go up super fast, 00:05:06.990 --> 00:05:07.780 something like that. 00:05:07.780 --> 00:05:09.220 It's going to go up real fast. 00:05:09.220 --> 00:05:10.640 Let's say I wanted to figure out the areas-- this side's 00:05:10.640 --> 00:05:13.930 going to go real fast too-- between that, and instead of 00:05:13.930 --> 00:05:19.180 figuring out the area between x-axis and that, f of x, I want 00:05:19.180 --> 00:05:28.730 to figure out the area between f of x and-- Instead 00:05:28.730 --> 00:05:31.530 of figuring out this bottom area, right? 00:05:31.530 --> 00:05:34.380 Like the normal problems we've done, we would figure out this 00:05:34.380 --> 00:05:36.950 type of area, you know, between two points. 00:05:36.950 --> 00:05:49.710 Let's say I want to figure out the area inside of the curve 00:05:49.710 --> 00:05:56.090 where the height here is 32. 00:05:56.090 --> 00:05:59.790 So I want to figure out this area inside the curve. 00:05:59.790 --> 00:06:02.070 How do we do that? 00:06:02.070 --> 00:06:05.010 Well one way we could do it is just like we did the last one, 00:06:05.010 --> 00:06:07.970 we can figure out some function that's essentially a line, a 00:06:07.970 --> 00:06:11.030 horizontal line that goes straight across here. 00:06:11.030 --> 00:06:14.800 And we'll essentially just be figuring out the area 00:06:14.800 --> 00:06:16.050 between the two functions. 00:06:16.050 --> 00:06:18.460 So what's a function that's a line that just goes 00:06:18.460 --> 00:06:20.820 straight at y equals 32? 00:06:20.820 --> 00:06:23.000 I think I just gave you the answer. 00:06:23.000 --> 00:06:25.720 Exactly. 00:06:25.720 --> 00:06:28.580 Let me stick with that greenish color. 00:06:28.580 --> 00:06:32.200 So we could say g of x is equal to 32. 00:06:32.200 --> 00:06:33.415 It's just a constant function. 00:06:33.415 --> 00:06:35.510 It just goes straight across. 00:06:35.510 --> 00:06:38.310 And then we need to figure out what the area is between the 00:06:38.310 --> 00:06:40.050 two, so we need to figure out what are these two points. 00:06:42.990 --> 00:06:47.240 So when does x to the fifth equal 32? 00:06:47.240 --> 00:06:50.000 I mean you could solve it algebraically, you know, 00:06:50.000 --> 00:06:53.954 you could say x to the fifth equals 32. 00:06:53.954 --> 00:07:03.420 x is equal to 2, and actually, you know what? 00:07:03.420 --> 00:07:05.020 I made a mistake. 00:07:05.020 --> 00:07:07.760 Let's say that this is not equal to x to the fifth. 00:07:07.760 --> 00:07:10.950 Let's say that f of x is equal to x to the absolute 00:07:10.950 --> 00:07:13.270 value of x to the fifth. 00:07:13.270 --> 00:07:15.590 Because the mistake, obviously x to the fifth is not a 00:07:15.590 --> 00:07:17.420 parabola looking thing. x to the fifth goes 00:07:17.420 --> 00:07:18.630 negative like this. 00:07:18.630 --> 00:07:22.070 But I have committed so much to this cup shape that I'll make 00:07:22.070 --> 00:07:24.350 it the absolute value of x to the fifth. 00:07:24.350 --> 00:07:27.290 So if I say the absolute value of x to the fifth is equal to 00:07:27.290 --> 00:07:30.170 32, I think you see where I realized my mistake. 00:07:30.170 --> 00:07:33.280 But if I say the absolute value of x to the fifth is 32, 00:07:33.280 --> 00:07:35.800 there's two places where that's true. 00:07:35.800 --> 00:07:38.430 It's x is equal to plus or minus 2. 00:07:38.430 --> 00:07:39.670 These are the two points. 00:07:39.670 --> 00:07:41.795 I should have done something with an even exponent so I 00:07:41.795 --> 00:07:43.520 could have had this cup shape, but anyway the absolute 00:07:43.520 --> 00:07:46.280 value solved my problem. 00:07:46.280 --> 00:07:48.740 So what is this area? 00:07:48.740 --> 00:07:51.180 We know it's between negative 2 and 2, so we just set up 00:07:51.180 --> 00:07:52.300 the indefinite integral. 00:07:52.300 --> 00:07:59.210 It's the indefinite integral for minus 2 to 2 of the top 00:07:59.210 --> 00:08:04.350 function, the top boundary, 32, minus the bottom boundary. 00:08:07.250 --> 00:08:11.270 Well, this will be a little bit tricky, but minus the absolute 00:08:11.270 --> 00:08:16.360 value of x to the fifth dx. 00:08:16.360 --> 00:08:18.660 And actually instead of doing this, I think you could see 00:08:18.660 --> 00:08:21.160 that there's symmetry here, so we could just figure out 00:08:21.160 --> 00:08:23.690 this area and multiply by 2. 00:08:23.690 --> 00:08:27.223 This problem's a little hairy, just because I had a bad 00:08:27.223 --> 00:08:29.160 choice of initial function. 00:08:29.160 --> 00:08:32.410 Not exactly what I wanted, but we'll work on forward. 00:08:32.410 --> 00:08:38.640 So instead of doing that, let's do the integral from 0 to 2 of 00:08:38.640 --> 00:08:45.020 32 minus x to the fifth dx. 00:08:45.020 --> 00:08:47.340 And then multiply that by 2. 00:08:47.340 --> 00:08:48.060 So what is that? 00:08:48.060 --> 00:08:56.060 That's 32x minus x to the sixth over 6. 00:08:56.060 --> 00:09:05.720 And we're going to evaluate it from 2 and 0 at 64 minus 64/6. 00:09:05.720 --> 00:09:13.860 2 to the sixth is 64, and then 32 times 0 is 0 and the 00:09:13.860 --> 00:09:20.370 next is 6, that's 0, so the answer is 64 minus 64/6. 00:09:20.370 --> 00:09:21.880 I'm about to run out of time. 00:09:21.880 --> 00:09:23.320 That's just a fraction problem there. 00:09:23.320 --> 00:09:25.230 Oh, and that's half of it, right? 00:09:25.230 --> 00:09:27.370 So we want to multiply that by 2. 00:09:27.370 --> 00:09:34.290 So if we multiply that by 2, we get 128 minus 128/6. 00:09:34.290 --> 00:09:35.980 I haven't figured out what it is. 00:09:35.980 --> 00:09:36.850 Well I guess we could figure it out. 00:09:36.850 --> 00:09:45.840 It's 128 times 1 minus 1/6 or 128 times 5/6. 00:09:45.840 --> 00:09:48.280 And I don't know what that is. 00:09:48.280 --> 00:09:51.100 I can multiply if I wanted to, but I have 10 seconds left 00:09:51.100 --> 00:09:52.400 so I'll leave you there. 00:09:52.400 --> 00:09:53.365 Hope I didn't confuse you. 00:09:53.365 --> 00:09:54.170 See you soon.

Indefinite Integration (part V)

https://www.youtube.com/watch?v=Pra6r20geXU

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https://www.youtube.com/api/timedtext?v=Pra6r20geXU&ei=eGeUZe2BMoq7vdIP6eqO6A8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249832&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2A6783B3ADF7D479BCB5C088ECAD31F58F1A6F7C.5F3450B0ED735E31DB6C07BB58F32F59CBBFE934&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.920 --> 00:00:03.830 I'm not going to do a presentation on a 00:00:03.830 --> 00:00:04.710 type of integral. 00:00:04.710 --> 00:00:07.680 I guess if you have this in your tool kit-- and actually 00:00:07.680 --> 00:00:12.910 you have it beyond the exam on this type of integral, and you 00:00:12.910 --> 00:00:15.290 actually keep it and you retain it, then you, I think, will 00:00:15.290 --> 00:00:17.890 become an integration jock. 00:00:17.890 --> 00:00:20.290 But anyway, let me show you what I'm talking about. 00:00:20.290 --> 00:00:23.020 So let's just remember what the product rule 00:00:23.020 --> 00:00:24.320 of differentiation was. 00:00:24.320 --> 00:00:29.990 So let's say I had two functions, let's say f of x 00:00:29.990 --> 00:00:36.505 times g of x, and I wanted to take the derivative of this. 00:00:39.700 --> 00:00:42.930 f of x times g of x. 00:00:42.930 --> 00:00:45.260 Well the chain rule just told us that this is just the same 00:00:45.260 --> 00:00:47.420 thing as, let's say, the derivative of the first 00:00:47.420 --> 00:00:58.590 function f prime of x times the second function g of x plus now 00:00:58.590 --> 00:01:04.350 the first function f of x times the derivative of the 00:01:04.350 --> 00:01:05.996 second function. 00:01:05.996 --> 00:01:10.210 And I'll show you where I'm going with this in a second. 00:01:10.210 --> 00:01:14.160 Now, if we were to integrate both sides of this equation, 00:01:14.160 --> 00:01:17.340 because we're still doing algebra on some levels, 00:01:17.340 --> 00:01:18.780 anything you do to one side of the equation, you 00:01:18.780 --> 00:01:20.140 can do to the other. 00:01:20.140 --> 00:01:22.160 So if we were to integrate both sides-- well if you integrate 00:01:22.160 --> 00:01:27.040 this side, you're taking the integral of a derivative, you 00:01:27.040 --> 00:01:29.430 just get back to what you took the derivative originally of, 00:01:29.430 --> 00:01:35.915 so this just becomes f of x times g of x, and then we have 00:01:35.915 --> 00:01:39.000 to integrate the right hand side, well that just becomes-- 00:01:39.000 --> 00:01:40.990 and we're doing the indefinite integral, kind of the 00:01:40.990 --> 00:01:43.360 antiderivative, but we can use this tool when we do 00:01:43.360 --> 00:01:45.540 definite integrals as well. 00:01:45.540 --> 00:01:55.960 So that's the integral of f prime of x g of x dx plus 00:01:55.960 --> 00:02:04.840 the integral of f of x g prime of x d of x. 00:02:04.840 --> 00:02:07.580 And now this might seem a little bit arbitrary, and it is 00:02:07.580 --> 00:02:11.560 a little bit arbitrary, let me take-- well, I could take 00:02:11.560 --> 00:02:16.290 either of these-- but let me just take this one and move it 00:02:16.290 --> 00:02:17.200 to this side of the equation. 00:02:17.200 --> 00:02:19.010 So I'm going to subtract this term from both 00:02:19.010 --> 00:02:20.210 sides of this equation. 00:02:20.210 --> 00:02:25.120 And so, we could say this-- so let me change colors, because 00:02:25.120 --> 00:02:28.330 this could get confusing-- this term right here, we could say 00:02:28.330 --> 00:02:43.785 that term f of x g prime of x d of x is equal to this term-- 00:02:43.785 --> 00:02:47.770 let me switch back to the yellow-- is equal to f of 00:02:47.770 --> 00:02:56.260 x g of x minus this term. 00:02:56.260 --> 00:02:58.105 Because I put it onto this side of the equation. 00:03:07.640 --> 00:03:08.930 So what did I just do? 00:03:08.930 --> 00:03:12.300 It looks like I just-- well, I am just essentially playing 00:03:12.300 --> 00:03:14.170 with the product rule from differentiation. 00:03:14.170 --> 00:03:15.020 That's all I did. 00:03:15.020 --> 00:03:16.880 And you probably wondering, well, Sal, this is all nice 00:03:16.880 --> 00:03:20.910 and it looks fancy, but what good is this going to do me? 00:03:20.910 --> 00:03:26.140 Well, what I essentially just did is I kind of proved this-- 00:03:26.140 --> 00:03:29.950 you could call this a formula, but I often forget it, 00:03:29.950 --> 00:03:32.450 especially once I haven't done it a long time, and then I 00:03:32.450 --> 00:03:35.350 actually just reprove it to myself just by remembering the 00:03:35.350 --> 00:03:39.250 product rule-- but this is called integration by parts. 00:03:39.250 --> 00:03:42.030 And I'll show you where this is useful. 00:03:42.030 --> 00:03:44.130 Let's say we want to take the indefinite integral 00:03:44.130 --> 00:03:58.590 of x cosine of x d of x. 00:03:58.590 --> 00:04:02.580 Well, everything we have in our integration tool kit so far I 00:04:02.580 --> 00:04:04.130 don't think help us here, right? 00:04:04.130 --> 00:04:06.680 Because we don't have a function and its derivative, so 00:04:06.680 --> 00:04:10.540 we can't do substitution or-- which is the same thing as the 00:04:10.540 --> 00:04:13.550 reverse chain rule-- this isn't a simple polynomial. 00:04:13.550 --> 00:04:17.550 So if you encounter this when you're doing integrals, kind of 00:04:17.550 --> 00:04:21.480 the last tool kit-- and this is pretty sophisticated-- is to 00:04:21.480 --> 00:04:22.770 do integration by parts. 00:04:22.770 --> 00:04:24.795 And so how can we use this for integration by parts? 00:04:24.795 --> 00:04:27.750 Well, integration by parts tells us that if we have an 00:04:27.750 --> 00:04:29.910 integral where we have a function and then the 00:04:29.910 --> 00:04:33.480 derivative of another function, then we could use this formula 00:04:33.480 --> 00:04:35.590 to hopefully simplify it. 00:04:35.590 --> 00:04:38.250 So what I'm going to do-- and you might view this as, well, 00:04:38.250 --> 00:04:39.580 Sal, how did you know to do this? 00:04:39.580 --> 00:04:42.370 And I'll tell you my thought process actually after 00:04:42.370 --> 00:04:44.270 I show you what I did. 00:04:44.270 --> 00:04:47.400 We always, in math in general, you always want to simplify. 00:04:47.400 --> 00:04:50.800 You always want to move from something that's complicated 00:04:50.800 --> 00:04:52.440 to something that's simpler. 00:04:52.440 --> 00:04:57.800 So in this situation, we could assume that x is f of x, and we 00:04:57.800 --> 00:05:00.730 could assume that g prime of x is cosine of x, or we could 00:05:00.730 --> 00:05:02.420 assume the other way around. 00:05:02.420 --> 00:05:08.070 The reason why I'm going to assume that x is f of x, and 00:05:08.070 --> 00:05:12.450 I'm going to assume g prime of x is cosine of x, is because 00:05:12.450 --> 00:05:15.540 later we want to take the derivative of f of x. 00:05:15.540 --> 00:05:18.400 The derivative of f of x simplifies things a lot. 00:05:18.400 --> 00:05:22.730 And we also want to take the integral of g prime of x. 00:05:22.730 --> 00:05:25.430 We want to take the antiderivative of g prime of x. 00:05:25.430 --> 00:05:29.680 And the antiderivative of cosine of x is sine of x, 00:05:29.680 --> 00:05:31.280 which is just as complicated. 00:05:31.280 --> 00:05:33.440 It's not making it any more complicated. 00:05:33.440 --> 00:05:36.510 And actually, try it the other way around, and you'll see that 00:05:36.510 --> 00:05:38.380 if you took the antiderivative of x, you would get something 00:05:38.380 --> 00:05:39.390 that's more complicated. 00:05:39.390 --> 00:05:40.740 You get x squared over 2. 00:05:40.740 --> 00:05:44.020 So that's the intuition, and let me just solve through it, 00:05:44.020 --> 00:05:46.920 and hopefully it'll make a little bit more sense. 00:05:46.920 --> 00:05:51.540 So if I assume that f of x is x, and g prime of x is cosine 00:05:51.540 --> 00:05:55.170 of x, then f of x-- this yellow term, let me write it in 00:05:55.170 --> 00:05:58.695 yellow, just for fun-- so f of x-- so I'm saying 00:05:58.695 --> 00:06:00.120 that f of x is x. 00:06:03.510 --> 00:06:05.300 So that's x. 00:06:05.300 --> 00:06:10.580 And let's say g of x is cosine of x, right? 00:06:10.580 --> 00:06:13.690 I'm sorry, g prime of x is cosine of x. 00:06:13.690 --> 00:06:16.840 So the derivative of g of x is cosine of x. 00:06:16.840 --> 00:06:18.340 So what's g of x? 00:06:18.340 --> 00:06:21.250 It was the antiderivative of cosine of x. 00:06:21.250 --> 00:06:22.350 So that's sine of x. 00:06:24.970 --> 00:06:26.310 I hope you understand what I'm saying. 00:06:26.310 --> 00:06:27.230 This can be a little confusing. 00:06:27.230 --> 00:06:29.340 Let me write it separately here. 00:06:29.340 --> 00:06:32.120 Actually, well I'm running out of space, but I'll write it in 00:06:32.120 --> 00:06:34.340 the corner right down here. 00:06:34.340 --> 00:06:38.600 See, I'm saying that f of x is x and I'm saying that 00:06:38.600 --> 00:06:43.300 g of x is sine of x. 00:06:43.300 --> 00:06:45.500 And the reason why I knew g of x is sine of x is because I 00:06:45.500 --> 00:06:51.630 said the derivative, I said g prime of x, is cosine of x. 00:06:51.630 --> 00:06:55.360 So you know, if the function-- if the derivative of a function 00:06:55.360 --> 00:06:57.970 is cosine, then we know the function itself is sine. 00:06:57.970 --> 00:06:59.450 That's just something you memorize. 00:06:59.450 --> 00:07:01.880 I haven't proven it to you yet, but it's usually in the inside 00:07:01.880 --> 00:07:02.920 cover of your calculus book. 00:07:02.920 --> 00:07:04.540 But let's just move forward. 00:07:04.540 --> 00:07:11.460 And then this is minus the integral of the 00:07:11.460 --> 00:07:12.750 derivative of f of x. 00:07:12.750 --> 00:07:14.045 Well what's the derivative of f of x? 00:07:14.045 --> 00:07:16.600 Well, we said f of x is x, right? 00:07:16.600 --> 00:07:21.870 So the derivative is just 1 times g of x. 00:07:21.870 --> 00:07:23.926 Well, I already said that g of x is sine of x. 00:07:27.470 --> 00:07:30.430 And I think you would agree that we've now simplified this 00:07:30.430 --> 00:07:32.340 a good bit, because this is just the integral of 00:07:32.340 --> 00:07:33.860 sine of x, right? 00:07:33.860 --> 00:07:39.290 So this is just equal to x sine of x-- this is just this first 00:07:39.290 --> 00:07:42.860 term right here-- minus-- and what's the integral 00:07:42.860 --> 00:07:44.540 of sine of x? 00:07:44.540 --> 00:07:49.960 Well, the derivative of-- well, let me make it even simpler. 00:07:49.960 --> 00:07:53.470 Well we can make this a minus, we can make this a minus sign 00:07:53.470 --> 00:07:55.930 of x and make this a plus. 00:07:55.930 --> 00:07:57.010 And now it's really easy. 00:07:57.010 --> 00:07:59.120 What's the antiderivative of minus sine of x? 00:07:59.120 --> 00:08:00.580 We can ignore this 1. 00:08:00.580 --> 00:08:02.800 What's the antiderivative of minus sine of x? 00:08:02.800 --> 00:08:06.510 Well, yeah, it's just cosine of x. 00:08:06.510 --> 00:08:10.810 And we should never forget the plus c. 00:08:10.810 --> 00:08:16.010 So we just used the product rule to derive this formula for 00:08:16.010 --> 00:08:18.240 integration by parts, and in a lot of calculus books they 00:08:18.240 --> 00:08:21.280 do this u and v and dvd. 00:08:21.280 --> 00:08:22.740 This is the same exact thing. 00:08:22.740 --> 00:08:27.350 I like this more, because it naturally makes 00:08:27.350 --> 00:08:27.970 more sense to me. 00:08:27.970 --> 00:08:31.570 It's easier for me to read, and I can derive at any time, 00:08:31.570 --> 00:08:32.820 just from the product rule. 00:08:32.820 --> 00:08:34.500 So I don't have to necessarily memorize it. 00:08:34.500 --> 00:08:36.290 You might want to memorize it for the exam, because it's 00:08:36.290 --> 00:08:38.920 faster for when you take the AP exam. 00:08:38.920 --> 00:08:43.910 But when you do calculus-- it's been about fifteen years since 00:08:43.910 --> 00:08:47.430 I learned it-- and I just remember integration by parts 00:08:47.430 --> 00:08:50.750 is just really a derivation of the product rule, and that 00:08:50.750 --> 00:08:51.590 gets me back to the format. 00:08:51.590 --> 00:08:52.920 I don't even have to look it up. 00:08:52.920 --> 00:08:54.260 And then I can use it. 00:08:54.260 --> 00:08:57.930 So that's an introduction to integration by parts. 00:08:57.930 --> 00:09:00.890 In the next presentation, I will do a bunch of examples-- 00:09:00.890 --> 00:09:03.330 well, as many as I can fit in to ten minutes of actually 00:09:03.330 --> 00:09:07.930 using integration by parts to solve fairly fancy integrals. 00:09:07.930 --> 00:09:09.850 I'll see you in the next presentation.

Indefinite Integration (part 7)

https://www.youtube.com/watch?v=F-OsMq7QKEQ

vtt

https://www.youtube.com/api/timedtext?v=F-OsMq7QKEQ&ei=dmeUZYTOKa-sp-oPjoal6AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=ECF5F7FAE5B73BC025E36E83B8EB7DA6BC4C3C27.3478E8C7807BAA35673CEF2DF083E1FE494B3ECC&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.930 --> 00:00:03.730 I'm now going to do it integration by parts problems. 00:00:03.730 --> 00:00:07.870 I think it's just a fun problem to see because one, it's the 00:00:07.870 --> 00:00:10.300 example a lot of people use, sometimes even a trick problem 00:00:10.300 --> 00:00:13.920 that's given on a really hard math exam, or if you go to 00:00:13.920 --> 00:00:17.560 calculus competitions like I used to in high school. 00:00:17.560 --> 00:00:21.470 Not to make myself too-- I was actually not that geeky as a 00:00:21.470 --> 00:00:25.650 high school student, but I have to admit, I was a mathlete. 00:00:25.650 --> 00:00:29.830 But anyway, this is just a fun integration by parts problem 00:00:29.830 --> 00:00:34.770 because you actually never have to evaluate the final integral. 00:00:34.770 --> 00:00:36.830 So let's say we want to take the integral-- it's 00:00:36.830 --> 00:00:38.690 a bit of a classic. 00:00:38.690 --> 00:00:41.660 I wouldn't be surprised if your math teacher does the same 00:00:41.660 --> 00:00:43.790 problem for you, just to show you integration by parts. 00:00:43.790 --> 00:00:47.080 Let's take the integral of e to the x-- you probably never 00:00:47.080 --> 00:00:49.960 heard of someone call a math problem a classic, but 00:00:49.960 --> 00:00:53.810 hopefully I will instill in you this love for mathematics and 00:00:53.810 --> 00:00:57.540 you will also consider this to be a classic problem. 00:00:57.540 --> 00:00:59.760 e to the x times cosine of x. 00:01:02.670 --> 00:01:04.490 I think you might already see where I'm going with this, 00:01:04.490 --> 00:01:07.330 because these are both fun functions, because e to the x 00:01:07.330 --> 00:01:08.870 you can take the derivative, you could take the 00:01:08.870 --> 00:01:11.280 anti-derivative and it still stays e to the x. 00:01:11.280 --> 00:01:14.510 Cosine of x you take the derivative, you go to minus 00:01:14.510 --> 00:01:16.980 sign of x, you take the derivative again then you to 00:01:16.980 --> 00:01:19.270 minus cosine of x, then you take the derivative again then 00:01:19.270 --> 00:01:20.420 you get a plus sign of x. 00:01:20.420 --> 00:01:21.450 It's like this cycle. 00:01:21.450 --> 00:01:23.540 The same thing happens when you take the anti-derivative. 00:01:23.540 --> 00:01:26.340 It's not as cool as e to the x, it doesn't stay exactly the 00:01:26.340 --> 00:01:29.390 same, but it kind of cycles. 00:01:29.390 --> 00:01:31.690 If you take two anti-derivatives you get back 00:01:31.690 --> 00:01:33.550 to the negative of itself. 00:01:33.550 --> 00:01:35.400 And if you take two derivatives, you get back 00:01:35.400 --> 00:01:37.780 to the negative of itself. 00:01:37.780 --> 00:01:40.320 It's also a pretty cool function and I think you can 00:01:40.320 --> 00:01:45.850 start to see how integration by parts might be cool here. 00:01:45.850 --> 00:01:48.730 Whenever I do integration by parts I always like to assume 00:01:48.730 --> 00:01:50.880 that this is the g prime of x. 00:01:50.880 --> 00:01:54.070 That e to the x is g prime of x, because e to the x 00:01:54.070 --> 00:01:55.090 literally doesn't change. 00:01:55.090 --> 00:01:58.690 Although we could do this problem the other way. 00:01:58.690 --> 00:02:00.820 Maybe I'll experiment doing it the other way. 00:02:00.820 --> 00:02:03.190 but let's assume this is g prime of x, and let's 00:02:03.190 --> 00:02:06.410 assume this f of x. 00:02:06.410 --> 00:02:08.450 So this is derivative. 00:02:08.450 --> 00:02:11.420 So integration by parts, as we take the original functions, 00:02:11.420 --> 00:02:14.430 g of x and f of x. 00:02:14.430 --> 00:02:16.590 If this is g prime of x, what's go of x. 00:02:16.590 --> 00:02:20.130 What's the anti-derivative of e to the x. 00:02:20.130 --> 00:02:21.270 It's just use e to the x. 00:02:21.270 --> 00:02:23.460 I'm going to switch colors, I don't like this blue. 00:02:23.460 --> 00:02:27.200 So this is g of x. 00:02:27.200 --> 00:02:29.450 I actually took the anti-derivative of it, but 00:02:29.450 --> 00:02:31.060 it's the same exact thing. 00:02:31.060 --> 00:02:35.320 And then times f of x. 00:02:35.320 --> 00:02:41.320 Then I want to subtract the indefinite integral 00:02:41.320 --> 00:02:46.100 of f prime of x. 00:02:46.100 --> 00:02:47.715 One, g of x. 00:02:51.200 --> 00:02:54.220 This is the same as this, which are both the anti-derivative of 00:02:54.220 --> 00:02:56.370 this, although they are all the same. 00:02:56.370 --> 00:03:01.830 So this is g of x and then I would take the derivative 00:03:01.830 --> 00:03:04.050 of f of x. f prime of x. 00:03:04.050 --> 00:03:05.770 What's the derivative of cosine of x? 00:03:05.770 --> 00:03:07.740 It's minus sine of x. 00:03:07.740 --> 00:03:13.410 So sine of x d x, it's minus sine of x. 00:03:13.410 --> 00:03:15.490 I could put the minus here, that'll make it look messy, I 00:03:15.490 --> 00:03:16.970 could put the minus here that'll make it messy or I 00:03:16.970 --> 00:03:19.240 could just put minus here and make these minuses cancel 00:03:19.240 --> 00:03:21.340 out and I get a plus here. 00:03:21.340 --> 00:03:25.370 So I get the integral of e to the x cosine of x d x is equal 00:03:25.370 --> 00:03:29.886 to e to the x cosine of x plus the integral of e to 00:03:29.886 --> 00:03:32.680 the x sine of x d x. 00:03:32.680 --> 00:03:34.410 Hopefully I haven't confused you too much. 00:03:34.410 --> 00:03:36.430 I should actually do some integration by parts problems 00:03:36.430 --> 00:03:37.280 without e to the x. 00:03:37.280 --> 00:03:40.460 It's very hard to keep track of what I've done here. 00:03:40.460 --> 00:03:41.250 This is the anti-derivative. 00:03:44.900 --> 00:03:47.000 This is the anti-derivative and this is also the 00:03:47.000 --> 00:03:48.000 anti-derivative. 00:03:48.000 --> 00:03:50.565 This is g prime of x, this is g of x. 00:03:55.670 --> 00:03:58.520 So once again we are not clear whether we've 00:03:58.520 --> 00:03:59.920 made any progress. 00:03:59.920 --> 00:04:00.070 We've 00:04:00.070 --> 00:04:03.780 gone from e to the x cosine of x to e to the x sine of x. 00:04:03.780 --> 00:04:08.150 Let's take integration by parts again, see what happens. 00:04:08.150 --> 00:04:10.550 I'm just going to write on the right side of the equal sign, 00:04:10.550 --> 00:04:13.900 because this might get a little long. 00:04:13.900 --> 00:04:18.620 I'm just going to write this first part x to the x cosine 00:04:18.620 --> 00:04:23.945 of x plus-- and now let's do integration by parts again. 00:04:32.520 --> 00:04:34.880 For this round of integration by parts this was g of x, but 00:04:34.880 --> 00:04:39.730 now, for this around, I'm going to assume it's g prime of x. 00:04:39.730 --> 00:04:42.080 Which doesn't really make a difference because whenever I 00:04:42.080 --> 00:04:44.440 take the anti-derivative of it to g of x, it stays the same. 00:04:44.440 --> 00:04:45.870 And then I'm going to assume that this is f of x. 00:04:48.750 --> 00:04:54.370 So integration by parts tells us we take f of x times g of x, 00:04:54.370 --> 00:04:57.270 so I take this function and the anti-derivative of 00:04:57.270 --> 00:04:59.130 this function. 00:04:59.130 --> 00:05:02.170 The anti-derivative of this function is once again just e 00:05:02.170 --> 00:05:06.940 to the x and then f times that function unchanged 00:05:06.940 --> 00:05:09.630 time sine of x. 00:05:09.630 --> 00:05:16.160 From that I subtract the integral of the anti-derivative 00:05:16.160 --> 00:05:21.410 of this or I take g of x which is e to the x, and then the 00:05:21.410 --> 00:05:24.390 derivative of f of x, f prime of x. 00:05:24.390 --> 00:05:25.680 What's the derivative of sine of x? 00:05:25.680 --> 00:05:28.605 It's cosine of x. 00:05:28.605 --> 00:05:31.590 Cosine of x d of x. 00:05:31.590 --> 00:05:32.650 Let's see if we're getting anywhere. 00:05:32.650 --> 00:05:35.560 It seems like I just keep adding terms, making it 00:05:35.560 --> 00:05:36.760 more and more complicated. 00:05:36.760 --> 00:05:38.980 In order to see if we're getting anywhere, let me just 00:05:38.980 --> 00:05:41.590 rewrite the whole thing and maybe get rid of these 00:05:41.590 --> 00:05:43.430 parenthesis, because it's just a plus, so we can get 00:05:43.430 --> 00:05:44.155 rid of the parenthesis. 00:05:48.250 --> 00:05:51.220 Let me use a new color. 00:05:51.220 --> 00:05:51.980 OK. 00:05:51.980 --> 00:06:00.020 So this is the original problem, e to the x cosine of x 00:06:00.020 --> 00:06:05.820 d x equals, and now let me switch back to this color, it 00:06:05.820 --> 00:06:11.920 equals e to the x cosine of x, and then I can just-- this 00:06:11.920 --> 00:06:13.970 parentheses doesn't matter because I'm just adding 00:06:13.970 --> 00:06:18.730 everything in the parentheses-- e to x cosine of x plus e to 00:06:18.730 --> 00:06:35.520 the x sine of x minus e to the x cosine x access d x. 00:06:35.520 --> 00:06:38.330 Now you might think that I arbitrarily switched colors 00:06:38.330 --> 00:06:42.360 here when I rewrote this, but if you look you might 00:06:42.360 --> 00:06:45.800 see why I actually did switch colors here. 00:06:45.800 --> 00:06:47.740 See anything interesting? 00:06:47.740 --> 00:06:48.890 Exactly. 00:06:48.890 --> 00:06:51.590 This is the same thing as this, just a minus right? 00:06:51.590 --> 00:06:55.060 So we're going to do something what I think to be fairly cool. 00:06:55.060 --> 00:06:59.110 Let's add this term to both sides of the equation. 00:06:59.110 --> 00:07:01.140 Let's take this and let's put it on to this 00:07:01.140 --> 00:07:02.420 side of the equation. 00:07:02.420 --> 00:07:03.800 If I take this and put it on this side of the 00:07:03.800 --> 00:07:05.500 equation, what happens? 00:07:05.500 --> 00:07:08.100 I then have two of these on the left side equation, so that 00:07:08.100 --> 00:07:16.250 becomes-- I mean I could write it out it's e to the x cosine 00:07:16.250 --> 00:07:19.250 of x d x plus, right? 00:07:19.250 --> 00:07:20.710 Because I'm taking this and I'm putting it on that side of the 00:07:20.710 --> 00:07:26.620 equation, e to the x cosine of x d x. 00:07:26.620 --> 00:07:30.470 That's just the same thing as 2 times the integral of e 00:07:30.470 --> 00:07:34.770 to the x cosine of x d x. 00:07:34.770 --> 00:07:37.560 And then that equals this term. 00:07:37.560 --> 00:07:45.300 Which equals e to the x cosine of x plus e to the x sine of x. 00:07:45.300 --> 00:07:47.020 I know it's really messy. 00:07:47.020 --> 00:07:49.810 All I have to do now to solve this integral is divide both 00:07:49.810 --> 00:07:52.670 sides by 2 and I'm done. 00:07:52.670 --> 00:07:55.240 So let me write it out, this is very exciting, it's 00:07:55.240 --> 00:07:56.800 the home stretch. 00:07:56.800 --> 00:07:59.750 If I divide both sides by 2, I get-- and I'm going to try to 00:07:59.750 --> 00:08:05.740 write it so you can see everything-- e to the x cosine 00:08:05.740 --> 00:08:16.730 of x d x equals and on that side I have e to the x cosine 00:08:16.730 --> 00:08:26.200 of x plus e to the x sine of x over 2. 00:08:26.200 --> 00:08:28.480 I think that's pretty neat. 00:08:28.480 --> 00:08:31.720 It's neat how integration by parts allowed us to do this. 00:08:31.720 --> 00:08:33.510 We actually never even have to evaluate this integral. 00:08:33.510 --> 00:08:36.080 We said, this integral is just the original problem again. 00:08:36.080 --> 00:08:37.700 And you can think about why that happened, right? 00:08:37.700 --> 00:08:39.540 Because these trick functions cycle. 00:08:39.540 --> 00:08:42.030 So we had to do integration by parts twice to get back 00:08:42.030 --> 00:08:43.560 to where we were before. 00:08:43.560 --> 00:08:50.010 And then we use that to solve it without actually having 00:08:50.010 --> 00:08:51.330 to evaluate the integral. 00:08:51.330 --> 00:08:53.520 And what I also think is cool is even if you just look 00:08:53.520 --> 00:08:57.410 at this solution, it's kind of neat, right? 00:08:57.410 --> 00:09:01.240 The anti-derivative of e to the x and-- actually never forget 00:09:01.240 --> 00:09:05.560 the plus c, that would've given me minus 1 point on the exam. 00:09:05.560 --> 00:09:08.320 What's kind of cool, the integral of e to the x cosine 00:09:08.320 --> 00:09:13.423 of x is this expression that's e to the x cosine of x plus e 00:09:13.423 --> 00:09:15.290 to the x sine of x divided by 2. 00:09:15.290 --> 00:09:19.520 It's the average of e to the x cosine of x and 00:09:19.520 --> 00:09:21.160 e to the x sine of x. 00:09:21.160 --> 00:09:24.790 I think that's a pretty neat property, and you might want 00:09:24.790 --> 00:09:30.210 to graph them and play with them, but it's kind of neat. 00:09:30.210 --> 00:09:33.980 Hopefully I have convinced you that is a classic of a problem, 00:09:33.980 --> 00:09:37.130 and you also find it neat, and I'll see you in the 00:09:37.130 --> 00:09:38.640 next presentation.

Integration by Parts (part 6 of Indefinite Integration)

https://www.youtube.com/watch?v=ouYZiIh8Ctc

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https://www.youtube.com/api/timedtext?v=ouYZiIh8Ctc&ei=dmeUZbCTLsevmLAPm4i-0Ag&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=50252D5B68451CE12687441F93D78CEE5A1B9426.29BFA594286B01799B7E0E67AEE46BDC33ADEF6A&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:01.900 Welcome back. 00:00:01.900 --> 00:00:05.920 Well I'm now going to do just a bunch of integration by parts 00:00:05.920 --> 00:00:09.110 problems, as many as I can do in ten minutes without 00:00:09.110 --> 00:00:09.750 confusing you. 00:00:09.750 --> 00:00:11.400 So let me just write the formula for integration by 00:00:11.400 --> 00:00:13.910 parts, and if you ever forget it-- I mean, it doesn't hurt to 00:00:13.910 --> 00:00:15.840 memorize it, but if you ever forget it-- you just really 00:00:15.840 --> 00:00:18.240 have to just derive it from the product rule of 00:00:18.240 --> 00:00:19.480 differentiation. 00:00:19.480 --> 00:00:27.080 But it just says that if we have an integral of f of x 00:00:27.080 --> 00:00:32.320 times g prime of x-- so if you see, within the integral, sign, 00:00:32.320 --> 00:00:34.590 one function and then you see the derivative of another 00:00:34.590 --> 00:00:38.640 function, and I think with practice-- integration by parts 00:00:38.640 --> 00:00:40.150 is really a bit of an art. 00:00:40.150 --> 00:00:45.480 It's not systematic-- that's g prime of x-- that equals f of 00:00:45.480 --> 00:00:51.650 access times g of x-- this is how it's the product rule in 00:00:51.650 --> 00:00:56.200 reverse-- minus the integral of the derivative of the first 00:00:56.200 --> 00:01:01.650 function, f prime of x, times the second function. 00:01:01.650 --> 00:01:03.570 And it kind of easy to memorize, because there's this 00:01:03.570 --> 00:01:04.605 symmetry to the formula. 00:01:08.110 --> 00:01:09.490 So let's see if we can apply this. 00:01:09.490 --> 00:01:12.140 And really, once you know that you should use integration by 00:01:12.140 --> 00:01:15.140 parts, I think you'll find that it's not that hard to do it. 00:01:15.140 --> 00:01:17.510 The hard part is to recognize when you should use 00:01:17.510 --> 00:01:18.890 integration by parts. 00:01:18.890 --> 00:01:22.940 From my point of view, it's kind of my last resort, or once 00:01:22.940 --> 00:01:24.650 you have a lot of practice, you might recognize, well, if 00:01:24.650 --> 00:01:27.250 there's an e to the x in it, or if there's a trig function in 00:01:27.250 --> 00:01:32.320 it, and I can't do the reverse chain rule or integration by 00:01:32.320 --> 00:01:37.370 substitution, then integration by parts is probably my best 00:01:37.370 --> 00:01:39.580 option, assuming I'm seeing this on an exam and 00:01:39.580 --> 00:01:40.070 not in real life. 00:01:40.070 --> 00:01:42.150 In real life, it might be an unsolvable integral, and 00:01:42.150 --> 00:01:44.800 you'd have to use a computer or some other technique. 00:01:44.800 --> 00:01:46.550 But if you're seeing it on an exam, you know it's a solvable 00:01:46.550 --> 00:01:48.410 integral, and if you can't solve it any other way, it's 00:01:48.410 --> 00:01:50.830 probably integration by parts. 00:01:50.830 --> 00:01:52.690 But let's just do some problems. 00:01:52.690 --> 00:01:59.600 Let's say I want to take the integral of x squared e to dx. 00:02:03.890 --> 00:02:05.920 So if I saw this out of the blue, and I didn't know that 00:02:05.920 --> 00:02:08.660 this was a presentation on the integration by parts, I would 00:02:08.660 --> 00:02:11.440 first-- clearly this isn't a polynomial, so I can't just do 00:02:11.440 --> 00:02:14.440 a simple polynomial antiderivative. 00:02:14.440 --> 00:02:18.190 Then I would try to see, is there the derivative of 00:02:18.190 --> 00:02:23.310 something, of one function, of kind of a composite function 00:02:23.310 --> 00:02:25.310 here so I can do the reverse chain rule. 00:02:25.310 --> 00:02:28.450 The derivative of x here is 1, so I can't do anything here. 00:02:28.450 --> 00:02:30.290 So I use the chain rule. 00:02:30.290 --> 00:02:33.700 And the way I think about the chain rule is I 00:02:33.700 --> 00:02:35.310 want to simplify it. 00:02:35.310 --> 00:02:38.040 So when I go into this term right here, I'm going to 00:02:38.040 --> 00:02:40.490 have to pick up my f of x. 00:02:40.490 --> 00:02:43.300 I have to pick my f of x out of probably one of these two 00:02:43.300 --> 00:02:47.500 functions, so that the f prime of x is simpler. 00:02:47.500 --> 00:02:53.700 And I need to pick my g prime of x, I would guess that either 00:02:53.700 --> 00:02:58.230 x squared is going to be my g prime of x, or e to the x is 00:02:58.230 --> 00:03:00.790 going to be my g prime of x, and I want to pick that so that 00:03:00.790 --> 00:03:02.855 when I take the antiderivative of it, it's going 00:03:02.855 --> 00:03:03.530 to be simpler. 00:03:03.530 --> 00:03:06.030 Or at least, not more complicated. 00:03:06.030 --> 00:03:07.570 I know that if I take the derivative of x squared, 00:03:07.570 --> 00:03:08.850 that simplifies it. 00:03:08.850 --> 00:03:11.820 And I also know that the-- and once again, this is one of my, 00:03:11.820 --> 00:03:15.510 to me, a very mind blowing idea-- but that the 00:03:15.510 --> 00:03:18.740 antiderivative of e to the x is e to the x. 00:03:18.740 --> 00:03:23.860 So it's probably a good idea to say that f of x is equal to-- 00:03:23.860 --> 00:03:27.280 it's probably, let me change colors-- it's probably a good 00:03:27.280 --> 00:03:30.650 idea to make f of x is equal to x squared, because later I'm 00:03:30.650 --> 00:03:32.310 going to take the derivative of it, and the derivative of it 00:03:32.310 --> 00:03:35.850 simplifies it, and it's probably a good idea to make g 00:03:35.850 --> 00:03:39.830 prime of x e to the x, because later, I'm going to take the 00:03:39.830 --> 00:03:41.760 antiderivative of it, and the antiderivative of e to 00:03:41.760 --> 00:03:42.770 the x is e to the x. 00:03:42.770 --> 00:03:45.240 It's not going to become any more complicated. 00:03:45.240 --> 00:03:52.120 So if we assume what I'm doing is right, then what did we say? 00:03:52.120 --> 00:03:56.890 Well here, we just multiply the two real functions, right? 00:03:56.890 --> 00:03:59.550 So when I say the real functions, I mean not 00:03:59.550 --> 00:04:01.030 the derivatives of e to one of them. 00:04:01.030 --> 00:04:04.460 So f of x, we're saying f of x is x squared. 00:04:04.460 --> 00:04:07.290 Let me try to stay color consistent. 00:04:09.840 --> 00:04:13.230 And we said g of x-- now don't get confused-- we're saying g 00:04:13.230 --> 00:04:16.790 prime of x-- let me write it in a corner down here-- we're 00:04:16.790 --> 00:04:22.360 saying that g prime of x is equal to e to the x. 00:04:22.360 --> 00:04:26.050 And of course, if g prime of x is equal to ex, then g of x 00:04:26.050 --> 00:04:30.520 is also equal to e to the x. 00:04:30.520 --> 00:04:32.840 So g of x-- I don't want you to think that I'm somehow 00:04:32.840 --> 00:04:34.420 putting g prime of x here. 00:04:34.420 --> 00:04:36.550 I've taken the antiderivative, it just happens to be 00:04:36.550 --> 00:04:37.705 the same function. 00:04:41.900 --> 00:04:47.500 And then from that, we subtract the integral, we take the 00:04:47.500 --> 00:04:56.890 derivative of x squared, so you get 2x, and then times the 00:04:56.890 --> 00:04:59.980 antiderivative of g prime of x. 00:04:59.980 --> 00:05:02.590 Well, the g prime of x is e to the x, you take the 00:05:02.590 --> 00:05:06.020 antiderivative, it's still e to the x. 00:05:06.020 --> 00:05:08.955 Actually I should probably stay at least keep the colors kind 00:05:08.955 --> 00:05:11.870 of consistent, so you know what I'm hopefully doing. 00:05:14.520 --> 00:05:16.540 The example using e to the x might be a little complicated, 00:05:16.540 --> 00:05:18.030 because it's hard to tell whether I've taken the 00:05:18.030 --> 00:05:19.130 derivative or not. 00:05:19.130 --> 00:05:21.330 And you can kind of keep reverting back to the top 00:05:21.330 --> 00:05:22.405 formula if you get confused. 00:05:29.130 --> 00:05:31.650 So it looks like I've simplified it a little bit. 00:05:31.650 --> 00:05:37.460 This integral looks easier to solve than this integral. 00:05:37.460 --> 00:05:39.760 But once again, when I look at this, I'm like, well, 00:05:39.760 --> 00:05:40.670 how do I solve this? 00:05:40.670 --> 00:05:45.230 I can't use the integration by substitution, because there's 00:05:45.230 --> 00:05:48.300 not an embedded function and then I have the derivative of 00:05:48.300 --> 00:05:50.190 it sitting right next to it, so. 00:05:50.190 --> 00:05:52.900 Maybe I need to do integration by parts again. 00:05:52.900 --> 00:05:54.030 So let's do that. 00:05:54.030 --> 00:05:59.260 So let's say-- let me do it separately-- let's assume-- I 00:05:59.260 --> 00:06:01.390 think you getting a little bit of the hang of it-- along the 00:06:01.390 --> 00:06:04.990 same vein, that this is f of x, that this is f of x, and that 00:06:04.990 --> 00:06:06.630 this is g prime of x now. 00:06:06.630 --> 00:06:09.320 We're kind of doing integration by parts within 00:06:09.320 --> 00:06:10.730 integration by parts. 00:06:10.730 --> 00:06:14.070 So if that is the case, that this integral is going to 00:06:14.070 --> 00:06:17.080 equal-- because we have the minus sign out front, this 00:06:17.080 --> 00:06:21.500 minus sign out front-- this integral is going to equal f 00:06:21.500 --> 00:06:24.540 of x times g of x where f of x is just 2x. 00:06:28.060 --> 00:06:30.530 Our g of x, this is g prime of x now. 00:06:30.530 --> 00:06:32.710 Remember, we're kind of doing a new problem within the 00:06:32.710 --> 00:06:34.560 original big problem. 00:06:34.560 --> 00:06:38.540 So this is g prime of x, but g of x is still just e of x. 00:06:38.540 --> 00:06:41.550 I took the antiderivative of it. 00:06:41.550 --> 00:06:47.810 And that's minus the integral of the derivative of 00:06:47.810 --> 00:06:50.050 the first function. 00:06:50.050 --> 00:06:51.830 That prime of x. 00:06:51.830 --> 00:06:53.530 So that's just 2. 00:06:53.530 --> 00:06:57.680 And then the antiderivative of the second function. 00:06:57.680 --> 00:06:58.290 Well, that's easy. 00:06:58.290 --> 00:07:00.010 The antiderivative of e to the x is just e to the x. 00:07:03.210 --> 00:07:04.310 Interesting. 00:07:04.310 --> 00:07:05.890 Now I think you see where we're going. 00:07:05.890 --> 00:07:08.850 This is actually-- let me write out the whole thing. 00:07:08.850 --> 00:07:11.870 Because this is-- x squared, e to the x. 00:07:11.870 --> 00:07:14.680 Just so we don't lose track of our original problem. 00:07:14.680 --> 00:07:15.310 Interesting. 00:07:15.310 --> 00:07:18.510 Now I think we have an integral that is pretty 00:07:18.510 --> 00:07:20.400 straightforward to solve. 00:07:20.400 --> 00:07:22.430 Don't want to forget my dx's. 00:07:22.430 --> 00:07:25.840 What's the integral of-- we could take this 2 out of 00:07:25.840 --> 00:07:27.880 this, and I think it becomes pretty obvious-- what's the 00:07:27.880 --> 00:07:29.120 integral of e to the x? 00:07:29.120 --> 00:07:30.880 This is a scratch-out, this says dex. 00:07:30.880 --> 00:07:32.830 Just so-- and it's a little messy. 00:07:32.830 --> 00:07:34.980 I don't like this color. 00:07:34.980 --> 00:07:36.230 Magenta. 00:07:36.230 --> 00:07:38.640 Well the integral of e to the x, or the antiderivative of e 00:07:38.640 --> 00:07:40.970 to the x is e to the x, right? 00:07:40.970 --> 00:07:41.780 So let's write that. 00:07:41.780 --> 00:07:43.940 So I'm going to rewrite everything we've done it. 00:07:43.940 --> 00:07:58.260 So there's x squared e to the x minus 2xe to the x, and then 00:07:58.260 --> 00:08:01.150 this minus, you distribute it so it becomes a plus, so then 00:08:01.150 --> 00:08:05.940 it's plus 2-- I just took the minus, I multiplied it times 00:08:05.940 --> 00:08:08.780 this minus, so I got a plus 2-- and then the antiderivative of 00:08:08.780 --> 00:08:12.210 e to the x is just e to the x. 00:08:12.210 --> 00:08:17.140 And then of course, we should never forget the plus c. 00:08:17.140 --> 00:08:18.560 Pretty fancy, no? 00:08:18.560 --> 00:08:24.130 We've figured out the antiderivative, the indefinite 00:08:24.130 --> 00:08:27.550 integral of x squared e to the x is this big fancy thing. 00:08:27.550 --> 00:08:29.520 I bet you, before listening to this video, you would never 00:08:29.520 --> 00:08:34.260 imagine that you could tackle integration like this. 00:08:34.260 --> 00:08:36.970 You could actually try x to the n times e to the x. 00:08:36.970 --> 00:08:38.580 You can try x to the tenth times e to the x. 00:08:38.580 --> 00:08:41.480 It actually turns out you'll just have to do this many many 00:08:41.480 --> 00:08:44.550 many many times, but every time you do integration by parts, 00:08:44.550 --> 00:08:48.670 that the exponent on the x-term just becomes smaller and 00:08:48.670 --> 00:08:51.490 smaller and smaller until you get to something that's really 00:08:51.490 --> 00:08:53.640 easy to integrate, and then you can do it, and you'll have kind 00:08:53.640 --> 00:08:55.430 of this big long expression. 00:08:55.430 --> 00:08:58.550 It might be tedious, it might be hairy, but at least you have 00:08:58.550 --> 00:09:00.840 a tool kit-- or you have something in your tool kit-- 00:09:00.840 --> 00:09:05.180 that you can tackle integral problems like this. 00:09:05.180 --> 00:09:07.850 I'll probably do one more video on integration by parts, just 00:09:07.850 --> 00:09:11.620 because I think this is one of the harder concepts to really 00:09:11.620 --> 00:09:14.930 grasp and feel comfortable with, and then I'll try to 00:09:14.930 --> 00:09:16.590 do a bunch of examples. 00:09:16.590 --> 00:09:19.560 Maybe not soon, but in the next couple of weeks, on 00:09:19.560 --> 00:09:20.670 just a lot of integration. 00:09:20.670 --> 00:09:23.210 And I'm going to mix it up so that hopefully you can get a 00:09:23.210 --> 00:09:27.680 sense of how I try to figure out which of my integration 00:09:27.680 --> 00:09:29.650 tools I should use for a certain problem when 00:09:29.650 --> 00:09:30.870 I see the problem. 00:09:30.870 --> 00:09:32.720 See you in the next presentation.

Definite Integrals (area under a curve) (part III)

https://www.youtube.com/watch?v=7wUHJ7JQ-gs

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en

WEBVTT Kind: captions Language: en 00:00:00.690 --> 00:00:01.410 Welcome back. 00:00:01.410 --> 00:00:05.400 I'm just continuing on with hopefully giving you, one, how 00:00:05.400 --> 00:00:07.660 to actually solve indefinite integrals and also giving you a 00:00:07.660 --> 00:00:10.950 sense of why you solve it the way you do. 00:00:10.950 --> 00:00:16.150 And I think that's often missing in some textbooks. 00:00:16.150 --> 00:00:17.780 But anyway, let's say that this is the distance and let me give 00:00:17.780 --> 00:00:21.250 you a formula, actually, for the distance, just for fun. 00:00:21.250 --> 00:00:23.200 Oh, my phone is ringing. 00:00:23.200 --> 00:00:26.590 Let me lower the volume, because you're more important. 00:00:26.590 --> 00:00:30.610 So, let's say that the distance, s -- this time I'll 00:00:30.610 --> 00:00:34.845 write it as a function -- let's say the distance is -- I said 00:00:34.845 --> 00:00:40.220 it started at five, so let's say it's 2t -- let's say this 00:00:40.220 --> 00:00:42.440 is actually a cubic function. 00:00:42.440 --> 00:00:47.290 You're not only accelerating, your rate of acceleration 00:00:47.290 --> 00:00:47.970 is increasing. 00:00:47.970 --> 00:00:50.300 I think, actually, the rate of acceleration, if I'm not 00:00:50.300 --> 00:00:52.340 mistaken, is actually called jerk, but I might have 00:00:52.340 --> 00:00:54.330 to Wikipedia that. 00:00:54.330 --> 00:00:57.690 Let's say its 2t to the third plus 5. 00:01:00.240 --> 00:01:08.980 And let's say I wanted to know how far I travel between t 00:01:08.980 --> 00:01:17.540 equals 2 seconds and t equals 5 seconds. 00:01:17.540 --> 00:01:18.100 Right? 00:01:18.100 --> 00:01:20.130 I'm not looking for the total distance I've traveled. 00:01:20.130 --> 00:01:23.570 I just want to know how far do I travel between time equals 2 00:01:23.570 --> 00:01:25.790 seconds and time equal 5 seconds. 00:01:25.790 --> 00:01:26.040 Right? 00:01:26.040 --> 00:01:30.610 So this might be 2 and this is 5. 00:01:30.610 --> 00:01:33.082 So an easy way to do that is I could just evaluate this 00:01:33.082 --> 00:01:36.920 function at t equals 5 -- let me use a different color. 00:01:36.920 --> 00:01:39.750 I think it's getting messy -- I could just evaluate this 00:01:39.750 --> 00:01:40.910 function at t equals 5. 00:01:44.890 --> 00:01:53.200 If t equals 5, 5 to the third power is 125, 250, it's 255, 00:01:53.200 --> 00:01:58.640 so the object has gone 255 feet at t equals 5, right? 00:01:58.640 --> 00:02:04.070 And then, at time equals 2, the object has gone how far? 00:02:04.070 --> 00:02:05.110 2 to the third is 8. 00:02:05.110 --> 00:02:05.550 16. 00:02:05.550 --> 00:02:07.180 It's gone 21 feet. 00:02:07.180 --> 00:02:09.590 Right? 00:02:09.590 --> 00:02:12.450 To figure out how far I travelled between time equals 2 00:02:12.450 --> 00:02:18.720 and time equals 5, I just say s of 5 minus s of 2, right? 00:02:18.720 --> 00:02:22.825 How far did I go after 5 seconds minus how far I 00:02:22.825 --> 00:02:24.000 already was after 2 seconds. 00:02:24.000 --> 00:02:33.520 And this is just 255 minus 21 and that's, what, 234. 00:02:33.520 --> 00:02:36.480 234 feet is how far I travelled between 2 00:02:36.480 --> 00:02:38.070 seconds and 5 seconds. 00:02:38.070 --> 00:02:39.040 Interesting. 00:02:39.040 --> 00:02:40.950 And I think you're starting to get a little intuition about 00:02:40.950 --> 00:02:43.420 why we evaluated that previous indefinite integral in the 00:02:43.420 --> 00:02:45.120 previous video the way we did. 00:02:45.120 --> 00:02:47.820 So let's actually draw the derivative of this function. 00:02:47.820 --> 00:02:49.220 So what's the derivative? 00:02:49.220 --> 00:02:53.950 So let me just call that v of t, I guess. 00:02:53.950 --> 00:02:56.690 v of t is just the derivative, right, because it's the rate 00:02:56.690 --> 00:02:59.420 of change of distance with respect to time. 00:02:59.420 --> 00:03:05.530 3 times 2 is 6t squared and the constant disappears, right? 00:03:05.530 --> 00:03:07.010 So it's just 6t squared. 00:03:07.010 --> 00:03:08.140 And that makes sense, right? 00:03:08.140 --> 00:03:10.010 Because your velocity doesn't care about where you 00:03:10.010 --> 00:03:11.500 started off from, right? 00:03:11.500 --> 00:03:13.210 You're going to be the same velocity if you started 00:03:13.210 --> 00:03:15.010 from 10 feet or if you started from 2 feet. 00:03:15.010 --> 00:03:17.520 Your velocity doesn't really matter about where your 00:03:17.520 --> 00:03:18.710 starting position is. 00:03:18.710 --> 00:03:20.420 So let's graph this. 00:03:20.420 --> 00:03:20.570 See? 00:03:20.570 --> 00:03:23.250 You're actually learning a little physics while 00:03:23.250 --> 00:03:24.310 you're learning calculus. 00:03:24.310 --> 00:03:25.800 Actually, I think it's silly that they're taught as 00:03:25.800 --> 00:03:26.630 two separate classes. 00:03:26.630 --> 00:03:29.060 I think physics and calculus should just be one 00:03:29.060 --> 00:03:30.960 fun 2-hour class. 00:03:30.960 --> 00:03:33.330 But I'll talk about that at another time. 00:03:36.040 --> 00:03:38.100 So, going back to this. 00:03:38.100 --> 00:03:38.730 Let me graph that. 00:03:38.730 --> 00:03:39.090 6t squared. 00:03:39.090 --> 00:03:41.120 Well, that's just going to look like a parabola. 00:03:41.120 --> 00:03:42.110 Right? 00:03:42.110 --> 00:03:45.440 It's going to look something like this. 00:03:45.440 --> 00:03:46.970 This is t. 00:03:46.970 --> 00:03:49.090 This is the velocity. 00:03:49.090 --> 00:03:53.990 And now, if we just had this velocity graph, if we didn't 00:03:53.990 --> 00:03:57.320 know all of this over here and I asked you the 00:03:57.320 --> 00:03:58.070 same question, though. 00:03:58.070 --> 00:04:02.370 I said, how far does this thing travel between 2 00:04:02.370 --> 00:04:08.420 seconds and 5 seconds? 00:04:08.420 --> 00:04:08.815 Right? 00:04:12.270 --> 00:04:14.990 Well, I could do it the way that we learned in the previous 00:04:14.990 --> 00:04:19.230 video where I draw a bunch of small rectangles, each of a 00:04:19.230 --> 00:04:23.530 really small width, and I multiply it times its 00:04:23.530 --> 00:04:27.565 instantaneous velocity at that exact moment, right? 00:04:30.630 --> 00:04:35.350 And then I sum up all of those rectangles -- look how pretty 00:04:35.350 --> 00:04:40.350 that is -- I sum up all of the rectangles. 00:04:40.350 --> 00:04:45.930 And I'll get a pretty good approximation for how far 00:04:45.930 --> 00:04:48.850 I've travelled between 2 and 5 seconds. 00:04:48.850 --> 00:04:54.070 Because remember, the area of each of these rectangles 00:04:54.070 --> 00:04:57.960 represents how far I traveled in that little 00:04:57.960 --> 00:04:59.960 amount of time, dt. 00:04:59.960 --> 00:05:05.310 Because time times a constant velocity is equal to distance. 00:05:05.310 --> 00:05:17.480 But as you can see this also tells me the area between 00:05:17.480 --> 00:05:20.130 t equals 2 and t equals 5. 00:05:20.130 --> 00:05:23.440 So, not only did I figure out the distance between how far I 00:05:23.440 --> 00:05:27.280 traveled from 2 seconds to 5 seconds, I also figured out the 00:05:27.280 --> 00:05:31.060 area under this curve from 2 seconds to 5 seconds. 00:05:31.060 --> 00:05:38.580 So, interestingly enough, if I just changed this from a to b, 00:05:38.580 --> 00:05:41.920 and, in general, if you want to figure out the area under a 00:05:41.920 --> 00:05:51.680 curve from a to b, it's just the indefinite integral from a 00:05:51.680 --> 00:05:53.790 to b -- actually, from b to a. 00:05:53.790 --> 00:05:55.650 The b should be the larger one. 00:05:55.650 --> 00:05:56.920 b to a. 00:05:56.920 --> 00:05:59.090 I guess a to b, depending on how you say it. 00:05:59.090 --> 00:06:00.360 Let me write that in a different color because I 00:06:00.360 --> 00:06:02.100 think I'm making it messier. 00:06:02.100 --> 00:06:04.950 From a to b of this velocity function. 00:06:04.950 --> 00:06:10.460 So, in this case, 6t squared d t, right? 00:06:10.460 --> 00:06:13.200 If these weren't 2 and 5, if this was just a and b. 00:06:13.200 --> 00:06:16.370 And the way you evaluate this is you figure out the 00:06:16.370 --> 00:06:21.760 antiderivative of this inside function, and then you evaluate 00:06:21.760 --> 00:06:25.010 the antiderivative at b, and then from that, you 00:06:25.010 --> 00:06:28.040 subtract it out at a. 00:06:28.040 --> 00:06:32.590 So in this case, the antiderivative of this is 2t to 00:06:32.590 --> 00:06:40.900 the third and we evaluated at b, and we evaluated at a. 00:06:40.900 --> 00:06:42.770 Actually, let me stick to the old numbers. 00:06:42.770 --> 00:06:46.510 We evaluated it at 5 and you evaluated it at 2. 00:06:46.510 --> 00:06:51.240 So if you evaluated it at 5, that's 255. 00:06:51.240 --> 00:06:52.750 If you evaluate it at 2, that's 21. 00:06:52.750 --> 00:06:55.020 So you're doing the exact same thing we did here when we 00:06:55.020 --> 00:06:57.480 actually had this graph. 00:06:57.480 --> 00:07:01.560 So I did all of this, not to confuse you further, but really 00:07:01.560 --> 00:07:04.580 just to give you an intuition of why one, why the 00:07:04.580 --> 00:07:08.290 antiderivative is the area under the curve, and then two, 00:07:08.290 --> 00:07:21.390 why-- let's say that a, b-- and then why we evaluate 00:07:21.390 --> 00:07:23.560 it this way. 00:07:23.560 --> 00:07:25.950 You might see this in your books. 00:07:29.040 --> 00:07:32.560 This is just saying, if I want to figure out the area under a 00:07:32.560 --> 00:07:37.820 curve from a to b of f of x, that we figure out 00:07:37.820 --> 00:07:38.600 the antiderivative. 00:07:38.600 --> 00:07:42.660 This capital F is just the antiderivative. 00:07:42.660 --> 00:07:45.740 We just figure out the antiderivative and we evaluated 00:07:45.740 --> 00:07:49.940 at b and we evaluated at a, and then we subtract 00:07:49.940 --> 00:07:50.540 the difference. 00:07:50.540 --> 00:07:51.680 And that's what we did here, right? 00:07:51.680 --> 00:07:53.150 This is what we did here intuitively when we 00:07:53.150 --> 00:07:56.110 worked with distance. 00:07:56.110 --> 00:07:58.220 The derivative and the antiderivative don't only apply 00:07:58.220 --> 00:08:00.020 to distance and velocity. 00:08:00.020 --> 00:08:03.870 But I did this to give you an intuition of why this works 00:08:03.870 --> 00:08:07.690 and why the antiderivative is the area under a curve. 00:08:07.690 --> 00:08:10.710 So let me clear this up and just rewrite that last thing 00:08:10.710 --> 00:08:13.490 I wrote, but maybe a little bit cleaner. 00:08:17.680 --> 00:08:18.650 OK. 00:08:18.650 --> 00:08:24.380 So let's say that F of x with a big, fat capital F is equal to 00:08:24.380 --> 00:08:30.440 -- actually, let me do it a better way -- let me say that 00:08:30.440 --> 00:08:36.930 the derivative of big fat F of x is equal to f of x. 00:08:36.930 --> 00:08:39.200 Right? 00:08:39.200 --> 00:08:40.920 I think, actually, this is the fundamental theorem of 00:08:40.920 --> 00:08:43.450 calculus, but I don't want to throw out things without 00:08:43.450 --> 00:08:43.920 knowing for sure. 00:08:43.920 --> 00:08:46.170 I have to go make sure. 00:08:46.170 --> 00:08:47.770 See, I haven't done math in a long time. 00:08:47.770 --> 00:08:50.230 I'm giving you all this based on intuition, not necessarily 00:08:50.230 --> 00:08:52.400 what I'm reading. 00:08:52.400 --> 00:08:56.020 So the derivative of big F is small f, and all we're saying 00:08:56.020 --> 00:09:06.470 is that if we take the integral of small f of x from a to b, 00:09:06.470 --> 00:09:13.480 dx, that this is big F, it's antiderivative, at b minus 00:09:13.480 --> 00:09:16.320 the antiderivative at a. 00:09:16.320 --> 00:09:18.190 In the next presentation, I'll use this. 00:09:18.190 --> 00:09:19.846 This is actually pretty easy to use once you know how 00:09:19.846 --> 00:09:20.580 to use antiderivatives. 00:09:20.580 --> 00:09:22.990 And we did these three videos really just to give you -- or 00:09:22.990 --> 00:09:25.165 actually, is this the third or the second -- just to give you 00:09:25.165 --> 00:09:28.720 an intuition of why this is, because I think that's really 00:09:28.720 --> 00:09:30.830 important if you're ever going to really use calculus in your 00:09:30.830 --> 00:09:33.160 life or write a computer program or whatever. 00:09:33.160 --> 00:09:36.280 And in the next couple videos I'll actually apply this to a 00:09:36.280 --> 00:09:38.640 bunch of problems and you'll hopefully see that it's a 00:09:38.640 --> 00:09:42.860 pretty straightforward thing to actually compute. 00:09:42.860 --> 00:09:44.880 I'll see you in the next presentation.

Definite integrals (part II)

https://www.youtube.com/watch?v=6PaFm_Je5A0

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en

WEBVTT Kind: captions Language: en 00:00:01.000 --> 00:00:02.110 Welcome back. 00:00:02.110 --> 00:00:06.800 So where I left off, we said that we had this, I guess you 00:00:06.800 --> 00:00:09.450 could call it, equation or this function, although I didn't 00:00:09.450 --> 00:00:11.110 write it with the function notation, where I said, the 00:00:11.110 --> 00:00:13.540 distance is equal to 16 t squared, and I graphed it, it's 00:00:13.540 --> 00:00:16.100 like a parabola, right, for positive time. 00:00:16.100 --> 00:00:17.960 And then we said, well, the velocity, if we know the 00:00:17.960 --> 00:00:21.190 distance, the velocity is just the change of the distance 00:00:21.190 --> 00:00:22.360 with respect to time. 00:00:22.360 --> 00:00:24.080 It's just, the velocity is always changing, you can't 00:00:24.080 --> 00:00:25.560 just take the slope, you actually have to take 00:00:25.560 --> 00:00:26.920 the derivative, right? 00:00:26.920 --> 00:00:29.760 So we took the derivative with respect to time of this 00:00:29.760 --> 00:00:32.915 function, or this equation, and we got 32t, and this 00:00:32.915 --> 00:00:33.210 is the velocity. 00:00:33.210 --> 00:00:35.060 And then we graphed it. 00:00:35.060 --> 00:00:36.130 And then I asked a question. 00:00:36.130 --> 00:00:39.160 I was like, well, we want to figure out, if we were given 00:00:39.160 --> 00:00:47.080 this, if we were given just this, and I asked you, what is 00:00:47.080 --> 00:00:49.920 the distance that this object travels after time, you 00:00:49.920 --> 00:00:52.130 know, after 10 seconds? 00:00:52.130 --> 00:00:56.240 Let's, you know, let's say this t0 is equal to 10 seconds. 00:00:56.240 --> 00:00:59.150 I want to know how far is this thing gone after 10 seconds. 00:00:59.150 --> 00:01:01.450 And let's say you didn't know that you could just take the 00:01:01.450 --> 00:01:03.605 antiderivative, let's say we didn't know this at all, and 00:01:03.605 --> 00:01:04.840 let's say you didn't know that you could just take the 00:01:04.840 --> 00:01:07.860 antiderivative, because we just showed that, you know, the 00:01:07.860 --> 00:01:10.990 derivative of distance is velocity, so the antiderivative 00:01:10.990 --> 00:01:12.670 of velocity is distance. 00:01:12.670 --> 00:01:14.700 So let's say you couldn't just take the antiderivative. 00:01:14.700 --> 00:01:18.140 What's a way that you could start to try to approximate 00:01:18.140 --> 00:01:20.780 how far you've traveled after, say, 10 seconds? 00:01:20.780 --> 00:01:21.330 Well [? as I said, ?] 00:01:21.330 --> 00:01:24.520 you graph this, and you say, let's assume over some 00:01:24.520 --> 00:01:31.580 change in time, velocity is roughly constant, right? 00:01:31.580 --> 00:01:33.390 Let's say velocity is right here. 00:01:33.390 --> 00:01:37.370 So you could approximate how far you travel over that small 00:01:37.370 --> 00:01:40.020 change in time by multiplying that change in time, let's say 00:01:40.020 --> 00:01:43.900 that's like, you know, a millionth of a second, times 00:01:43.900 --> 00:01:46.940 the velocity at roughly that time, or maybe even the average 00:01:46.940 --> 00:01:51.940 velocity over that time, and you'd get the distance you've 00:01:51.940 --> 00:01:58.390 traveled over that very small fraction of time, right? 00:01:58.390 --> 00:02:02.770 But if you look at it visually, that also happens to be the 00:02:02.770 --> 00:02:06.470 area of this rectangle, right? 00:02:06.470 --> 00:02:08.320 And what we said, is if you want to know how far you travel 00:02:08.320 --> 00:02:12.160 after 10 seconds, you just draw a bunch of these rectangles, 00:02:12.160 --> 00:02:14.520 and you sum up the area, right? 00:02:14.520 --> 00:02:16.550 And you could imagine, and you don't have to imagine, it's 00:02:16.550 --> 00:02:21.380 actually true, the smaller the bases of these rectangles, and 00:02:21.380 --> 00:02:26.610 the more of these rectangles you have, the more accurate 00:02:26.610 --> 00:02:30.840 your approximation will be, and you'll approach 2 things. 00:02:30.840 --> 00:02:34.640 You'll approach the area under this curve, right, almost the 00:02:34.640 --> 00:02:37.740 exact area under this curve, and you'd also get almost the 00:02:37.740 --> 00:02:47.030 exact value of the distance after, say, 10 seconds 00:02:47.030 --> 00:02:48.130 in this case, right? 00:02:48.130 --> 00:02:50.100 But 10 didn't have to be an exact number. 00:02:50.100 --> 00:02:51.290 It could have been a variable. 00:02:51.290 --> 00:02:53.310 So this is something pretty interesting. 00:02:53.310 --> 00:02:57.100 All of a sudden, we see that the antiderivative 00:02:57.100 --> 00:03:00.450 is pretty darn similar to the area under the curve. 00:03:00.450 --> 00:03:01.800 And it actually turns out that they're the same thing. 00:03:01.800 --> 00:03:04.600 And this is where I'm going to teach you the 00:03:04.600 --> 00:03:05.930 indefinite integral. 00:03:05.930 --> 00:03:07.860 So the indefinite integral, I don't know how comfortable you 00:03:07.860 --> 00:03:10.010 are with summation, I remember the first time l learned 00:03:10.010 --> 00:03:13.170 calculus, I wasn't that comfortable with summation, but 00:03:13.170 --> 00:03:17.030 it's really, all the indefinite integral, is is you can kind of 00:03:17.030 --> 00:03:19.590 view it as a sum, right? 00:03:19.590 --> 00:03:23.130 So now, you'll maybe understand a little bit more why this 00:03:23.130 --> 00:03:24.860 symbol looks kind of like a sigma. 00:03:24.860 --> 00:03:26.270 That's actually how I view it. 00:03:26.270 --> 00:03:28.320 And please look it up so you can see properly 00:03:28.320 --> 00:03:30.250 drawn integrals. 00:03:30.250 --> 00:03:33.830 But in this case, the indefinite integral is just 00:03:33.830 --> 00:03:38.700 saying, well, I'm going to take the sum from t equals 0, right, 00:03:38.700 --> 00:03:43.910 so from t equals 0, to let's say in this example, t equals 00:03:43.910 --> 00:03:45.560 10, right, because I said 10. 00:03:45.560 --> 00:03:47.660 From t equals 0 to t equals 10. 00:03:47.660 --> 00:03:53.150 and I'm going to take the sum of each of the heights, the 00:03:53.150 --> 00:03:58.810 height at any given point, which is the velocity. 00:04:02.280 --> 00:04:03.720 And then, what's the formula for the velocity? 00:04:03.720 --> 00:04:13.140 It's 32t and then I'm at times the base at each 00:04:13.140 --> 00:04:16.620 of these rectangles, dt. 00:04:16.620 --> 00:04:18.440 And so this is the definite integral. 00:04:18.440 --> 00:04:21.620 The definite integral is literally, and they never do 00:04:21.620 --> 00:04:24.050 this in math texts, and that's what always kind of confused 00:04:24.050 --> 00:04:26.560 me, is that you can kind of view it like a sum, like this. 00:04:29.140 --> 00:04:33.090 It's kind of the sum of each of these rectangles, but it's the 00:04:33.090 --> 00:04:38.680 limit, as-- if these were discrete rectangles, you could 00:04:38.680 --> 00:04:41.110 just do a sum, and you could make the rectangle bases 00:04:41.110 --> 00:04:42.870 smaller and smaller, and have more and more rectangles, 00:04:42.870 --> 00:04:44.210 and just do a regular sum. 00:04:44.210 --> 00:04:45.870 And actually, that's how, if you ever write a computer 00:04:45.870 --> 00:04:49.770 program to approximate an integral, or approximate the 00:04:49.770 --> 00:04:51.640 area under a curve, that's the way a computer program 00:04:51.640 --> 00:04:53.020 would actually do it. 00:04:53.020 --> 00:04:57.480 But the actual indefinite integral says, well, this is a 00:04:57.480 --> 00:05:01.310 sum, but it's the limit as the bases of these rectangles get 00:05:01.310 --> 00:05:05.230 smaller and smaller and smaller and smaller, and we have more 00:05:05.230 --> 00:05:06.940 and more and more of these rectangles. 00:05:06.940 --> 00:05:11.310 So as these dt's approach 0, the number of rectangles 00:05:11.310 --> 00:05:12.880 actually approach infinity. 00:05:12.880 --> 00:05:14.950 So I'm actually going to, I'll do that more rigorously later, 00:05:14.950 --> 00:05:16.780 but I think it's very important to get this intuitive feel 00:05:16.780 --> 00:05:17.680 of just what an integral is. 00:05:17.680 --> 00:05:20.930 It isn't just this voodoo that happens to be there. 00:05:20.930 --> 00:05:24.160 But anyway, so going back to the problem. 00:05:24.160 --> 00:05:29.360 So the integral from-- this is now a definite integral, 00:05:29.360 --> 00:05:32.670 extending from t equals 0 to t equals 10. 00:05:32.670 --> 00:05:34.020 This tells us 2 things. 00:05:34.020 --> 00:05:38.300 This tells us the area of the curve from t equals zero to t 00:05:38.300 --> 00:05:41.360 equals 10, right, it tells us this whole area, and it also 00:05:41.360 --> 00:05:46.580 tells us how far the object has gone after 10 seconds. 00:05:46.580 --> 00:05:47.320 Right? 00:05:47.320 --> 00:05:48.450 So it's very interesting. 00:05:48.450 --> 00:05:50.770 The indefinite integral tells us 2 things. 00:05:50.770 --> 00:05:55.160 It tells us area, and it also tells us the antiderivative. 00:05:55.160 --> 00:05:55.360 Right? 00:05:55.360 --> 00:05:58.130 We're already familiar with it as an antiderivative. 00:05:58.130 --> 00:06:00.220 So let me give you another example. 00:06:00.220 --> 00:06:01.760 Actually, maybe I'll stick with this example, but 00:06:01.760 --> 00:06:03.650 I'll clear it a bit. 00:06:03.650 --> 00:06:05.310 Actually, maybe I should erase. 00:06:05.310 --> 00:06:08.810 Erasing might be a good option with this one, 00:06:08.810 --> 00:06:12.190 since it's fairly messy. 00:06:12.190 --> 00:06:13.830 I think you know all this stuff now. 00:06:13.830 --> 00:06:14.760 I just need space. 00:06:17.830 --> 00:06:20.372 Maybe, OK, so we have that indefinite integral. 00:06:20.372 --> 00:06:22.040 And we could actually figure it out, too. 00:06:22.040 --> 00:06:25.470 I mean, well, after t seconds, [UNINTELLIGIBLE]. 00:06:25.470 --> 00:06:27.920 So and the way you evaluate an indefinite integral, and let me 00:06:27.920 --> 00:06:31.780 show you that first, is that you figure out the integral. 00:06:31.780 --> 00:06:33.740 So let me just say, let me continue with the 00:06:33.740 --> 00:06:35.060 problem, actually. 00:06:35.060 --> 00:06:38.790 As you can tell, I don't plan much for these presentations. 00:06:38.790 --> 00:06:41.170 So the way you figure out the indefinite integral, is you 00:06:41.170 --> 00:06:46.300 say, and sometimes they won't write t equals 0 to t equals t. 00:06:46.300 --> 00:06:54.610 They'll just say from 0 to 10 of 32t dt. 00:06:54.610 --> 00:06:55.370 Right? 00:06:55.370 --> 00:06:57.770 And the way you evaluate this, is you figure out the 00:06:57.770 --> 00:06:59.780 antiderivative, and you really don't have to do the plus c 00:06:59.780 --> 00:07:03.330 here, so the antiderivative, we know, is 16t squared, right? 00:07:03.330 --> 00:07:05.440 It's one half t squared times 32. 00:07:05.440 --> 00:07:06.690 So that's 16t squared. 00:07:09.300 --> 00:07:14.010 And we evaluate this at ten, and we evaluate it at 0, and 00:07:14.010 --> 00:07:15.680 then we subtract the difference. 00:07:15.680 --> 00:07:20.250 So we evaluate this at 10, so 16 times 100, right? 00:07:20.250 --> 00:07:23.060 That's evaluated at 10, and then we subtract 00:07:23.060 --> 00:07:24.155 it, evaluate at 0. 00:07:24.155 --> 00:07:26.320 So 16 times 0 is 0. 00:07:26.320 --> 00:07:30.960 So after 10 seconds, we would have gone 1600 feet. 00:07:30.960 --> 00:07:41.230 And also, the area under this curve is 1600. 00:07:41.230 --> 00:07:44.200 So let's use this to do a couple more examples. 00:07:44.200 --> 00:07:47.990 And actually, I want to show you why we do this subtraction. 00:07:47.990 --> 00:07:51.560 Actually, I'm going to do that right now. 00:07:51.560 --> 00:07:53.110 Let me clear it. 00:08:00.600 --> 00:08:01.850 Oh, that's ugly. 00:08:07.520 --> 00:08:09.100 I'll now do it more general, actually. 00:08:13.700 --> 00:08:20.600 Let me draw this twice, once for the distance, and 00:08:20.600 --> 00:08:21.480 once for its derivative. 00:08:28.860 --> 00:08:34.580 So let's say that the distance, yeah, well, let's just say it 00:08:34.580 --> 00:08:36.670 looks something like this. 00:08:36.670 --> 00:08:40.580 Let's say you start at some distance, and then it 00:08:40.580 --> 00:08:42.350 goes off like that. 00:08:42.350 --> 00:08:43.410 Right? 00:08:43.410 --> 00:08:47.190 So let's say we call this distance b. 00:08:47.190 --> 00:08:51.160 Well, let's just call this, you know, I don't know, 5. 00:08:51.160 --> 00:08:51.450 Right? 00:08:51.450 --> 00:08:56.600 We start at 5 feet, and then we moved forward from there. 00:08:56.600 --> 00:09:00.600 And this axis is of course time, this axis, maybe I 00:09:00.600 --> 00:09:02.590 shouldn't do 5, because it looks so much like s. 00:09:02.590 --> 00:09:04.280 That's 5, 5 feet. 00:09:04.280 --> 00:09:09.420 And this is the s, or distance, axis. 00:09:09.420 --> 00:09:11.270 And actually, I just looked at the clock. 00:09:11.270 --> 00:09:12.310 I'm running out of time. 00:09:12.310 --> 00:09:15.120 So let me continue this in the next presentation.

Introduction to definite integrals

https://www.youtube.com/watch?v=0RdI3-8G4Fs

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https://www.youtube.com/api/timedtext?v=0RdI3-8G4Fs&ei=dmeUZfLdLai5vdIPueGoCA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249830&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B12E2FA68075769E64B736AEAFDD264724481D1F.B1943F379C6286015B659141CCA4329E0FC9E7D8&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.000 --> 00:00:02.040 Welcome back. 00:00:02.040 --> 00:00:03.980 In this presentation, I actually want to show you how 00:00:03.980 --> 00:00:06.730 we can use the antiderivative to figure out the 00:00:06.730 --> 00:00:08.340 area under a curve. 00:00:08.340 --> 00:00:09.695 Actually I'm going to focus more a little bit more 00:00:09.695 --> 00:00:10.770 on the intuition. 00:00:10.770 --> 00:00:12.960 So let actually use an example from physics. 00:00:12.960 --> 00:00:15.640 I'll use distance and velocity. 00:00:15.640 --> 00:00:17.550 And actually this could be a good review for derivatives, 00:00:17.550 --> 00:00:19.590 or actually an application of derivatives. 00:00:19.590 --> 00:00:22.610 So let's say that I described the position 00:00:22.610 --> 00:00:23.500 of something moving. 00:00:23.500 --> 00:00:26.160 Let's say it's x. 00:00:26.160 --> 00:00:35.520 Let's say that x is equal to, I don't know, 16t squared. 00:00:35.520 --> 00:00:35.850 Right? 00:00:35.850 --> 00:00:36.880 So s is distance. 00:00:36.880 --> 00:00:38.120 Let me write this in the corner. 00:00:38.120 --> 00:00:41.220 I don't know why the convention is to use s as 00:00:41.220 --> 00:00:42.250 the variable for distance. 00:00:42.250 --> 00:00:45.400 One would think, well actually, I know, why won't they use d? 00:00:45.400 --> 00:00:48.740 Because d is the letter used for differential, I guess. 00:00:48.740 --> 00:00:55.800 So s is equal to distance, and then t is equal to time. 00:00:58.990 --> 00:01:02.580 So this is just a formula that tells us the position, kind of 00:01:02.580 --> 00:01:06.210 how far has something gone, after x many, let's 00:01:06.210 --> 00:01:07.140 say, seconds, right? 00:01:07.140 --> 00:01:10.620 So after like, 4 seconds, we would have gone, let's say 00:01:10.620 --> 00:01:12.750 the distance is in feet, this is in seconds. 00:01:12.750 --> 00:01:15.690 After 4 seconds, we would have gone 256 feet. 00:01:15.690 --> 00:01:16.660 That's all that says. 00:01:16.660 --> 00:01:21.250 And let me graph that as well. 00:01:21.250 --> 00:01:23.120 Graph it. 00:01:23.120 --> 00:01:28.620 That's a horrible line. 00:01:28.620 --> 00:01:30.430 Have to use the line tool, might have better luck. 00:01:33.400 --> 00:01:35.730 It's slightly better. 00:01:35.730 --> 00:01:38.110 Actually, let me undo that too, because I just want to do 00:01:38.110 --> 00:01:40.200 it for positive t, right? 00:01:40.200 --> 00:01:42.480 Because you can't really go back in time. 00:01:42.480 --> 00:01:45.340 For the purposes of this lecture, you can't 00:01:45.340 --> 00:01:47.520 go back in time. 00:01:47.520 --> 00:01:51.810 So that'll have to do. 00:01:51.810 --> 00:01:55.820 So this curve will essentially just be a parabola, right? 00:01:55.820 --> 00:01:56.720 It'll look something like this. 00:02:01.700 --> 00:02:02.790 So actually, if you look at it, I mean you 00:02:02.790 --> 00:02:04.100 could just eyeball it. 00:02:04.100 --> 00:02:06.590 The object, every second you go, it's going a little 00:02:06.590 --> 00:02:07.410 bit further, right? 00:02:07.410 --> 00:02:08.990 So it's actually accelerating. 00:02:08.990 --> 00:02:11.880 And so what if we wanted to figure out what the velocity 00:02:11.880 --> 00:02:13.960 of this object, right? 00:02:13.960 --> 00:02:18.580 This is, let's see, this is d, this is t, right? 00:02:18.580 --> 00:02:20.630 And this is, I don't know if it's clear, but this is 00:02:20.630 --> 00:02:22.780 kind of 1/2 a parabola. 00:02:22.780 --> 00:02:24.900 So this is the distance function. 00:02:24.900 --> 00:02:26.330 What would the velocity be? 00:02:26.330 --> 00:02:29.170 Well the velocity is just, what's velocity? 00:02:29.170 --> 00:02:31.590 It's distance divided by time, right? 00:02:31.590 --> 00:02:33.490 And since this velocity is always changing, we 00:02:33.490 --> 00:02:35.570 want to figure out the instantaneous velocity. 00:02:35.570 --> 00:02:38.620 And that's actually one of the initial uses of what made 00:02:38.620 --> 00:02:39.930 derivatives so useful. 00:02:39.930 --> 00:02:43.430 So we want to find the change, the instantaneous change 00:02:43.430 --> 00:02:45.450 with respect to time of this formula, right? 00:02:45.450 --> 00:02:47.350 Because this is the distance formula. 00:02:47.350 --> 00:02:50.410 So if we know the instant rate of change of distance with 00:02:50.410 --> 00:02:53.310 respect to time, we'll know the velocity, right? 00:02:53.310 --> 00:03:02.040 So ds, dt, is equal to? 00:03:02.040 --> 00:03:03.550 So what's the derivative here? 00:03:03.550 --> 00:03:09.280 It's 32t, right? 00:03:09.280 --> 00:03:10.320 And this is the velocity. 00:03:14.060 --> 00:03:16.660 Maybe I should switch back to, let me write that, 00:03:16.660 --> 00:03:20.360 v equals velocity. 00:03:20.360 --> 00:03:21.880 I don't know why I switched colors, but I'll stick 00:03:21.880 --> 00:03:23.250 with the yellow. 00:03:23.250 --> 00:03:24.510 So let's graph this function. 00:03:24.510 --> 00:03:28.680 This will actually be a fairly straightforward graph to draw. 00:03:33.670 --> 00:03:35.270 It's pretty straight. 00:03:35.270 --> 00:03:37.160 And then we draw the x-axis. 00:03:41.910 --> 00:03:43.390 I'm doing pretty good. 00:03:43.390 --> 00:03:43.790 OK. 00:03:48.010 --> 00:03:56.370 So this, I'll draw it in red, this is this going 00:03:56.370 --> 00:03:57.420 to be a line, right? 00:03:57.420 --> 00:03:59.450 32t it's a line with slope 32. 00:03:59.450 --> 00:04:00.530 So it's actually a pretty steep line. 00:04:00.530 --> 00:04:02.640 I won't draw it that steep because I'm going to use 00:04:02.640 --> 00:04:05.880 this for an illustration. 00:04:05.880 --> 00:04:06.855 So this is the velocity. 00:04:09.990 --> 00:04:11.580 This is velocity. 00:04:11.580 --> 00:04:17.330 This is that graph, and this is distance, right? 00:04:17.330 --> 00:04:19.970 So in case you hadn't learned already, and maybe I'll do a 00:04:19.970 --> 00:04:22.470 whole presentation on kind of using calculus for physics, and 00:04:22.470 --> 00:04:24.000 using derivatives for physics. 00:04:24.000 --> 00:04:27.460 But if you have to distance formula, it's derivative 00:04:27.460 --> 00:04:28.730 is just velocity. 00:04:28.730 --> 00:04:30.830 And I guess if you view it the other way, if you 00:04:30.830 --> 00:04:33.920 have the velocity, it's antiderivative is distance. 00:04:33.920 --> 00:04:37.800 Although you won't know where, at what position, 00:04:37.800 --> 00:04:38.770 the object started. 00:04:38.770 --> 00:04:42.080 In this case, the object started at position of 0, 00:04:42.080 --> 00:04:44.420 but it could be, you know, at any constant, right? 00:04:44.420 --> 00:04:46.210 You could have started here and then curved up. 00:04:46.210 --> 00:04:48.140 But anyway, let's just assume we started at 0. 00:04:48.140 --> 00:04:51.170 So the derivative of distance is velocity, the antiderivative 00:04:51.170 --> 00:04:52.350 of velocity is distance. 00:04:52.350 --> 00:04:54.020 Keep that in mind. 00:04:54.020 --> 00:04:56.130 Well let's look at this. 00:04:56.130 --> 00:05:03.880 Let's assume that we were only given this graph. 00:05:03.880 --> 00:05:05.520 And we said, you know, this is the graph of the 00:05:05.520 --> 00:05:08.850 velocity of some object. 00:05:08.850 --> 00:05:11.930 And we want to figure out what the distance is after, you 00:05:11.930 --> 00:05:13.220 know, t seconds, right? 00:05:13.220 --> 00:05:17.340 So this is the t-axis, this is the velocity axis, right? 00:05:17.340 --> 00:05:19.490 So let's say we were only given this, and let's say we didn't 00:05:19.490 --> 00:05:22.590 know that the antiderivative of the velocity function is 00:05:22.590 --> 00:05:23.250 the distance function. 00:05:23.250 --> 00:05:27.340 How would we figure out, how would we figure out what 00:05:27.340 --> 00:05:29.360 the distance would be at any given time? 00:05:29.360 --> 00:05:31.530 Well let's think about it. 00:05:31.530 --> 00:05:34.080 If we have a constant, this red is kind of bloody. 00:05:34.080 --> 00:05:37.150 Let me switch to something more pleasant. 00:05:37.150 --> 00:05:40.340 If we have, over any small period of time, right, or if we 00:05:40.340 --> 00:05:44.090 have a constant velocity, when you have a constant velocity, 00:05:44.090 --> 00:05:46.990 distance is just velocity times time, right? 00:05:46.990 --> 00:05:50.030 So let's say we had a very small time 00:05:50.030 --> 00:05:52.090 fragment here, right? 00:05:52.090 --> 00:05:54.190 I'll draw it big, but let's say this time fragment 00:05:54.190 --> 00:05:55.640 it is really small. 00:05:55.640 --> 00:05:59.330 And let's called this very small time fragment, let call 00:05:59.330 --> 00:06:02.480 this delta t, or dt actually. 00:06:02.480 --> 00:06:05.120 The way I've used dt is like, it's like a change in time 00:06:05.120 --> 00:06:07.040 that's unbelievably small, right? 00:06:07.040 --> 00:06:09.490 So it's like almost instantaneous, but not quite. 00:06:09.490 --> 00:06:11.410 Or you can actually view it as instantaneous. 00:06:11.410 --> 00:06:13.710 So this is how much time goes by. 00:06:13.710 --> 00:06:16.390 You can kind of view this as a very small change in time. 00:06:16.390 --> 00:06:20.040 So if we have a very small change of time, and over that 00:06:20.040 --> 00:06:22.510 very small change in time, we have a roughly constant 00:06:22.510 --> 00:06:26.500 velocity, let's say the roughly constant velocity is this. 00:06:31.250 --> 00:06:34.600 Right, this is the velocity, so say we had over this very small 00:06:34.600 --> 00:06:37.210 change in time, we have this roughly constant velocity 00:06:37.210 --> 00:06:38.210 that's on this graph. 00:06:38.210 --> 00:06:41.720 Actually, let me take do it here. 00:06:41.720 --> 00:06:43.400 We have this roughly constant velocity. 00:06:43.400 --> 00:06:47.870 So the distance that the object travels over the small time 00:06:47.870 --> 00:06:50.650 would be the small time times the velocity, right? 00:06:50.650 --> 00:06:54.150 It would be whatever the value of this red line is, times the 00:06:54.150 --> 00:06:57.340 width of this distance, right? 00:06:57.340 --> 00:06:59.230 So what's another way? 00:06:59.230 --> 00:07:01.950 Visually I kind of did it ahead of time, but 00:07:01.950 --> 00:07:02.900 what's happening here? 00:07:02.900 --> 00:07:08.120 If I take this change in time, right, which is kind of the 00:07:08.120 --> 00:07:12.890 base of this rectangle, and I multiply it times the velocity 00:07:12.890 --> 00:07:15.750 which is really just the height of this rectangle, what 00:07:15.750 --> 00:07:16.510 have I figured out? 00:07:16.510 --> 00:07:20.790 Well I figured out the area of this rectangle, right? 00:07:20.790 --> 00:07:23.390 Right, the velocity this moment, times the change in 00:07:23.390 --> 00:07:26.040 time at this moment, is nothing but the area of 00:07:26.040 --> 00:07:28.130 this very skinny rectangle. 00:07:28.130 --> 00:07:29.210 Skinny and tall, right? 00:07:29.210 --> 00:07:33.080 It's almost infinitely skinny, but it's, we're assuming for 00:07:33.080 --> 00:07:37.040 these purposes it has some very notional amount of width. 00:07:37.040 --> 00:07:39.990 So there we figured out the area of this column, right? 00:07:39.990 --> 00:07:44.510 Well, if we wanted to figure out the distance that you 00:07:44.510 --> 00:07:50.960 travel after, let's say, you know, I don't know, let's say 00:07:50.960 --> 00:07:54.010 t, let's say t sub nought, right? 00:07:54.010 --> 00:07:55.710 This is just a particular t. 00:07:55.710 --> 00:07:57.980 After t sub nought seconds, right? 00:07:57.980 --> 00:08:00.840 Well then, all we would have to do is, we would have to just 00:08:00.840 --> 00:08:04.010 figure, we would just do a bunch of dt's, right? 00:08:04.010 --> 00:08:08.900 You'd do another one here, you'd figure out the area of 00:08:08.900 --> 00:08:12.630 this column, you'd figure out the area of this column, the 00:08:12.630 --> 00:08:15.490 area of this column, right? 00:08:15.490 --> 00:08:18.970 Because each of these areas of each of these columns 00:08:18.970 --> 00:08:21.690 represents the distance that the object travels 00:08:21.690 --> 00:08:24.610 over that dt, right? 00:08:24.610 --> 00:08:28.506 So if you wanted to know how far you traveled after t sub 00:08:28.506 --> 00:08:33.340 zero seconds, you'd essentially get, or an approximation would 00:08:33.340 --> 00:08:36.210 be, the sum of all of these areas. 00:08:36.210 --> 00:08:40.110 And as you got more and more, as you made the dt's smaller 00:08:40.110 --> 00:08:41.430 and smaller, skinnier, skinnier, skinnier. 00:08:41.430 --> 00:08:43.810 And you had more and more and more and more of these 00:08:43.810 --> 00:08:47.930 rectangles, then your approximation will get pretty 00:08:47.930 --> 00:08:50.700 close to, well, two things. 00:08:50.700 --> 00:08:53.320 It'll get pretty close to, as you can imagine, the area 00:08:53.320 --> 00:08:56.230 under this curve, or in this case a line. 00:08:56.230 --> 00:09:01.870 But it would also get you pretty much the exact amount 00:09:01.870 --> 00:09:06.720 of distance you've traveled after t sub nought seconds. 00:09:06.720 --> 00:09:12.410 So I think I'm running into the ten minute wall, so I'm just 00:09:12.410 --> 00:09:15.600 going to pause here, and I'm going to continue this in 00:09:15.600 --> 00:09:17.280 the next presentation.

Indefinite Integration (part IV)

https://www.youtube.com/watch?v=VJ9VRUDQyK8

vtt

https://www.youtube.com/api/timedtext?v=VJ9VRUDQyK8&ei=eWeUZejtDJi_mLAPz86R8As&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249833&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=6D6CFE3B38B17F3C8373D544DE4F348E2DBB6D79.0AAA21A7B68A64838487EB5D1C701560C86E3B64&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.820 --> 00:00:01.850 Welcome back. 00:00:01.850 --> 00:00:04.510 In the last presentation, I showed you how to essentially 00:00:04.510 --> 00:00:07.660 reverse the chain rule when you're doing an integral. 00:00:07.660 --> 00:00:10.370 And you could also do this by integral, it's called 00:00:10.370 --> 00:00:11.890 integration by substitution. 00:00:11.890 --> 00:00:12.690 And I'll show you why. 00:00:12.690 --> 00:00:15.030 And this is essentially just a reverse of the chain rule. 00:00:15.030 --> 00:00:23.770 The last problem we did in that last video, I said, sine of x 00:00:23.770 --> 00:00:29.490 to the third power times cosine of x, and I took the integral 00:00:29.490 --> 00:00:30.980 of the whole thing. 00:00:30.980 --> 00:00:34.850 And I did it kind of, you know, just telling you that, well, 00:00:34.850 --> 00:00:36.090 we're just reversing the chain rule. 00:00:36.090 --> 00:00:41.110 So we see the derivative of the sine x here, right, which is 00:00:41.110 --> 00:00:44.170 cosine of x, so we can just treat sine of x like a variable 00:00:44.170 --> 00:00:45.550 and take its integral. 00:00:45.550 --> 00:00:51.890 And I said that that is equal to sine of x to 00:00:51.890 --> 00:00:54.260 the fourth, times 1/4. 00:00:54.260 --> 00:00:54.600 Right? 00:00:54.600 --> 00:00:56.940 And the reason why we could just treat the sine of x like 00:00:56.940 --> 00:00:59.880 it's just like kind of a variable instead of a function, 00:00:59.880 --> 00:01:02.120 is because we had its derivative sitting right here. 00:01:02.120 --> 00:01:04.640 And if you keep doing it back and forth between the chain 00:01:04.640 --> 00:01:06.100 rule and what I just did, I think it'll make 00:01:06.100 --> 00:01:07.080 a lot of sense. 00:01:07.080 --> 00:01:08.430 So this might have been a little confusing. 00:01:08.430 --> 00:01:11.595 So I'll show you a technique called integration by 00:01:11.595 --> 00:01:13.130 substitution, which is essentially the 00:01:13.130 --> 00:01:14.410 exact same thing. 00:01:14.410 --> 00:01:17.200 So let's start with this exact same integral here, and let's 00:01:17.200 --> 00:01:19.970 pretend like we don't know the answer. 00:01:19.970 --> 00:01:26.250 We say, well, we have a function and its derivative, so 00:01:26.250 --> 00:01:34.510 let me let u equal the function that we have the derivative of. 00:01:34.510 --> 00:01:34.846 Right? 00:01:38.830 --> 00:01:42.070 u is equal to sine of x, right? 00:01:42.070 --> 00:01:43.510 u is sine of x. 00:01:43.510 --> 00:01:45.180 Well, what's the derivative of u? 00:01:45.180 --> 00:01:47.000 du, dx. 00:01:49.910 --> 00:01:52.610 Well, we know what du dx is, right? 00:01:52.610 --> 00:01:55.540 du of dx is equal to cosine of x. 00:01:55.540 --> 00:01:58.440 We memorized that, and maybe in a future presentation I'll 00:01:58.440 --> 00:02:00.060 actually prove it to you. 00:02:00.060 --> 00:02:03.400 So what we can now do is substitute these 2 things 00:02:03.400 --> 00:02:07.240 into this integral. 00:02:07.240 --> 00:02:11.080 So the integral now becomes, instead of writing sine of x 00:02:11.080 --> 00:02:15.830 to the third power, we can write u to the third power. 00:02:15.830 --> 00:02:17.590 And what's cosine of x? 00:02:17.590 --> 00:02:21.700 Well, we just showed, cosine of x is just du dx, right? 00:02:21.700 --> 00:02:32.860 So it's times du dx, and then we have times dx, right? 00:02:32.860 --> 00:02:37.100 And I know you're probably not too comfortable with the 00:02:37.100 --> 00:02:40.740 differentials at this point, but they behave in just the way 00:02:40.740 --> 00:02:42.520 regular numbers do with a fraction. 00:02:42.520 --> 00:02:46.400 So this dx and this dx actually do cancel out, and you're left 00:02:46.400 --> 00:02:53.160 with, that this is equal to the integral of u to the third du. 00:02:53.160 --> 00:02:54.320 And now this is an easy integral. 00:02:54.320 --> 00:02:56.850 The only thing different than what you might have seen 00:02:56.850 --> 00:02:59.195 recently is that instead of an x, we have a u here. 00:02:59.195 --> 00:03:02.060 And while we know that the answer of this integral, this 00:03:02.060 --> 00:03:09.020 is equal to 1/4 u to the fourth, and then, of course, 00:03:09.020 --> 00:03:10.600 we should add the plus c. 00:03:10.600 --> 00:03:11.390 Are we done? 00:03:11.390 --> 00:03:13.730 Well, I mean, almost, but it would be nice to just take that 00:03:13.730 --> 00:03:17.630 u, and unwind it, and put the sine of x back in for it. 00:03:17.630 --> 00:03:20.840 So let's do that. 00:03:20.840 --> 00:03:22.190 So u is the sine of x. 00:03:22.190 --> 00:03:23.390 That's what we said at the beginning. 00:03:23.390 --> 00:03:30.180 1/4 sine of x to the fourth, plus c. 00:03:30.180 --> 00:03:31.520 Done. 00:03:31.520 --> 00:03:33.440 Actually, this might be an easier way to think about these 00:03:33.440 --> 00:03:36.290 type of integrals than what I did in the last presentation. 00:03:36.290 --> 00:03:39.000 But you know, every now and then you have to bear with me. 00:03:39.000 --> 00:03:40.540 I do things in the wrong order. 00:03:40.540 --> 00:03:42.940 Let's do a couple more problems like this. 00:03:47.060 --> 00:04:06.770 Let's take the integral of 2x plus 3 times x squared plus 3x 00:04:06.770 --> 00:04:16.140 plus 15 to the fifth power dx. 00:04:16.140 --> 00:04:18.340 That looks complicated to you, doesn't it? 00:04:18.340 --> 00:04:21.840 Well, just like we said, this is a pattern, like we saw 00:04:21.840 --> 00:04:23.120 in the previous examples. 00:04:23.120 --> 00:04:28.780 We have this expression here, x squared plus 3x plus 15, 00:04:28.780 --> 00:04:30.590 and well, what's the derivative of this? 00:04:30.590 --> 00:04:32.540 x squared plus 3x plus 15? 00:04:32.540 --> 00:04:34.930 Well, it's 2x plus 3, right? 00:04:34.930 --> 00:04:38.700 Notice that I'm engineering it so it works cleanly, but most 00:04:38.700 --> 00:04:40.850 textbooks and tests tend to do that. 00:04:40.850 --> 00:04:42.550 So let's make the substitution. 00:04:42.550 --> 00:04:45.430 Because we have a u that we can use, and then we have 00:04:45.430 --> 00:04:46.630 its derivative, right? 00:04:46.630 --> 00:04:56.380 So we can say u is equal to x squared plus 3x plus 15, and we 00:04:56.380 --> 00:05:00.320 can say then, the derivative of u, we know the derivative 00:05:00.320 --> 00:05:04.060 of u is 2x plus 3, right? 00:05:04.060 --> 00:05:05.980 Because the derivative of 15 is 0. 00:05:05.980 --> 00:05:08.220 So now we can make our substitutions. 00:05:08.220 --> 00:05:10.480 I'm just going to switch the orders of these two 00:05:10.480 --> 00:05:11.650 around, no different. 00:05:11.650 --> 00:05:13.770 So this is just u to the fifth, right? 00:05:13.770 --> 00:05:15.950 Because this is this. 00:05:15.950 --> 00:05:17.300 So this is just u to the fifth. 00:05:20.050 --> 00:05:24.640 And then this is du dx times du dx, right? 00:05:24.640 --> 00:05:26.940 I just switched the orders. 00:05:26.940 --> 00:05:31.655 And then I multiply that times dx. 00:05:31.655 --> 00:05:32.730 And these cancel. 00:05:32.730 --> 00:05:35.120 And I know you're not completely comfortable yet with 00:05:35.120 --> 00:05:38.115 even this integration notation, why is this dx sitting there in 00:05:38.115 --> 00:05:39.616 the first place, but when we do the definite integrals it 00:05:39.616 --> 00:05:42.110 will make more sense. 00:05:42.110 --> 00:05:48.730 But this is just equal to the integral of u to the fifth du. 00:05:48.730 --> 00:05:50.220 And the integral of this, well, this is easy. 00:05:50.220 --> 00:05:55.640 This is just equal to 1/6 u to the sixth, right, plus c. 00:05:55.640 --> 00:05:58.590 And now we can just unwind this. 00:05:58.590 --> 00:06:00.830 I'll do it up here to make it extra messy. 00:06:00.830 --> 00:06:05.920 This is just equal to 1/6 times u, which is this right here, 00:06:05.920 --> 00:06:09.570 right, we just set u to equal this expression, 1/6 x 00:06:09.570 --> 00:06:14.380 squared plus 3x plus 15. 00:06:14.380 --> 00:06:20.010 All of this to the sixth power, plus c. 00:06:20.010 --> 00:06:21.060 Let's do one more. 00:06:21.060 --> 00:06:23.500 I think we have time for one more. 00:06:23.500 --> 00:06:27.970 Image, clear image, image, invert. 00:06:27.970 --> 00:06:28.460 OK. 00:06:28.460 --> 00:06:33.210 I will also switch colors, just to keep things interesting. 00:06:33.210 --> 00:06:48.210 Let's take the integral of e to the x times e 00:06:48.210 --> 00:06:53.590 to the x to the fifth. 00:06:53.590 --> 00:06:55.040 I keep using that. 00:06:55.040 --> 00:06:59.250 Let's say to the minus third power. 00:06:59.250 --> 00:07:00.820 dx. 00:07:00.820 --> 00:07:04.565 Well, once again, we have this expression e to the x, and 00:07:04.565 --> 00:07:05.630 what's the derivative of e to the x? 00:07:05.630 --> 00:07:07.550 Well, the derivative of e to the x, as we learned, which 00:07:07.550 --> 00:07:12.750 is one of these things that amazes me, is e to the x. 00:07:12.750 --> 00:07:16.050 Actually, that's one definition for e, is number which, when 00:07:16.050 --> 00:07:17.810 it's raised to the x power, it's the derivative of 00:07:17.810 --> 00:07:19.600 the same expression. 00:07:19.600 --> 00:07:21.140 But anyway, I don't want to confuse you too much. 00:07:21.140 --> 00:07:26.070 But we can say then that u is equal to e to the x, and we 00:07:26.070 --> 00:07:31.090 know that du dx is equal to e to the x as well, which is, 00:07:31.090 --> 00:07:32.510 once again, mind blowing. 00:07:32.510 --> 00:07:36.150 So if we rewrite this top integral, this is just equal 00:07:36.150 --> 00:07:38.230 to, I won't switch this time. 00:07:38.230 --> 00:07:39.790 So this is du dx, right? 00:07:42.530 --> 00:07:49.690 du dx times u to the minus 3 dx. 00:07:49.690 --> 00:07:50.800 And I know what you're thinking, Sal. 00:07:50.800 --> 00:07:53.310 Well, du dx is e to the x. 00:07:53.310 --> 00:07:54.420 u is also e to the x. 00:07:54.420 --> 00:07:56.390 Why didn't I substitute it the other way around? 00:07:56.390 --> 00:07:57.830 Why didn't I say this? 00:07:57.830 --> 00:08:04.470 Why didn't I say that this one is u, and why didn't I say this 00:08:04.470 --> 00:08:14.750 one is du dx to the minus 3? 00:08:14.750 --> 00:08:16.730 Well, as you can see, this would have been useless, right? 00:08:16.730 --> 00:08:18.590 Because then I can't multiply it times a dx, and it 00:08:18.590 --> 00:08:19.570 gets all confusing. 00:08:19.570 --> 00:08:22.830 And actually, I just realized, I constructed a very silly 00:08:22.830 --> 00:08:24.890 problem for you, because you could simplify this before even 00:08:24.890 --> 00:08:25.870 doing it with substitution. 00:08:25.870 --> 00:08:28.060 But we'll continue doing it with substitution. 00:08:28.060 --> 00:08:30.260 But anyway, you see, if you did it in this way, it becomes very 00:08:30.260 --> 00:08:32.420 complicated, so we don't want to do it like that. 00:08:32.420 --> 00:08:39.060 So this, as we see, simplifies to the integral of u to 00:08:39.060 --> 00:08:44.100 the minus 3 du, and that that equals, let's see. 00:08:44.100 --> 00:08:50.510 You raise exponent 1 minus 1/2 u to the minus two, and that's 00:08:50.510 --> 00:09:00.320 the same thing as minus 1/2 e to the x to the minus 2, or we 00:09:00.320 --> 00:09:06.600 could view that as minus 1/2 e to the minus 2x, and of 00:09:06.600 --> 00:09:09.230 course, plus c at the end. 00:09:09.230 --> 00:09:11.720 Now why was my problem that I gave you silly? 00:09:11.720 --> 00:09:13.630 Well, I could have simplified this before even doing 00:09:13.630 --> 00:09:15.070 the substitution, right? 00:09:15.070 --> 00:09:16.890 I could have said that that's the same thing as the integral 00:09:16.890 --> 00:09:22.950 of e to the x times e to the minus three x dx, which is the 00:09:22.950 --> 00:09:29.530 same thing as the integral of e to the minus 2x dx. 00:09:29.530 --> 00:09:32.920 And actually, it's good that it by substitution, because this 00:09:32.920 --> 00:09:35.550 probably wouldn't have been completely intuitive for you 00:09:35.550 --> 00:09:37.670 to do just yet, as well. 00:09:37.670 --> 00:09:42.190 But anyway, that's integration by substitution. 00:09:42.190 --> 00:09:44.340 I might do another presentation where I do slightly harder 00:09:44.340 --> 00:09:46.780 problems, using this same technique. 00:09:46.780 --> 00:09:48.750 I'll see you soon.

Indefinite Integration (part III)

https://www.youtube.com/watch?v=77-najNh4iY

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https://www.youtube.com/api/timedtext?v=77-najNh4iY&ei=eWeUZemVBq2UhcIPyYq5qA4&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249833&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=586427A44799282532FB07ABB8BE6877C6ED2B8D.2C76BB81587C8F304973DA0A5C5254BECDC4D681&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.840 --> 00:00:01.890 Welcome back. 00:00:01.890 --> 00:00:04.030 Well I'm now going to do a presentation on how to 00:00:04.030 --> 00:00:07.330 essentially invert the chain rule or reverse the chain 00:00:07.330 --> 00:00:09.730 rule, because we're doing integration, which is the 00:00:09.730 --> 00:00:11.910 opposite of taking the derivative. 00:00:11.910 --> 00:00:13.570 So let's just take a review of what the chain 00:00:13.570 --> 00:00:14.610 rule told us before. 00:00:19.220 --> 00:00:23.710 If I were to take the derivative of f of g of x-- 00:00:23.710 --> 00:00:25.180 hopefully this doesn't confuse you too much. 00:00:25.180 --> 00:00:29.060 I'll give another example with a concrete f of x 00:00:29.060 --> 00:00:29.986 and a concrete g of x. 00:00:29.986 --> 00:00:31.820 If I want to take the derivative of that, the chain 00:00:31.820 --> 00:00:34.830 rule just says the derivative of this composite function is 00:00:34.830 --> 00:00:38.410 just the derivative of the inside function. 00:00:38.410 --> 00:00:45.930 g prime of x times the derivative of the outer, or 00:00:45.930 --> 00:00:49.840 kind of the parent function, but still having g 00:00:49.840 --> 00:00:51.610 of x in at times. 00:00:51.610 --> 00:00:55.885 f prime of g of x. 00:00:55.885 --> 00:00:59.850 I know this might seem complicated if you aren't too 00:00:59.850 --> 00:01:02.693 comfortable with this type of notation, but done in kind 00:01:02.693 --> 00:01:04.790 of an example form it makes a lot of sense. 00:01:04.790 --> 00:01:13.790 If I said what is the derivative of let's 00:01:13.790 --> 00:01:20.100 say sin of x squared. 00:01:22.800 --> 00:01:26.490 Well in this situation, f of x is sin of x, right? 00:01:26.490 --> 00:01:28.870 And then g of x is x squared, right? 00:01:28.870 --> 00:01:33.920 And sin of x squared is essentially f of g of x. 00:01:33.920 --> 00:01:35.680 And this review of chain rule. 00:01:35.680 --> 00:01:38.530 You could go watch the video on the chain rule as well, but I 00:01:38.530 --> 00:01:40.210 don't mind doing a couple of problems here. 00:01:40.210 --> 00:01:42.730 All this is saying that the derivative of this is you take 00:01:42.730 --> 00:01:45.330 the derivative of the inside function-- g of x in this 00:01:45.330 --> 00:01:49.845 example, which is 2x-- and you multiply it times the 00:01:49.845 --> 00:01:51.370 derivative of the outer function or the 00:01:51.370 --> 00:01:52.270 parent function. 00:01:52.270 --> 00:01:56.420 And we memorize I guess that the derivative of sin of x 00:01:56.420 --> 00:02:02.470 is cosine of x, so it's times cosine of g of x. 00:02:02.470 --> 00:02:06.900 So we keep the x squared there. 00:02:06.900 --> 00:02:08.840 If it confuses you, just think about the inside 00:02:08.840 --> 00:02:10.160 and the outside function. 00:02:10.160 --> 00:02:12.700 If you take the derivative of kind of this composite 00:02:12.700 --> 00:02:16.260 function, it's the same thing that equals the derivative of 00:02:16.260 --> 00:02:19.970 the inside function, which is 2x times the derivative 00:02:19.970 --> 00:02:21.230 of the outside function. 00:02:21.230 --> 00:02:23.775 But we keep this inside function in it, and we 00:02:23.775 --> 00:02:24.960 keep this x right there. 00:02:24.960 --> 00:02:27.130 So that's a review of the chain rule. 00:02:27.130 --> 00:02:31.050 So what happens if we want to reverse the chain rule? 00:02:31.050 --> 00:02:35.260 Well if we wanted to reverse it, we're essentially saying 00:02:35.260 --> 00:02:40.560 that we want to take the integral of something where we 00:02:40.560 --> 00:02:45.650 have the derivative of kind of the inner function, and then we 00:02:45.650 --> 00:02:52.980 have the derivative of a larger composite function. 00:02:52.980 --> 00:02:55.300 I'm just rewriting the chain rule, but I'm writing in 00:02:55.300 --> 00:02:58.330 an integral form that this is equal to f of g of x. 00:03:00.870 --> 00:03:06.140 This statement up here is the exact same thing as 00:03:06.140 --> 00:03:08.800 the statement down here. 00:03:08.800 --> 00:03:12.010 All I did is I took the integral of both sides. 00:03:12.010 --> 00:03:15.240 I'm saying the integral of this is equal to the 00:03:15.240 --> 00:03:17.680 integral of this right here. 00:03:17.680 --> 00:03:19.700 I probably shouldn't switched equal signs like that 00:03:19.700 --> 00:03:23.930 with you, but let's use this formula I guess. 00:03:23.930 --> 00:03:26.285 But all you have to know is this the reverse of the chain 00:03:26.285 --> 00:03:27.290 rule to solve some problems. 00:03:32.270 --> 00:03:36.125 Image invert colors. 00:03:36.125 --> 00:03:37.440 Let me rewrite that. 00:03:37.440 --> 00:03:45.180 The integral-- if I have g prime of x times f prime of 00:03:45.180 --> 00:03:52.390 g of x dx, then that is equal to f of g of x. 00:03:52.390 --> 00:03:54.880 This is just the chain rule in reverse. 00:03:54.880 --> 00:03:57.010 And I know it's very complicated sometimes when you 00:03:57.010 --> 00:03:58.420 have it in this notation, but I'll give you a 00:03:58.420 --> 00:04:00.070 couple of examples. 00:04:00.070 --> 00:04:08.160 What if I had the integral of let's say-- this is actually 00:04:08.160 --> 00:04:11.400 one that's often kind of viewed as a trick, but you'll see 00:04:11.400 --> 00:04:15.180 it's actually not that tricky of a trick. 00:04:15.180 --> 00:04:15.810 OK. 00:04:15.810 --> 00:04:29.650 So let's say I have the natural log squared over x dx. 00:04:29.650 --> 00:04:33.100 And if you saw an integral like this, you'd probably be 00:04:33.100 --> 00:04:35.760 daunted, and you'd be surprised, many people well 00:04:35.760 --> 00:04:38.270 into college calculus courses are still daunted 00:04:38.270 --> 00:04:39.500 by this problem. 00:04:39.500 --> 00:04:41.060 But all you have to recognize is this is 00:04:41.060 --> 00:04:41.870 the reverse chain rule. 00:04:41.870 --> 00:04:43.200 Why is this the reverse chain rule? 00:04:43.200 --> 00:04:51.610 Well, this is the same thing as the integral of 1/x times the 00:04:51.610 --> 00:04:56.260 natural log-- whoops, this should be nlx, right-- the 00:04:56.260 --> 00:05:00.770 natural log of x squared dx. 00:05:00.770 --> 00:05:03.400 These are the same thing, I just took the 1/x out. 00:05:03.400 --> 00:05:05.550 Now this might look a little familiar. 00:05:05.550 --> 00:05:09.610 Well, what's the derivative of the natural log of x? 00:05:09.610 --> 00:05:10.960 If you remember from the derivative module, 00:05:10.960 --> 00:05:12.530 it's 1/x, right? 00:05:12.530 --> 00:05:15.560 Let me write that down in the corner here. 00:05:15.560 --> 00:05:22.170 The derivative of the natural log of x is equal to 1/x. 00:05:22.170 --> 00:05:25.130 So right here we have the derivative of the 00:05:25.130 --> 00:05:26.640 natural log of x. 00:05:26.640 --> 00:05:31.360 So now we can just say that we could essentially treat this 00:05:31.360 --> 00:05:34.875 natural log of x as kind of a variable by itself. 00:05:34.875 --> 00:05:37.520 And essentially what I'm going to be doing if I could 00:05:37.520 --> 00:05:38.900 actually substitute for. 00:05:38.900 --> 00:05:40.440 Actually let's do that. 00:05:40.440 --> 00:05:42.850 Well no, no, no I don't do that now, that'll confuse you. 00:05:42.850 --> 00:05:44.930 Although my flip-flopping is probably confusing 00:05:44.930 --> 00:05:46.690 you even more. 00:05:46.690 --> 00:05:49.990 I have the derivative of the natural log of x, so I can then 00:05:49.990 --> 00:05:52.250 say well I have the derivative there, so this is a 00:05:52.250 --> 00:05:54.410 composite function. 00:05:54.410 --> 00:05:58.460 This is essentially f prime of g of x. 00:05:58.460 --> 00:06:02.400 So then I can say well that integral must be 00:06:02.400 --> 00:06:07.260 equal to this thing. 00:06:07.260 --> 00:06:09.580 This is something squared, right? 00:06:09.580 --> 00:06:11.330 So what's the integral of something squared? 00:06:11.330 --> 00:06:16.200 Well the integral of something squared is 1/3. 00:06:16.200 --> 00:06:17.920 That's something to the third power. 00:06:17.920 --> 00:06:20.660 We learned in the previous indefinite integral 00:06:20.660 --> 00:06:22.960 module, right? 00:06:22.960 --> 00:06:26.470 And then it's 1/3 something to the third power, and then we 00:06:26.470 --> 00:06:30.150 know from the chain rule that something is the ln of x. 00:06:32.960 --> 00:06:35.120 And I don't know if I've already forgotten to do it 00:06:35.120 --> 00:06:39.000 once, but don't forget to do the plus c. 00:06:39.000 --> 00:06:41.810 Now you say, Sal, this completely confused me, 00:06:41.810 --> 00:06:43.590 because it probably did. 00:06:43.590 --> 00:06:45.360 And if it completely confused you, let's just take the 00:06:45.360 --> 00:06:47.620 derivative of this and I think you'll see it happening the 00:06:47.620 --> 00:06:50.120 other way around and it might make a little sense. 00:06:50.120 --> 00:06:52.960 When you take the derivative, we just use the chain rule. 00:06:52.960 --> 00:06:55.300 You take the derivative of the inside first. 00:06:55.300 --> 00:07:01.370 The derivative of the inside is 1/x and you multiply that times 00:07:01.370 --> 00:07:03.670 the derivative of the outside function, and then you 00:07:03.670 --> 00:07:05.330 keep the inside the same. 00:07:05.330 --> 00:07:08.150 So the derivative of the outside function is 3 times it 00:07:08.150 --> 00:07:15.090 coefficient, so it's 3 times 1/3 times the whole thing 00:07:15.090 --> 00:07:17.180 to one less exponent. 00:07:17.180 --> 00:07:21.050 So the whole thing is ln of x. 00:07:21.050 --> 00:07:22.420 And then of course plus 0, right. 00:07:22.420 --> 00:07:24.490 The derivative of c is 0. 00:07:24.490 --> 00:07:27.090 Well this is just equal 3, 3 cancel out. 00:07:27.090 --> 00:07:36.150 This is equal to 1/x times the ln of x squared, which 00:07:36.150 --> 00:07:39.150 is our original problem. 00:07:39.150 --> 00:07:42.600 Let me do another problem because I probably started 00:07:42.600 --> 00:07:44.360 off with something a little bit too hard. 00:07:47.540 --> 00:08:02.070 What is the integral of let's say sin of x to 00:08:02.070 --> 00:08:05.500 the third power dx. 00:08:05.500 --> 00:08:07.040 That's often written like this. 00:08:07.040 --> 00:08:09.330 That's often written like sin of x. 00:08:11.960 --> 00:08:15.010 Same thing, but I like to think of it this way because 00:08:15.010 --> 00:08:18.750 it's not a new notation. 00:08:18.750 --> 00:08:20.610 Actually this is a mistake. 00:08:20.610 --> 00:08:23.310 Clearly I'm making up these problems on the fly. 00:08:23.310 --> 00:08:25.250 Actually I don't want to do that. 00:08:25.250 --> 00:08:26.410 That is the wrong problem. 00:08:26.410 --> 00:08:28.380 I want to take the integral-- and actually you can see kind 00:08:28.380 --> 00:08:31.020 of how I'm thinking about these problems-- I'm going to take 00:08:31.020 --> 00:08:41.880 the integral of cosine of x times the sin of x to 00:08:41.880 --> 00:08:44.960 the third power dx. 00:08:44.960 --> 00:08:49.790 Well, we have this kind of more complicated part, the sin of x, 00:08:49.790 --> 00:08:53.410 and we have the derivative sin of x because we learned the 00:08:53.410 --> 00:08:56.190 derivative sin of x is cosine of x. 00:08:56.190 --> 00:09:00.420 So if we have a function inside of a larger composite function, 00:09:00.420 --> 00:09:03.530 and we have it's derivative, we can just treat this function as 00:09:03.530 --> 00:09:05.720 kind of like a single entity. 00:09:05.720 --> 00:09:09.290 Like if this was just one variable and then we 00:09:09.290 --> 00:09:10.550 take integral of it. 00:09:10.550 --> 00:09:18.450 So this just equal to sin of x and we raise this one more 00:09:18.450 --> 00:09:23.880 power to the fourth and we multiply times 1/4. 00:09:23.880 --> 00:09:25.030 And how did we do that? 00:09:25.030 --> 00:09:28.430 Because we know that the integral of say x to the fourth 00:09:28.430 --> 00:09:31.570 dx is equal to-- I mean x to the third dx-- is equal 00:09:31.570 --> 00:09:33.970 to 1/4 x to the fourth. 00:09:33.970 --> 00:09:35.910 Instead of an x we had a sin here. 00:09:35.910 --> 00:09:38.390 And remember the reason why we did that is because the 00:09:38.390 --> 00:09:41.330 derivative of the sin function is sitting right here. 00:09:41.330 --> 00:09:44.660 In the next presentation, I'll show you why this can also be 00:09:44.660 --> 00:09:47.750 done using substitution, or why they're the same thing. 00:09:47.750 --> 00:09:49.090 I'll see you in the next presentation.

Indefinite integrals (part II)

https://www.youtube.com/watch?v=mHvSYRUEWnE

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WEBVTT Kind: captions Language: en 00:00:00.930 --> 00:00:01.990 Welcome back. 00:00:01.990 --> 00:00:04.410 In this presentation I'm just going to do a bunch of examples 00:00:04.410 --> 00:00:08.200 of taking the antiderivative or the indefinite integral of 00:00:08.200 --> 00:00:11.610 polynomial expressions, and hopefully I'll show you that 00:00:11.610 --> 00:00:13.740 it's a pretty straightforward thing to do. 00:00:13.740 --> 00:00:15.952 So let's get started. 00:00:15.952 --> 00:00:22.755 If I wanted to take indefinite integral-- and you could do a 00:00:22.755 --> 00:00:24.810 web search for integral and you'll see this drawn 00:00:24.810 --> 00:00:29.190 properly-- take the indefinite integral-- let me make 00:00:29.190 --> 00:00:31.060 a big expression. 00:00:31.060 --> 00:00:37.930 Let's say I want to take the indefinite integral of 3x to 00:00:37.930 --> 00:00:49.650 the negative 5 minus 7x to the third plus 3 00:00:49.650 --> 00:00:54.250 minus x to the ninth. 00:00:54.250 --> 00:00:56.900 So you might already be intimidated by 00:00:56.900 --> 00:00:58.670 what I wrote down. 00:00:58.670 --> 00:01:01.700 Well, one, if you saw the last presentation or if you 00:01:01.700 --> 00:01:03.035 understood presentation, you probably realize, well the 00:01:03.035 --> 00:01:04.690 indefinite integral even though it looks like fancy 00:01:04.690 --> 00:01:06.770 math isn't that fancy. 00:01:06.770 --> 00:01:10.020 Or at least it isn't that difficult to perform. 00:01:10.020 --> 00:01:15.610 And all you have to realize now is if we took the derivative of 00:01:15.610 --> 00:01:17.790 a polynomial, it was just the sum of the derivatives 00:01:17.790 --> 00:01:20.490 of each of the terms. 00:01:20.490 --> 00:01:23.880 it actually it turns out is the same way the other way around. 00:01:23.880 --> 00:01:27.990 The antiderivative of this entire expression is just the 00:01:27.990 --> 00:01:32.160 sum of the antidervatives of each of the individual terms. 00:01:32.160 --> 00:01:33.420 So we can just take the [? integers ?] 00:01:33.420 --> 00:01:35.570 of each term and we'll get the answer. 00:01:35.570 --> 00:01:37.960 So what does this equal? 00:01:37.960 --> 00:01:41.800 Well in this case 3x to the minus 5 power. 00:01:41.800 --> 00:01:46.000 So we take the exponent, we add 1 to the exponent, so now we 00:01:46.000 --> 00:01:52.470 get x to the negative 4, and then we multiply the 00:01:52.470 --> 00:01:58.380 coefficient times 1 over the new exponent. 00:01:58.380 --> 00:02:02.160 So 1 over the new exponent is minus 1/4. 00:02:02.160 --> 00:02:05.933 So 3 times minus 1/4 is minus 3/4. 00:02:09.320 --> 00:02:09.860 And let's see. 00:02:09.860 --> 00:02:12.670 Here we have x to the third. 00:02:12.670 --> 00:02:16.050 So instead of x to the third, let's raise it by one number. 00:02:16.050 --> 00:02:19.520 So we get x to the fourth. 00:02:19.520 --> 00:02:21.560 And then we multiply the coefficient. 00:02:21.560 --> 00:02:23.770 You know, we could either just keep the minus and say the 00:02:23.770 --> 00:02:25.353 coefficient's 7, or we could just say the coefficient 00:02:25.353 --> 00:02:27.290 is minus 7. 00:02:27.290 --> 00:02:31.420 We multiply the coefficient times 1 over the new exponent. 00:02:31.420 --> 00:02:36.370 So the new exponent is 4, so we multiply 1/4 times 00:02:36.370 --> 00:02:39.160 minus 7, so minus 7/4. 00:02:42.140 --> 00:02:45.730 And now this is interesting. 00:02:45.730 --> 00:02:46.880 3, just 3. 00:02:46.880 --> 00:02:48.270 Well how do we apply this? 00:02:48.270 --> 00:02:53.760 Well isn't 3 the same thing as 3 times x to the 0? 00:02:53.760 --> 00:02:55.710 Right, because x to the 0 is just 1. 00:02:55.710 --> 00:02:57.300 And that's how you should view it. 00:02:57.300 --> 00:03:00.280 It shows you that this rule is actually very consistent. 00:03:00.280 --> 00:03:02.740 So what's the answer derivative of 3? 00:03:02.740 --> 00:03:07.360 Well if we view 3 as 3 x to the 0, we raise the exponent by 00:03:07.360 --> 00:03:12.520 1, so now we're going to have x to the 1. 00:03:12.520 --> 00:03:14.145 And x to the 1 is just x, so I'm just going 00:03:14.145 --> 00:03:16.180 to leave it as an x. 00:03:16.180 --> 00:03:20.410 And we multiply it, the old coefficient-- this 3 or you 00:03:20.410 --> 00:03:21.830 know the derivative coefficient-- we multiply that 00:03:21.830 --> 00:03:25.600 times 1 over the inverse of the new exponent. 00:03:25.600 --> 00:03:28.580 So the exponent's 1, so the inverse of 1 is 1, 00:03:28.580 --> 00:03:33.670 so it just stays 3. 00:03:33.670 --> 00:03:37.670 We've multiplied 3 times 1/1, which is still just 3. 00:03:37.670 --> 00:03:40.090 And then finally x to the ninth-- I think you're getting 00:03:40.090 --> 00:03:43.460 the hang of this-- we raise the exponent by one, 00:03:43.460 --> 00:03:46.270 x to the tenth. 00:03:46.270 --> 00:03:47.920 And then we multiply the current coefficient. 00:03:47.920 --> 00:03:49.860 Well the current coefficient is minus 1, right. 00:03:49.860 --> 00:03:51.430 We just didn't write the 1 there. 00:03:51.430 --> 00:03:54.530 We multiply the current coefficient minus 1 times 00:03:54.530 --> 00:04:01.880 1 over the new exponent, so it's minus 1/10. 00:04:01.880 --> 00:04:02.810 There we did it. 00:04:02.810 --> 00:04:05.520 That wasn't too difficult of taking the antiderivative 00:04:05.520 --> 00:04:09.710 or-- I always forget. 00:04:09.710 --> 00:04:11.270 Plus c, right? 00:04:11.270 --> 00:04:12.835 Because when you take the derivative of any constant 00:04:12.835 --> 00:04:15.030 it becomes 0, so it might have disappeared here. 00:04:15.030 --> 00:04:17.050 So plus c where this is any constant. 00:04:17.050 --> 00:04:19.310 This could be a 10, could be a million, could 00:04:19.310 --> 00:04:21.305 be a minus trillion. 00:04:21.305 --> 00:04:23.400 It's any constant. 00:04:23.400 --> 00:04:26.160 And just to really hit the point home, let's take the 00:04:26.160 --> 00:04:28.300 derivative of this and just make sure we got 00:04:28.300 --> 00:04:29.890 this expression. 00:04:29.890 --> 00:04:32.590 And hopefully this is second nature to you by now. 00:04:32.590 --> 00:04:34.550 And you know if you ever run out of practice problems 00:04:34.550 --> 00:04:37.350 in your book because you love doing math so much, 00:04:37.350 --> 00:04:38.280 just make up problems. 00:04:38.280 --> 00:04:39.083 That's what I'm doing. 00:04:42.210 --> 00:04:44.860 I do this even when I'm not recording videos, just for fun. 00:04:47.500 --> 00:04:48.930 So let's take the derivative of this. 00:04:48.930 --> 00:04:50.840 Minus 4 times this coefficient. 00:04:50.840 --> 00:04:54.910 Minus 4 times minus 3/4 is 3x. 00:04:54.910 --> 00:04:59.740 Then we subtract 1 from this exponent, minus 5. 00:04:59.740 --> 00:05:09.090 And then 4 times 4 is minus 7 x to the-- we take 1 from this 00:05:09.090 --> 00:05:10.300 exponent-- x to the third. 00:05:10.300 --> 00:05:11.920 And I promise you I'm not even looking up here. 00:05:11.920 --> 00:05:14.000 I know you might think, well Sal, he's just looking up here, 00:05:14.000 --> 00:05:17.250 but no I'm actually in my head at least working through this. 00:05:17.250 --> 00:05:19.540 And then plus the derivative of 3x. 00:05:19.540 --> 00:05:23.140 Well the derivative of 3x is 3-- is almost second nature 00:05:23.140 --> 00:05:25.510 now, but you can kind of do this-- is 3x to the 1. 00:05:25.510 --> 00:05:31.820 And you say 1 times 3 is 3 times x to the 0. 00:05:31.820 --> 00:05:35.400 And then 10 times minus 1/10. 00:05:35.400 --> 00:05:37.220 Well that's just minus 1. 00:05:37.220 --> 00:05:43.670 x to the 1 less than 10, so x to the ninth, plus-- what's the 00:05:43.670 --> 00:05:45.690 derivative of any constant? 00:05:45.690 --> 00:05:48.330 Right, it's 0. 00:05:48.330 --> 00:05:51.810 You could almost do this constant as some number 00:05:51.810 --> 00:05:53.850 times x to the 0. 00:05:53.850 --> 00:05:56.310 And if you took the derivative, well you multiply the 0 00:05:56.310 --> 00:05:58.600 times c and you get 0. 00:05:58.600 --> 00:06:00.130 Well, you might get minus 1 depending on how 00:06:00.130 --> 00:06:00.520 you're doing it. 00:06:00.520 --> 00:06:02.060 But that's actually kind of an interesting question. 00:06:02.060 --> 00:06:04.120 OK I'll stop digressing. 00:06:04.120 --> 00:06:06.190 But you get a 0 here, and if you simplify that, that just 00:06:06.190 --> 00:06:12.450 equals 3x to the minus 5 minus 7x to the third plus 00:06:12.450 --> 00:06:15.720 3 minus x to the ninth. 00:06:15.720 --> 00:06:18.480 Think we have time for one more problem like this. 00:06:18.480 --> 00:06:19.630 I think you probably got this. 00:06:19.630 --> 00:06:21.890 This is probably one of the more straightforward things 00:06:21.890 --> 00:06:22.970 you'll learn in mathematics. 00:06:22.970 --> 00:06:25.200 And in future presentations I'll give you more of an 00:06:25.200 --> 00:06:30.400 intuition of why the antiderivative is useful. 00:06:30.400 --> 00:06:32.580 We're learning the indefinite integral, but we could learn to 00:06:32.580 --> 00:06:35.045 use the definite integral, which we'll learn in a couple 00:06:35.045 --> 00:06:38.500 of presentations to figure out things like the area under 00:06:38.500 --> 00:06:41.050 curve, or the volume of a rotational figure. 00:06:41.050 --> 00:06:42.800 Well I don't confuse you too much. 00:06:42.800 --> 00:06:45.630 Let's do one more problem. 00:06:45.630 --> 00:06:47.820 I won't make this one as hairy. 00:06:47.820 --> 00:06:55.870 So the integral of negative 1/2x to the minus 3 00:06:55.870 --> 00:06:59.760 plus 7x to the fifth. 00:07:06.350 --> 00:07:09.340 Let's start with this term of the polynomial. 00:07:09.340 --> 00:07:14.790 We raise the exponent one, so x to the minus 2 now, 00:07:14.790 --> 00:07:18.710 right, because we added one to negative 3. 00:07:18.710 --> 00:07:21.620 And then we multiply 1 over this new exponent times 00:07:21.620 --> 00:07:22.940 the old coefficient. 00:07:22.940 --> 00:07:24.480 And actually I'll write out all the steps. 00:07:24.480 --> 00:07:26.170 So the old coefficient is minus 1/2. 00:07:30.395 --> 00:07:34.230 So this is a minus 2. 00:07:34.230 --> 00:07:37.930 Minus 2 so we multiply it times minus 1/2. 00:07:41.680 --> 00:07:44.000 Let me switch colors back. 00:07:44.000 --> 00:07:49.410 Plus we raise the exponent by one, x to the sixth, and we 00:07:49.410 --> 00:07:54.140 multiply the old coefficient times 1 over the new 00:07:54.140 --> 00:07:56.550 coefficient, times 1/6. 00:08:00.820 --> 00:08:02.620 And so what's the answer? 00:08:02.620 --> 00:08:05.430 Well what's minus 1/2 times minus 1/2? 00:08:05.430 --> 00:08:11.030 Well that's positive 1/4 x to the minus 2. 00:08:11.030 --> 00:08:13.350 Oh, and of course, plus c. 00:08:13.350 --> 00:08:15.820 As you can tell, this is my main source of missing 00:08:15.820 --> 00:08:20.980 points on calculus quizzes. 00:08:20.980 --> 00:08:30.790 1/4 x to the minus 2 plus 7/6 x to the sixth plus c. 00:08:30.790 --> 00:08:31.700 There you go. 00:08:31.700 --> 00:08:35.160 And if you wanted to take the derivative, minus 2 times 1/4 00:08:35.160 --> 00:08:40.420 is minus 2/4 which is minus 1/2 x to the minus 3. 00:08:40.420 --> 00:08:44.320 And then 6 times 7/6 is 7x. 00:08:44.320 --> 00:08:46.440 And then you decrease the exponent by one, 00:08:46.440 --> 00:08:47.430 x to the fifth. 00:08:47.430 --> 00:08:49.900 And the derivative of our constant is 0. 00:08:49.900 --> 00:08:53.200 And then we get our original expression. 00:08:53.200 --> 00:08:57.000 Hopefully at this point you're pretty comfortable taking a 00:08:57.000 --> 00:09:00.540 derivative of a polynomial, and then given a polynomial you can 00:09:00.540 --> 00:09:02.500 actually take the antiderivative, go 00:09:02.500 --> 00:09:03.980 the other way. 00:09:03.980 --> 00:09:06.770 And never forget to do your plus c. 00:09:06.770 --> 00:09:08.500 And I hope you understand why we have to put that constant 00:09:08.500 --> 00:09:14.460 there, because when you take an antiderivative, you don't know 00:09:14.460 --> 00:09:17.350 whether the original thing that you the derivative of I guess 00:09:17.350 --> 00:09:20.600 had a constant there, because the constant's derivative is 0. 00:09:20.600 --> 00:09:22.470 Hopefully I confused you with that last statement. 00:09:22.470 --> 00:09:24.320 I'll see you in the next presentation and I'll show you 00:09:24.320 --> 00:09:26.450 how to reverse the chain rule. 00:09:26.450 --> 00:09:26.930 See you soon.

The Indefinite Integral or Anti-derivative

https://www.youtube.com/watch?v=xRspb-iev-g

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WEBVTT Kind: captions Language: en 00:00:00.750 --> 00:00:03.210 Welcome to the presentation on the indefinite integral 00:00:03.210 --> 00:00:04.430 or the antiderivative. 00:00:04.430 --> 00:00:06.750 So let's begin with a bit of a review of the 00:00:06.750 --> 00:00:07.270 actual derivative. 00:00:07.270 --> 00:00:10.700 So if I were to take the derivative d/dx. 00:00:10.700 --> 00:00:13.450 It's just the derivative operator. 00:00:13.450 --> 00:00:16.700 If I were to take the derivative of the expression 00:00:16.700 --> 00:00:20.140 x squared-- this is an easy one if you remember the 00:00:20.140 --> 00:00:21.860 derivative presentation. 00:00:21.860 --> 00:00:23.400 Well, this is pretty straightforward. 00:00:23.400 --> 00:00:24.640 You just take the exponent. 00:00:24.640 --> 00:00:27.100 That becomes the new coefficient, right. 00:00:27.100 --> 00:00:29.040 You actually multiply it times the old coefficient, but in 00:00:29.040 --> 00:00:32.310 this case the old coefficient is 1, so 2 times 1 is 2. 00:00:32.310 --> 00:00:35.130 And you take the variable 2x. 00:00:35.130 --> 00:00:37.170 And then the new exponent will be one less than 00:00:37.170 --> 00:00:38.460 the old exponent. 00:00:38.460 --> 00:00:41.930 So it'll be 2x to the 1, or just 2x. 00:00:41.930 --> 00:00:42.580 So that was easy. 00:00:42.580 --> 00:00:46.000 If I had y equals x squared we now know that the slope at any 00:00:46.000 --> 00:00:50.240 point on that curve, it would be 2x. 00:00:50.240 --> 00:00:52.040 So what if we wanted to go the other way? 00:00:52.040 --> 00:00:55.720 Let's say if we wanted to start with 2x, and I wanted to say 00:00:55.720 --> 00:01:07.330 2x is the derivative of what. 00:01:07.330 --> 00:01:09.420 Well, we know the answer this question, right? 00:01:09.420 --> 00:01:10.755 Because we just took the derivative of x squared 00:01:10.755 --> 00:01:12.170 and we figured out 2x. 00:01:12.170 --> 00:01:14.680 But let's say we didn't know this already. 00:01:14.680 --> 00:01:18.170 You could probably figure it out intuitively, how you can 00:01:18.170 --> 00:01:21.070 kind of do this operation that we did here, how 00:01:21.070 --> 00:01:23.450 you can do it backwards. 00:01:23.450 --> 00:01:27.830 So in this case the notation-- well we know it's x squared-- 00:01:27.830 --> 00:01:31.870 but the notation for trying to figure out 2x is the derivative 00:01:31.870 --> 00:01:35.920 of what, we could say that-- let's say 2x is the 00:01:35.920 --> 00:01:39.410 derivative of y. 00:01:39.410 --> 00:01:43.050 So 2x is the derivative of y. 00:01:43.050 --> 00:01:46.150 Let's get rid of this of what. 00:01:46.150 --> 00:01:47.270 Then we can say this. 00:01:47.270 --> 00:01:51.260 We can say that y is equal to-- and I'm going to throw some 00:01:51.260 --> 00:01:55.990 very fancy notation at you and actually I'll explain why we 00:01:55.990 --> 00:01:59.500 use this notation in a couple presentations down the road. 00:01:59.500 --> 00:02:01.680 But you just have to know at this point what the notation 00:02:01.680 --> 00:02:03.925 means or what it tells you to really do, which really is 00:02:03.925 --> 00:02:06.070 just the antiderivative or the indefinite integral. 00:02:06.070 --> 00:02:10.340 So we could say that y is equal to the indefinite 00:02:10.340 --> 00:02:14.350 integral 2x dx. 00:02:14.350 --> 00:02:17.220 And I'm going to explain what this squiggly line here is and 00:02:17.220 --> 00:02:20.690 dx, but all you have to know is when you see the squiggly line 00:02:20.690 --> 00:02:24.700 and this dx and then something in between, all they're asking 00:02:24.700 --> 00:02:28.370 is they want you to figure out what the antiderivative 00:02:28.370 --> 00:02:30.060 of this expression is. 00:02:30.060 --> 00:02:32.550 And I'll explain later why this is called the 00:02:32.550 --> 00:02:33.340 indefinite integral. 00:02:33.340 --> 00:02:36.350 And actually this notation will make a lot more sense 00:02:36.350 --> 00:02:39.970 when I show you what a definite integral is. 00:02:39.970 --> 00:02:42.000 But let's just take it for granted right now that an 00:02:42.000 --> 00:02:44.000 indefinite integral-- which I just drew here, it's kind of 00:02:44.000 --> 00:02:47.450 like a little squirrely thing-- is just the antiderivative. 00:02:47.450 --> 00:02:52.350 So y is equal to the antiderivative essentially, 00:02:52.350 --> 00:02:56.150 or the indefinite integral of the expression 2x. 00:02:56.150 --> 00:02:57.270 So what is y equal to? 00:02:57.270 --> 00:03:02.210 Well y is obviously equal to x squared. 00:03:02.210 --> 00:03:03.220 Let me ask you a question. 00:03:03.220 --> 00:03:06.830 Is y just equal to x squared? 00:03:06.830 --> 00:03:08.660 Because we took the derivative, and clearly the derivative 00:03:08.660 --> 00:03:10.575 of x squared is 2x. 00:03:10.575 --> 00:03:14.320 But what's the derivative of x squared-- what's the 00:03:14.320 --> 00:03:15.880 derivative x squared plus 1? 00:03:21.090 --> 00:03:24.500 Well, the derivative of x squared is still 2x. 00:03:24.500 --> 00:03:26.100 What's the derivative of 1? 00:03:26.100 --> 00:03:28.460 Right, derivative of 1 is 0, so it's 2x plus 00:03:28.460 --> 00:03:30.540 0, or still just 2x. 00:03:30.540 --> 00:03:37.570 Similarly, what's the derivative of x squared plus 2? 00:03:37.570 --> 00:03:39.050 Well the derivative of x squared plus 2 once 00:03:39.050 --> 00:03:42.620 again is 2x plus 0. 00:03:42.620 --> 00:03:45.200 So notice the derivative of x squared plus 00:03:45.200 --> 00:03:47.890 any constant is 2x. 00:03:47.890 --> 00:03:52.390 So really y could be x squared plus any constant. 00:03:52.390 --> 00:03:55.420 And for any constant we put a big c there. 00:03:55.420 --> 00:03:56.960 So x squared plus c. 00:03:56.960 --> 00:03:59.100 And you'll meet many calculus teachers that will mark this 00:03:59.100 --> 00:04:01.600 problem wrong if you forget to put the plus c when you do 00:04:01.600 --> 00:04:03.340 an indefinite integral. 00:04:03.340 --> 00:04:07.360 So you're saying Sal, OK, you've showed me some notation, 00:04:07.360 --> 00:04:10.880 you've reminded me that the derivative of any constant 00:04:10.880 --> 00:04:14.640 number is 0, but this really doesn't help you solve 00:04:14.640 --> 00:04:15.270 an indefinite integral. 00:04:15.270 --> 00:04:18.950 Well let's think about a way-- a systematic way if I didn't do 00:04:18.950 --> 00:04:21.200 it for you already-- that we could solve an 00:04:21.200 --> 00:04:23.480 indefinite integral. 00:04:23.480 --> 00:04:24.540 Let me clear this. 00:04:30.440 --> 00:04:33.615 A bolder color I think would make this more interesting. 00:04:36.300 --> 00:04:45.300 Let's say we said y is equal to the indefinite integral of-- 00:04:45.300 --> 00:04:47.220 let me throw something interesting in there. 00:04:47.220 --> 00:04:54.350 Let's say the indefinite integral of x cubed dx. 00:04:54.350 --> 00:04:58.510 So we want to figure out some function whose derivative 00:04:58.510 --> 00:05:01.470 is x to the third. 00:05:01.470 --> 00:05:02.620 Well how can we figure that out? 00:05:02.620 --> 00:05:05.540 Well just from your intuition, you probably think, well it's 00:05:05.540 --> 00:05:10.420 probably something times x to the something, right? 00:05:10.420 --> 00:05:19.116 So let's say that y is equal to a x to the n. 00:05:19.116 --> 00:05:27.910 So then what is dy/dx, or the derivative of y is n. 00:05:27.910 --> 00:05:29.390 Well we learned this in the derivative module. 00:05:29.390 --> 00:05:32.320 You take the exponent, multiply it by the coefficient. 00:05:32.320 --> 00:05:34.480 So it's a times n. 00:05:37.890 --> 00:05:42.820 And then it's x to the n minus 1. 00:05:42.820 --> 00:05:46.810 Well in this situation we're saying that x to the third is 00:05:46.810 --> 00:05:50.330 this expression, it's the derivative of y. 00:05:50.330 --> 00:05:52.500 This is equal to x to the third. 00:05:52.500 --> 00:05:58.220 So if this is equal to x to third, what's a and what's n. 00:05:58.220 --> 00:06:00.360 Well, n is easy to figure out. 00:06:00.360 --> 00:06:02.670 n minus 1 is equal to 3. 00:06:02.670 --> 00:06:07.430 So that means that n is equal to 4. 00:06:07.430 --> 00:06:10.190 And then what is a equal to? 00:06:10.190 --> 00:06:14.770 Well a times n is equal to 1, right, because we just have a 1 00:06:14.770 --> 00:06:18.410 in this coefficient, this has a starting coefficient of 1. 00:06:18.410 --> 00:06:20.255 So a times n is 1. 00:06:20.255 --> 00:06:23.210 If n is 4, than a must be 1/4. 00:06:26.206 --> 00:06:30.780 So just using this definition of a derivative, I think we now 00:06:30.780 --> 00:06:33.340 figured out what y is equal to. 00:06:33.340 --> 00:06:41.620 y is equal to 1/4 x to the fourth. 00:06:41.620 --> 00:06:44.220 I think you might start seeing a pattern here. 00:06:44.220 --> 00:06:46.230 Well how did we get from x to the third to 00:06:46.230 --> 00:06:47.640 1/4 x to the fourth? 00:06:47.640 --> 00:06:51.940 Well, we increased the exponent by 1, and whatever the new 00:06:51.940 --> 00:06:56.050 exponent is, we multiply it times 1 over that new exponent. 00:06:56.050 --> 00:06:59.920 So let's think if we can do a generalized rule here. 00:07:02.870 --> 00:07:05.810 Oh, and of course, plus c. 00:07:05.810 --> 00:07:08.360 I would have failed this exam. 00:07:08.360 --> 00:07:13.260 So let's make a general rule that if I have the integral 00:07:13.260 --> 00:07:18.210 of-- well, since we already used a, let's say-- b 00:07:18.210 --> 00:07:23.670 times x to the n dx. 00:07:23.670 --> 00:07:24.650 What is this integral? 00:07:24.650 --> 00:07:27.420 This is an integral sign. 00:07:27.420 --> 00:07:33.630 Well my new rule is, I raise the exponent on x by 1, so it's 00:07:33.630 --> 00:07:36.880 going to be x to the n plus 1. 00:07:36.880 --> 00:07:40.980 And then I multiply x times the inverse of this number. 00:07:40.980 --> 00:07:45.380 So times 1 over n plus 1. 00:07:45.380 --> 00:07:47.580 And of course I had that b there all the time. 00:07:47.580 --> 00:07:50.310 And one day I'll do a more vigorous-- more rigorous proof 00:07:50.310 --> 00:07:53.520 and maybe it will be vigorous as well-- as to why this b 00:07:53.520 --> 00:07:56.490 just stays multiplying. 00:07:56.490 --> 00:07:59.390 Actually I don't have to do too rigorous of a proof if you just 00:07:59.390 --> 00:08:04.030 remember how a derivative is done, you just multiply this 00:08:04.030 --> 00:08:05.830 times the exponent minus 1. 00:08:05.830 --> 00:08:10.190 So here we multiply the coefficient times 1 over 00:08:10.190 --> 00:08:11.530 the exponent plus 1. 00:08:11.530 --> 00:08:13.630 It's just the inverse operation. 00:08:13.630 --> 00:08:16.460 So let's do a couple of examples like this really fast. 00:08:16.460 --> 00:08:18.820 I have a little time left. 00:08:18.820 --> 00:08:22.420 I think the examples, at least for me, really 00:08:22.420 --> 00:08:23.200 hit the point home. 00:08:23.200 --> 00:08:25.520 So let's say I wanted to figure out the integral 00:08:25.520 --> 00:08:31.310 of 5 x to the seventh dx. 00:08:31.310 --> 00:08:35.850 Well, I take the exponent, increase it by one. 00:08:35.850 --> 00:08:39.910 So I get x to the eighth, and then I multiply the coefficient 00:08:39.910 --> 00:08:42.100 times 1 over the new exponent. 00:08:42.100 --> 00:08:45.920 So it's 5/8 x to the eighth. 00:08:45.920 --> 00:08:48.250 And if you don't trust me, take the derivative of this. 00:08:48.250 --> 00:08:56.740 Take the derivative d/dx of 5/8 x to the eighth. 00:08:56.740 --> 00:08:59.970 Well you multiply 8 times 5/8. 00:08:59.970 --> 00:09:04.450 Well that equals 5 x to the-- and now the new exponent will 00:09:04.450 --> 00:09:08.600 be 8 minus 1-- 5 x to the seventh. 00:09:08.600 --> 00:09:10.880 Oh, and of course, plus c. 00:09:10.880 --> 00:09:13.090 Don't want to forget the plus c. 00:09:13.090 --> 00:09:15.680 So I think you have a sense of how this works. 00:09:15.680 --> 00:09:17.990 In the next presentation I'm going to do a bunch more 00:09:17.990 --> 00:09:19.960 examples, and I'll also show you how to kind of 00:09:19.960 --> 00:09:21.320 reverse the chain rule. 00:09:21.320 --> 00:09:23.270 And then we'll learn integration by parts, which is 00:09:23.270 --> 00:09:25.720 essentially just reversing the product rule. 00:09:25.720 --> 00:09:26.330 See you in the next presentation.

Derivatives (part 9)

https://www.youtube.com/watch?v=aEP4C_kvcO4

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en

WEBVTT Kind: captions Language: en 00:00:00.970 --> 00:00:03.630 Now that you've been introduced into some of the other 00:00:03.630 --> 00:00:05.665 functions that we can take a derivative of, we can now 00:00:05.665 --> 00:00:07.700 apply them using the chain and the product rule. 00:00:07.700 --> 00:00:10.770 So let's do some fun derivatives. 00:00:10.770 --> 00:00:12.750 And I think derivatives is all about exposure, it's 00:00:12.750 --> 00:00:13.690 all about practice. 00:00:13.690 --> 00:00:15.780 So I just encourage you to do as much practice as possible. 00:00:15.780 --> 00:00:19.540 And it's actually in some ways a pretty mechanical thing to do 00:00:19.540 --> 00:00:21.190 and it's easier than a lot of the math that you've 00:00:21.190 --> 00:00:21.740 learned before. 00:00:21.740 --> 00:00:24.360 Just maybe initially looks a little abstract. 00:00:24.360 --> 00:00:37.590 So let's say that f of x is equal to let's say the sin 00:00:37.590 --> 00:00:43.910 of 3x to the fifth plus 2x. 00:00:43.910 --> 00:00:45.520 So what is f prime of x? 00:00:45.520 --> 00:00:48.940 What is the derivative of this function? 00:00:48.940 --> 00:00:50.410 Well, we use the chain rule again. 00:00:50.410 --> 00:00:52.860 We take the derivative of the inside. 00:00:52.860 --> 00:00:54.140 So what's the derivative of the inside? 00:00:54.140 --> 00:01:03.800 Well, that's just 5 times 3 is 15 x to the fourth plus 2. 00:01:03.800 --> 00:01:06.100 And then we take the derivative of the larger function. 00:01:06.100 --> 00:01:08.030 In the last presentation we learned the derivative 00:01:08.030 --> 00:01:09.700 of sin is what? 00:01:09.700 --> 00:01:10.600 It's cosine. 00:01:10.600 --> 00:01:17.370 So it's times the cosine of this expression right here. 00:01:17.370 --> 00:01:23.010 3x to the fifth plus 2x. 00:01:23.010 --> 00:01:24.330 Pretty painless, no? 00:01:24.330 --> 00:01:25.380 Let's mix it up even more. 00:01:25.380 --> 00:01:28.030 Let's say that-- let me switch colors just to 00:01:28.030 --> 00:01:30.700 not be monotonous. 00:01:30.700 --> 00:01:32.290 I'll pick powder blue. 00:01:32.290 --> 00:01:33.530 Very nice. 00:01:33.530 --> 00:01:36.450 Let's say that y-- and I'm going to switch notation on 00:01:36.450 --> 00:01:38.700 purpose that you get used to the various notations 00:01:38.700 --> 00:01:40.150 you can use. 00:01:40.150 --> 00:01:44.310 Let's say that y is equal to-- let me think of something 00:01:44.310 --> 00:01:59.580 good-- e to the x times cosine to the fifth of x. 00:01:59.580 --> 00:02:01.460 That looks daunting to me. 00:02:01.460 --> 00:02:03.570 Let's see if we can break it down using the product 00:02:03.570 --> 00:02:05.550 and the chain rules. 00:02:05.550 --> 00:02:07.760 We want to figure out dy/dx. 00:02:07.760 --> 00:02:10.740 We want to figure out the rate at which y 00:02:10.740 --> 00:02:13.680 changes relative to x. 00:02:13.680 --> 00:02:15.500 Or the derivative. 00:02:15.500 --> 00:02:16.750 Find the derivative of both sides. 00:02:16.750 --> 00:02:19.800 Well, let's use the product rule. 00:02:19.800 --> 00:02:20.830 Well, we're going to have to use the chain and 00:02:20.830 --> 00:02:22.130 the product rules. 00:02:22.130 --> 00:02:26.060 So first we take the derivative of this first term, and once 00:02:26.060 --> 00:02:28.320 again we learned in the last presentation the most amazing 00:02:28.320 --> 00:02:31.070 fact, one of the most amazing facts in the universe that the 00:02:31.070 --> 00:02:33.670 derivative of either the x is what? 00:02:33.670 --> 00:02:36.480 It is e to the x. 00:02:36.480 --> 00:02:38.490 Blows my mind. 00:02:38.490 --> 00:02:40.330 e to the x. 00:02:40.330 --> 00:02:42.560 Once again I've taken the derivative, and it's 00:02:42.560 --> 00:02:43.720 the same expression. 00:02:43.720 --> 00:02:45.050 Amazing. 00:02:45.050 --> 00:02:48.070 And then I multiply it times a second expression. 00:02:48.070 --> 00:02:51.420 Cosine to the fifth of x. 00:02:51.420 --> 00:02:54.800 And now to that I add the derivative of the 00:02:54.800 --> 00:02:55.730 second expression. 00:02:55.730 --> 00:02:57.500 Now this will be a little bit more interesting. 00:02:57.500 --> 00:03:03.150 So this is cosine of x to the fifth. 00:03:03.150 --> 00:03:05.780 This is just another way of writing cosine of 00:03:05.780 --> 00:03:08.310 x to the fifth power. 00:03:08.310 --> 00:03:09.930 And I think that'll make it a little bit more clear, that 00:03:09.930 --> 00:03:13.050 this cosine superscript 5, this is really just 00:03:13.050 --> 00:03:14.120 cosine to the fifth x. 00:03:14.120 --> 00:03:15.980 This means cosine of x to the fifth. 00:03:15.980 --> 00:03:17.730 So now the derivative is a little bit clearer. 00:03:17.730 --> 00:03:19.750 We can use the chain rule-- and once again, we're just 00:03:19.750 --> 00:03:21.270 working on this right half. 00:03:21.270 --> 00:03:23.060 We take the derivative of the inside. 00:03:23.060 --> 00:03:25.480 What's the derivative of cosine of x? 00:03:25.480 --> 00:03:26.760 Yep you're right. 00:03:26.760 --> 00:03:28.230 Well, I don't know, I didn't hear you so I don't know. 00:03:28.230 --> 00:03:29.175 I'll assume you're right. 00:03:29.175 --> 00:03:32.070 The derivative of cosine of x is minus sin of x, and that's 00:03:32.070 --> 00:03:33.526 something you should memorize, although you should prove 00:03:33.526 --> 00:03:35.085 it to yourself as well. 00:03:35.085 --> 00:03:39.360 So we take the derivative of the inside minus sin. 00:03:39.360 --> 00:03:41.680 Derivative of cosine of x is minus sin of x, and then we 00:03:41.680 --> 00:03:42.900 take the derivative of the outside. 00:03:42.900 --> 00:03:44.220 We're just doing the chain rule. 00:03:44.220 --> 00:03:50.190 So it's 5 cosine to the fourth of x. 00:03:50.190 --> 00:03:54.640 So there we took the derivative of this piece, and then we have 00:03:54.640 --> 00:03:57.830 to multiply times this first piece. 00:03:57.830 --> 00:04:02.340 So that times e to the x. 00:04:02.340 --> 00:04:03.130 Interesting. 00:04:03.130 --> 00:04:05.230 You can simplify this if you want, but you get the point. 00:04:05.230 --> 00:04:07.030 I mean simplifying it from this point is really 00:04:07.030 --> 00:04:08.460 just kind of algebra. 00:04:08.460 --> 00:04:09.990 And I think you get the idea. 00:04:09.990 --> 00:04:12.590 Actually everything we're doing is algebra. 00:04:12.590 --> 00:04:14.340 If you realize it looks like something fairly complicated, 00:04:14.340 --> 00:04:16.870 but we just use the chain and the product rules. 00:04:16.870 --> 00:04:17.850 Let's do some more. 00:04:22.030 --> 00:04:23.600 I will now switch to magenta. 00:04:26.130 --> 00:04:31.260 We want to take the derivative dy/dx of-- let's see, 00:04:31.260 --> 00:04:32.300 some big expression. 00:04:32.300 --> 00:04:34.570 let me do something creative. 00:04:34.570 --> 00:04:49.920 Let's say the natural log of x over 3x plus 10. 00:04:54.830 --> 00:04:57.500 So the natural log of x over 3x plus 10. 00:04:57.500 --> 00:05:00.470 So you could use the quotient rule if you took the time to 00:05:00.470 --> 00:05:02.770 memorize it, which I've never taught you because it's really 00:05:02.770 --> 00:05:03.550 just the product rule. 00:05:03.550 --> 00:05:06.240 So I like to just rewrite this as the product rule. 00:05:06.240 --> 00:05:08.310 So they're the same thing. 00:05:08.310 --> 00:05:10.280 Once again we're taking the derivative, so I'm not going to 00:05:10.280 --> 00:05:12.830 keep rewriting this, but this is the same thing as taking the 00:05:12.830 --> 00:05:21.825 derivative of the natural log of x times 3x plus ten to 00:05:21.825 --> 00:05:23.700 the negative 1 power. 00:05:23.700 --> 00:05:26.695 3x plus 10 in the denominator is the same thing as 1 over 3x 00:05:26.695 --> 00:05:29.720 plus 10, which is the same thing as 3x plus 10 to 00:05:29.720 --> 00:05:30.990 the negative 1 power. 00:05:30.990 --> 00:05:33.610 Now we can use the combination of the product and the 00:05:33.610 --> 00:05:35.800 chain rules, and we can solve this sucker. 00:05:35.800 --> 00:05:38.160 So let's do it. 00:05:38.160 --> 00:05:44.150 So we take the derivative of this first term the natural log 00:05:44.150 --> 00:05:46.142 of x-- and we learned in the last presentation the 00:05:46.142 --> 00:05:49.330 derivative of the natural log of x is 1/x, which is 00:05:49.330 --> 00:05:50.980 pretty cool in of itself. 00:05:50.980 --> 00:05:53.560 And we multiply that times a second term. 00:05:53.560 --> 00:06:00.480 So time 3x plus 10 to the negative 1 power. 00:06:00.480 --> 00:06:03.270 And to that we add the derivative of the second term, 00:06:03.270 --> 00:06:06.430 and we're going to multiply that times the first term. 00:06:06.430 --> 00:06:08.370 So first we're going to have to use the chain rule. 00:06:08.370 --> 00:06:09.710 We take the derivative of the inside. 00:06:09.710 --> 00:06:10.800 Well the derivative of the inside's easy. 00:06:10.800 --> 00:06:12.990 The derivative of 3x x plus 10. 00:06:12.990 --> 00:06:13.900 That's just 3. 00:06:16.880 --> 00:06:18.750 And then we take the derivative of the whole thing, 00:06:18.750 --> 00:06:22.010 so it's negative 1. 00:06:22.010 --> 00:06:25.930 That's 3 times negative 1 times that whole 00:06:25.930 --> 00:06:28.340 expression to the minus 2. 00:06:28.340 --> 00:06:30.800 3x plus 10. 00:06:30.800 --> 00:06:35.420 And of course this whole thing times the natural log of x. 00:06:35.420 --> 00:06:36.980 We could simplify that. 00:06:36.980 --> 00:06:37.290 Let's see. 00:06:37.290 --> 00:06:40.170 This is 1/x and 3x plus 10 to negative 1. 00:06:40.170 --> 00:06:48.090 So we could rewrite this as 1 over x 3x plus 10. 00:06:48.090 --> 00:06:49.200 Let's see. 00:06:49.200 --> 00:06:51.550 Plus 3 times minus 1. 00:06:51.550 --> 00:06:57.808 So we could say that's the same thing as minus 3 ln of 00:06:57.808 --> 00:07:03.910 x over 3x plus 10 squared. 00:07:03.910 --> 00:07:05.620 I think you see how I got from here to here. 00:07:05.620 --> 00:07:08.180 I just manipulated the exponents and multiplied some 00:07:08.180 --> 00:07:11.760 numbers, et cetera, et cetera. 00:07:11.760 --> 00:07:13.080 Let's do one more. 00:07:13.080 --> 00:07:14.250 Just hit the point home. 00:07:14.250 --> 00:07:18.710 You really have the tools now at your disposal to do a 00:07:18.710 --> 00:07:21.180 lot of derivative problems. 00:07:21.180 --> 00:07:23.640 Probably most of the derivative problems you'll face in the 00:07:23.640 --> 00:07:26.770 first 1/2 year of calculus. 00:07:26.770 --> 00:07:28.870 I'm going to switch to green. 00:07:28.870 --> 00:07:32.720 Let's say y-- actually I'm tired of y. 00:07:32.720 --> 00:07:44.550 Let's say that p is equal to-- I don't know. 00:07:44.550 --> 00:07:51.510 Sin of x over cosine of x. 00:07:51.510 --> 00:07:52.996 Let's figure out what dp/dx. 00:07:55.580 --> 00:07:58.110 The rate at which p changes to x. 00:07:58.110 --> 00:08:04.510 So once again this is the same thing as sin of x times cosine 00:08:04.510 --> 00:08:07.600 of x to the negative 1. 00:08:07.600 --> 00:08:10.140 So we can just do the product and chain rules. 00:08:10.140 --> 00:08:12.200 So the derivative of the first term. 00:08:12.200 --> 00:08:16.370 Derivative of sin of x is cosine of x, times 00:08:16.370 --> 00:08:17.990 the second term. 00:08:17.990 --> 00:08:23.270 Times cosine of x to the minus 1. 00:08:23.270 --> 00:08:25.560 And then to that we add the derivative of the second term. 00:08:25.560 --> 00:08:27.460 We have to use the chain real here. 00:08:27.460 --> 00:08:29.590 So we take the derivative of the inside. 00:08:29.590 --> 00:08:35.070 Well derivative of cosine x is minus sin of x. 00:08:35.070 --> 00:08:37.910 And then times the derivative of the outside. 00:08:37.910 --> 00:08:46.580 well that's minus 1 cosine of x to the minus 2, and then we 00:08:46.580 --> 00:08:50.810 multiply that times the first term, sin of x. 00:08:50.810 --> 00:08:51.890 So let's simplify that. 00:08:51.890 --> 00:08:56.080 So this is cosine of x divided by cosine of x. 00:08:56.080 --> 00:08:57.320 You see how this cancels out? 00:08:57.320 --> 00:09:01.830 This is cosine of x over cosine of x, so this is equal to 1. 00:09:01.830 --> 00:09:04.160 Cosine of x divided cosine of x is 1. 00:09:04.160 --> 00:09:11.830 And then this minus sin cancels out with this minus sin, and 00:09:11.830 --> 00:09:16.165 we have sin times sin over cosine squared. 00:09:16.165 --> 00:09:17.480 So this is equal to 1. 00:09:17.480 --> 00:09:19.440 I'm going kind of fast because I'm about to run out of time, 00:09:19.440 --> 00:09:20.720 but I think you get what I'm doing. 00:09:20.720 --> 00:09:28.470 So this is sin squared x over cosine squared x, which 00:09:28.470 --> 00:09:32.430 is actually equal to-- what's sin over cosine? 00:09:32.430 --> 00:09:36.800 1 plus 10 squared x. 00:09:36.800 --> 00:09:40.480 And if you know your trig identities, that equals 1 00:09:40.480 --> 00:09:43.980 over cosine squared of x. 00:09:43.980 --> 00:09:46.890 And, of course, what we just proved is that the derivative 00:09:46.890 --> 00:09:51.200 of the tangent of x is equal to the secant squared of x. 00:09:51.200 --> 00:09:53.930 I hope I thoroughly confused you in that last problem. 00:09:53.930 --> 00:09:54.730 I'll see you in the next presentation.

Quotient rule and common derivatives

https://www.youtube.com/watch?v=E_1gEtiGPNI

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en

WEBVTT Kind: captions Language: en 00:00:00.800 --> 00:00:01.470 Welcome back. 00:00:01.470 --> 00:00:03.400 Let's do some more derivative problems. 00:00:03.400 --> 00:00:11.910 Let's say I want to figure out the derivative d over dx of-- 00:00:11.910 --> 00:00:15.856 and let me give something that looks a little bit different-- 00:00:15.856 --> 00:00:27.070 x to the third minus 5x to the fifth, all of that to the third 00:00:27.070 --> 00:00:41.820 power over 2x plus 5 to the fifth power. 00:00:41.820 --> 00:00:42.850 This is a parentheses. 00:00:42.850 --> 00:00:44.610 This is just saying that I want to take the derivative of 00:00:44.610 --> 00:00:45.840 this entire expression. 00:00:45.840 --> 00:00:48.080 So you're saying Sal, we've never learn how to do this, 00:00:48.080 --> 00:00:50.090 you have something in the numerator, you have something 00:00:50.090 --> 00:00:51.710 in the denominator I don't know what to do next. 00:00:51.710 --> 00:00:53.050 Well let's just rewrite this. 00:00:53.050 --> 00:00:55.310 Actually in your calculus textbooks there's something 00:00:55.310 --> 00:00:59.350 called the quotient rule, which I think is mildly lame, because 00:00:59.350 --> 00:01:03.290 the quotient rule is just the product rule where you have a 00:01:03.290 --> 00:01:05.860 negative exponent and they make it another rule, and they 00:01:05.860 --> 00:01:06.720 clutter your brain. 00:01:06.720 --> 00:01:08.470 So instead of using the quotient rule, we're just 00:01:08.470 --> 00:01:11.780 going to rewrite this bottom expression as a product, 00:01:11.780 --> 00:01:13.130 and then we can use the product rule. 00:01:13.130 --> 00:01:17.500 So this is the same thing as taking the derivative of x to 00:01:17.500 --> 00:01:22.830 the third minus 5x to the fifth, all of that to the third 00:01:22.830 --> 00:01:28.120 power, times 2x plus 5 to what? 00:01:28.120 --> 00:01:30.380 The minus fifth power. 00:01:30.380 --> 00:01:32.060 And now we can use the product rule. 00:01:32.060 --> 00:01:35.140 Take the derivative of the first term-- and the derivative 00:01:35.140 --> 00:01:37.220 of the first term isn't a joke-- you take the derivative 00:01:37.220 --> 00:01:39.190 of the inside first, let's do the chain rule, derivative 00:01:39.190 --> 00:01:40.760 of the inside first. 00:01:40.760 --> 00:01:52.010 That is 3x squared minus 25x to the fourth times the derivative 00:01:52.010 --> 00:01:57.700 of the outside, 3 times this entire expression x to the 00:01:57.700 --> 00:02:02.580 third minus 5x to the fifth. 00:02:02.580 --> 00:02:04.610 And then all of that, take this exponent down one to the 00:02:04.610 --> 00:02:08.510 squared, and then multiply it times this whole term. 00:02:08.510 --> 00:02:14.630 So 2x plus 5 to the minus fifth. 00:02:14.630 --> 00:02:17.240 And then to that we add the derivative of 00:02:17.240 --> 00:02:19.570 this term, so plus. 00:02:19.570 --> 00:02:22.400 So the derivative of this term we take the derivative of the 00:02:22.400 --> 00:02:23.600 inside, which is pretty easy. 00:02:23.600 --> 00:02:29.060 It's just 2 times the derivative of the outside, 00:02:29.060 --> 00:02:31.065 which is minus 5. 00:02:31.065 --> 00:02:33.620 And just so you know I didn't skip a step, the derivative of 00:02:33.620 --> 00:02:36.610 2x plus 5, the derivative of 2x is 2, derivative of 5 is 0. 00:02:36.610 --> 00:02:38.270 So the derivative of 2x plus 5 is just 2. 00:02:38.270 --> 00:02:42.990 So it's 2 times minus 5 2x plus 5. 00:02:42.990 --> 00:02:46.410 We just keep that the same to the minus fifth power, and then 00:02:46.410 --> 00:02:50.360 we multiply it times this first expression, x to the third 00:02:50.360 --> 00:02:55.340 minus 5x to the fifth to the third power. 00:02:55.340 --> 00:02:57.270 I know that's really messy and you'll probably not see 00:02:57.270 --> 00:02:59.570 problems this messy, but I just wanted to show you that the 00:02:59.570 --> 00:03:02.630 product rule we learned-- it's actually the product and the 00:03:02.630 --> 00:03:04.430 chain rule-- they can apply to a lot of different problems, 00:03:04.430 --> 00:03:06.696 and even though you hadn't seen something like this where you 00:03:06.696 --> 00:03:08.880 had numerator and a denominator, you can easily 00:03:08.880 --> 00:03:12.150 rewrite what you had in the denominator as a 00:03:12.150 --> 00:03:13.190 negative exponent. 00:03:13.190 --> 00:03:15.080 And then of course it's just the product for when you don't 00:03:15.080 --> 00:03:19.090 have to memorize that silly thing called the quotient rule. 00:03:19.090 --> 00:03:21.460 So with that out of the way, I'm now going to introduce you 00:03:21.460 --> 00:03:23.450 to some common derivatives of other functions. 00:03:23.450 --> 00:03:29.840 And these things are actually normally included in the inside 00:03:29.840 --> 00:03:31.340 cover of your calculus book, and they're just good to 00:03:31.340 --> 00:03:33.820 know, good things to know. 00:03:33.820 --> 00:03:36.030 And maybe in a later presentation I'll actually 00:03:36.030 --> 00:03:37.000 prove these things. 00:03:37.000 --> 00:03:38.710 You should never take things at face value. 00:03:38.710 --> 00:03:42.290 So you should to some degree memorize these, although you 00:03:42.290 --> 00:03:44.640 should prove it to yourself first. 00:03:44.640 --> 00:03:49.140 So the derivative of e to the x-- and I find this to be 00:03:49.140 --> 00:03:52.855 amazing. e shows up all sorts of crazy places in mathematics, 00:03:52.855 --> 00:03:56.770 and it's you know the strange number 2.7 whatever, whatever 00:03:56.770 --> 00:03:58.560 and it has all sorts of strange properties. 00:03:58.560 --> 00:04:02.660 And I think this is one of the most bizarre properties of e. 00:04:02.660 --> 00:04:03.930 The derivative of e to the x. 00:04:03.930 --> 00:04:07.880 So if I want to figure out the slope of any point along 00:04:07.880 --> 00:04:11.820 the curve e to the x-- this just might blow your mind. 00:04:11.820 --> 00:04:13.610 I think the more you think about it, the more it'll blow 00:04:13.610 --> 00:04:17.010 your mind-- is e to the x. 00:04:17.010 --> 00:04:17.950 That's amazing. 00:04:17.950 --> 00:04:21.810 At any point along the curve e to the x, the slope of 00:04:21.810 --> 00:04:25.900 that point is e to the x. 00:04:25.900 --> 00:04:27.230 Just to hit the point home. 00:04:27.230 --> 00:04:29.150 I'm diverging, a little bit. 00:04:29.150 --> 00:04:34.640 But if I said f of x is equal to e to the x, right? 00:04:34.640 --> 00:04:39.590 And let's say f of 2 is equal to e squared. 00:04:39.590 --> 00:04:44.540 And I asked you, friend-- I don't know your name-- what 00:04:44.540 --> 00:04:52.010 is the slope of e to the x at the point 2,e squared. 00:04:52.010 --> 00:04:54.540 And you could say Sal, the slope at that 00:04:54.540 --> 00:04:57.510 point is e squared. 00:04:57.510 --> 00:05:01.720 That blows my mind that it's a function where the slope at 00:05:01.720 --> 00:05:04.960 any point on that line is equal to the function. 00:05:04.960 --> 00:05:07.570 And it's e. e shows up all sorts of places. 00:05:07.570 --> 00:05:11.070 I might do a whole series of presentations called the 00:05:11.070 --> 00:05:15.530 magic of e, because e shows up all over the place. 00:05:15.530 --> 00:05:17.360 Well I don't want to diverge too much, so that's 00:05:17.360 --> 00:05:18.530 pretty amazing. 00:05:18.530 --> 00:05:21.760 Next I'm going to show you what I think is probably the second 00:05:21.760 --> 00:05:24.980 most amazing derivative-- and I don't think this has been fully 00:05:24.980 --> 00:05:27.740 explored in mathematics yet, because this also blows my 00:05:27.740 --> 00:05:34.870 mind-- is that the derivative of the natural log of x, right. 00:05:34.870 --> 00:05:37.980 So the natural log is just the logarithm with base e, and I 00:05:37.980 --> 00:05:39.440 hope you remember your logarithms. 00:05:39.440 --> 00:05:41.220 So what's the derivative of the natural log of x? 00:05:41.220 --> 00:05:43.480 So once again this is e related. 00:05:43.480 --> 00:05:46.430 Well it's 1/x. 00:05:46.430 --> 00:05:48.730 That also blows my mind. 00:05:48.730 --> 00:05:50.800 Because think about it. 00:05:50.800 --> 00:05:52.270 Let's draw a bunch of functions. 00:05:52.270 --> 00:06:00.940 If I said the derivative of x to the minus 3 is 00:06:00.940 --> 00:06:04.460 minus 3x to the minus 4. 00:06:04.460 --> 00:06:08.690 The derivative of x to the minus 2 is minus 00:06:08.690 --> 00:06:14.480 2x to the minus 3. 00:06:14.480 --> 00:06:20.850 The derivative of x to the minus 1 is minus 00:06:20.850 --> 00:06:23.870 1 x to the minus 2. 00:06:23.870 --> 00:06:30.750 The derivative of x to the 0-- well this is just 1, right? 00:06:30.750 --> 00:06:35.920 The derivative of x to the 0 is just 1, so the derivative is 0. 00:06:35.920 --> 00:06:39.382 The derivative of x is 1, derivative of x squared 00:06:39.382 --> 00:06:42.470 is 2x and so on, right? 00:06:42.470 --> 00:06:44.420 So it's interesting. 00:06:44.420 --> 00:06:48.130 We have this pattern from all the derivatives of all of the 00:06:48.130 --> 00:06:50.940 of kind of the exponents in increasing order where you go 00:06:50.940 --> 00:06:56.360 from x to the minus 4 x to the minus 3, x to the minus 2 and 00:06:56.360 --> 00:06:59.100 then there's no x to the minus 1 here. 00:06:59.100 --> 00:07:01.920 We go straight to x to the 0. 00:07:01.920 --> 00:07:06.040 What happened to x the minus 1? 00:07:06.040 --> 00:07:07.590 What happened to this? 00:07:07.590 --> 00:07:10.290 What function's derivative is x to the minus 1? 00:07:10.290 --> 00:07:11.860 This is bizarre to me. 00:07:11.860 --> 00:07:12.490 Where did it go? 00:07:12.490 --> 00:07:16.260 And it turns out that it's a natural log. 00:07:16.260 --> 00:07:19.260 This I still think about before I go to bed sometimes because 00:07:19.260 --> 00:07:21.830 it is kind of mind blowing. 00:07:21.830 --> 00:07:23.370 And later in another presentation I might 00:07:23.370 --> 00:07:24.240 actually prove this to you. 00:07:24.240 --> 00:07:26.370 But just to know that this is true, that the derivative of 00:07:26.370 --> 00:07:29.810 the natural log of x is 1/x I think is mind blowing. 00:07:29.810 --> 00:07:31.870 And so for now you can just memorize it. 00:07:31.870 --> 00:07:33.150 But both of these are mind blowing. 00:07:33.150 --> 00:07:35.830 The derivative of e to the x is e to the x, and the derivative 00:07:35.830 --> 00:07:38.780 of the natural log of x is 1/x. 00:07:38.780 --> 00:07:41.700 And I'll just do a couple of more just to present them to 00:07:41.700 --> 00:07:43.950 you, and then in the next presentation we'll actually use 00:07:43.950 --> 00:07:46.680 them using the product rule and the chain rule and et 00:07:46.680 --> 00:07:47.170 cetera, et cetera. 00:07:47.170 --> 00:07:50.120 And you might want to rewatch this and memorize them. 00:07:50.120 --> 00:07:53.760 I want to clear image. 00:07:53.760 --> 00:07:54.910 OK. 00:07:54.910 --> 00:07:57.920 And now I'll just do the basic trig functions, and you should 00:07:57.920 --> 00:07:59.240 memorize these as well. 00:07:59.240 --> 00:08:02.510 The derivative of sin of x-- this is pretty easy to 00:08:02.510 --> 00:08:05.140 remember-- is cosine of x. 00:08:05.140 --> 00:08:06.850 So the slope at any point along the [? line ?] 00:08:06.850 --> 00:08:09.260 sin of x is actually the cosine of that point. 00:08:09.260 --> 00:08:10.050 That's also interesting. 00:08:10.050 --> 00:08:11.460 One day I'm going to do this holographically because I 00:08:11.460 --> 00:08:14.060 think that might not be sinking in properly. 00:08:14.060 --> 00:08:19.610 The derivative of cosine of x is minus sin of x. 00:08:19.610 --> 00:08:21.710 There are good to memorize though, because you'll be 00:08:21.710 --> 00:08:24.500 able to recall is quickly on a test and then use it. 00:08:24.500 --> 00:08:31.620 And then finally the derivative of tan of x is equal to 1 over 00:08:31.620 --> 00:08:35.900 cosine square of x which you could also write as the 00:08:35.900 --> 00:08:38.580 secant squared of x. 00:08:38.580 --> 00:08:41.170 You might want to memorize these now, and actually I 00:08:41.170 --> 00:08:43.790 encourage you to explore these things, I encourage you to 00:08:43.790 --> 00:08:45.980 graph each of these functions. 00:08:45.980 --> 00:08:49.160 Graph a function, graph its derivative and look at them, 00:08:49.160 --> 00:08:53.080 and really intuitively understand why the derivative 00:08:53.080 --> 00:08:57.040 function actually does describe the slope of 00:08:57.040 --> 00:08:57.820 the original function. 00:08:57.820 --> 00:08:59.610 And actually I'll probably do a presentation on that. 00:08:59.610 --> 00:09:01.065 But I'm almost out of time in this presentation, 00:09:01.065 --> 00:09:03.950 so just memorize these. 00:09:03.950 --> 00:09:06.380 And memorize the derivative of e to the x, e to the x, and 00:09:06.380 --> 00:09:08.940 the natural log of x is 1/x. 00:09:08.940 --> 00:09:11.720 And in the next presentation we're going to start mixing and 00:09:11.720 --> 00:09:13.910 matching all of these functions, and we can use the 00:09:13.910 --> 00:09:17.490 product and chain rule on them to solve kind of arbitrarily 00:09:17.490 --> 00:09:19.520 complex derivatives. 00:09:19.520 --> 00:09:23.320 Between what we've just seen, we could probably solve 95% of 00:09:23.320 --> 00:09:26.100 the derivative problems you'll see on say the 00:09:26.100 --> 00:09:28.140 calculus AP test. 00:09:28.140 --> 00:09:28.810 I'll see you in the next presentation.

Product rule

https://www.youtube.com/watch?v=h78GdGiRmpM

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https://www.youtube.com/api/timedtext?v=h78GdGiRmpM&ei=eWeUZbGnOeiAp-oP_Z-awAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249833&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=736014D8BE9A6DB61FD34111CDB902250B19F406.1274710351309B867721C75590F98DEACCABB354&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.990 --> 00:00:02.920 Welcome back. 00:00:02.920 --> 00:00:06.530 I'm now going to introduce you to a new tool for 00:00:06.530 --> 00:00:08.370 solving derivatives. 00:00:08.370 --> 00:00:12.010 Really between this rule, which is the product rule, and the 00:00:12.010 --> 00:00:16.000 chain rule and just knowing a lot of function derivatives, 00:00:16.000 --> 00:00:19.330 you'll be ready to tackle almost any derivative problem. 00:00:19.330 --> 00:00:20.420 Let's start with the chain rule. 00:00:20.420 --> 00:00:31.810 Let's say that f of x is equal to h of x times g of x. 00:00:31.810 --> 00:00:32.780 This is the product rule. 00:00:32.780 --> 00:00:36.080 In the chain rule it was f of x is equal to h of g of x. 00:00:36.080 --> 00:00:36.310 Right? 00:00:36.310 --> 00:00:37.730 I don't know if you remember that. 00:00:37.730 --> 00:00:40.980 In this case, f of x is equal to h of x times g of x. 00:00:40.980 --> 00:00:45.570 If that's the case, then f prime of x is equal to the 00:00:45.570 --> 00:00:48.940 derivative of the first function times a second 00:00:48.940 --> 00:00:55.450 function plus the first function times the derivative 00:00:55.450 --> 00:00:57.030 of the second function. 00:00:57.030 --> 00:00:57.990 Pretty straightforward. 00:00:57.990 --> 00:00:59.350 Let's apply it. 00:00:59.350 --> 00:01:02.660 Let's say that-- I don't like this brown color, let me pick 00:01:02.660 --> 00:01:05.190 something more pleasant. 00:01:05.190 --> 00:01:08.250 Maybe mauve. 00:01:08.250 --> 00:01:17.270 Let's say that f of x is equal to 5x to the fifth minus x to 00:01:17.270 --> 00:01:29.850 the seventh times 20x squared plus 3x to them mine 7. 00:01:29.850 --> 00:01:31.610 So one way we could have done it, we could just 00:01:31.610 --> 00:01:32.520 multiply this out. 00:01:32.520 --> 00:01:36.310 This wouldn't be too bad, and then take the derivative 00:01:36.310 --> 00:01:37.200 like any polynomial. 00:01:37.200 --> 00:01:40.400 But let's use this product rule that I've just shown you. 00:01:40.400 --> 00:01:43.470 So the product rules that says, let me take the derivative of 00:01:43.470 --> 00:01:46.890 the first expression, or h of x if we wanted to map 00:01:46.890 --> 00:01:48.240 it into this rule. 00:01:48.240 --> 00:01:50.360 The derivative of that is pretty straightforward. 00:01:50.360 --> 00:01:52.470 5 times 5 is 25. 00:01:52.470 --> 00:01:57.160 25x to the fourth, right? 00:01:57.160 --> 00:02:01.480 Then minus 7, x to the sixth. 00:02:01.480 --> 00:02:04.780 We're just going to multiply it times this second expression, 00:02:04.780 --> 00:02:07.290 doing nothing different to it. 00:02:07.290 --> 00:02:09.980 Maybe I'll just do it in a different color. 00:02:09.980 --> 00:02:18.920 Times 20x plus 3x minus 7. 00:02:18.920 --> 00:02:24.270 And then to that we will add the derivative of 00:02:24.270 --> 00:02:27.110 this second function. 00:02:27.110 --> 00:02:35.000 The derivative of that second function is 40x minus 00:02:35.000 --> 00:02:38.590 21x to the minus eighth. 00:02:38.590 --> 00:02:41.570 And that times this first function. 00:02:41.570 --> 00:02:42.870 I guess I'll switch back to mauve, I think 00:02:42.870 --> 00:02:44.140 you get the point. 00:02:44.140 --> 00:02:51.480 5x to the fifth minus x to the seventh. 00:02:51.480 --> 00:02:54.660 All we did here was we said OK, f of x is made of these two 00:02:54.660 --> 00:02:56.730 expressions and they are multiplied by each other. 00:02:56.730 --> 00:02:58.930 If I want to take the derivative of it, I take the 00:02:58.930 --> 00:03:03.250 derivative of the first one and multiply it by the second one. 00:03:03.250 --> 00:03:05.530 And then I add that to the derivative of the second 00:03:05.530 --> 00:03:07.360 one and multiply it by the first one. 00:03:07.360 --> 00:03:10.050 Let's do some more examples and I think that will 00:03:10.050 --> 00:03:11.860 hit the point home. 00:03:11.860 --> 00:03:14.752 Clear image. 00:03:14.752 --> 00:03:18.850 Change the colors and I'm back in business. 00:03:18.850 --> 00:03:21.440 Let me think of a good problem. 00:03:21.440 --> 00:03:23.410 Let me do another one like this, and then I'll actually 00:03:23.410 --> 00:03:27.790 introduce ones and the product rule and the chain rule. 00:03:27.790 --> 00:03:40.630 So let's say that f of x is equal to 10x to the third plus 00:03:40.630 --> 00:03:53.180 5x squared minus 7 times 20x to the eighth minus 7. 00:03:53.180 --> 00:03:56.970 Then we say f prime of x, what's the derivative of 00:03:56.970 --> 00:03:58.710 this first expression. 00:03:58.710 --> 00:04:05.900 It's 30x squared plus 10x. 00:04:05.900 --> 00:04:09.290 And I just multiply it times this expression, right? 00:04:09.290 --> 00:04:13.570 20x to the eighth minus 7. 00:04:13.570 --> 00:04:16.240 And I add that to the derivative of this second 00:04:16.240 --> 00:04:21.130 expression, this is all on one line but I ran out of space, 00:04:21.130 --> 00:04:24.600 160x to the seventh, right? 00:04:24.600 --> 00:04:27.490 8 times 20 is 160. 00:04:27.490 --> 00:04:29.520 And then the derivative of 7 is zero. 00:04:29.520 --> 00:04:33.440 So it's just 160x to the seventh times this 00:04:33.440 --> 00:04:35.060 first expression. 00:04:35.060 --> 00:04:42.720 10x to the third plus 5x squared minus seven. 00:04:42.720 --> 00:04:43.290 There we go. 00:04:43.290 --> 00:04:44.300 And you could simplify it. 00:04:44.300 --> 00:04:46.190 You could multiply this out if you wanted or you could 00:04:46.190 --> 00:04:48.590 distribute this out if you wanted, maybe try to 00:04:48.590 --> 00:04:49.730 condense the terms. 00:04:49.730 --> 00:04:51.420 But that's really just algebra. 00:04:51.420 --> 00:04:54.030 So this is using the product rule. 00:04:54.030 --> 00:04:55.880 I'm going to do one more example where I'll show you, 00:04:55.880 --> 00:04:58.140 I'm going to use the product and the chain rule and 00:04:58.140 --> 00:05:02.170 I think this will optimally confuse you. 00:05:02.170 --> 00:05:03.225 I want to make sure I have some space. 00:05:07.910 --> 00:05:09.260 Here I'm going to use a slightly different notation. 00:05:09.260 --> 00:05:12.110 Instead of saying f of x and then what's f prime of x, I'm 00:05:12.110 --> 00:05:27.110 going to say y is equal to x squared plus 2x to the fifth 00:05:27.110 --> 00:05:40.200 times 3x to the minus three plus x squared to the minus 7. 00:05:40.200 --> 00:05:44.480 And I want to find the rate at which y changes relative to x. 00:05:44.480 --> 00:05:48.130 So I want to find dy over dx. 00:05:48.130 --> 00:05:49.950 This is just like, if this was f of x, it's just 00:05:49.950 --> 00:05:52.360 like saying f prime of x. 00:05:52.360 --> 00:05:53.290 This is just a [UNINTELLIGIBLE] 00:05:53.290 --> 00:05:54.470 notation. 00:05:54.470 --> 00:05:55.560 So what do I do in the chain rule? 00:05:55.560 --> 00:05:58.630 First I want the derivative of this term. 00:05:58.630 --> 00:06:02.460 Let me use colors to make it not too confusing. 00:06:02.460 --> 00:06:05.950 So what's the derivative of this term? 00:06:05.950 --> 00:06:08.540 We are going to use the chain rule first. 00:06:08.540 --> 00:06:15.990 So we take the derivative of the inside which is 2x plus 2 00:06:15.990 --> 00:06:18.790 and multiply times the derivative of the 00:06:18.790 --> 00:06:20.330 larger expression. 00:06:20.330 --> 00:06:26.920 But we keep x squared plus 3x there so it's times 5 times 00:06:26.920 --> 00:06:28.470 something to the fourth. 00:06:28.470 --> 00:06:33.320 And that something is x squared plus 2x. 00:06:33.320 --> 00:06:36.460 So there we took the derivative of this first term right here 00:06:36.460 --> 00:06:38.300 and then the product rules says we take the derivative of the 00:06:38.300 --> 00:06:40.740 first term, we just multiply it by the second term. 00:06:40.740 --> 00:06:49.120 So the second term is just 3x to the minus 3 plus x squared 00:06:49.120 --> 00:06:51.080 and all that to the minus 7. 00:06:51.080 --> 00:06:57.240 We did that and then to that we add plus the derivative of this 00:06:57.240 --> 00:06:59.870 second term times this first term. 00:06:59.870 --> 00:07:01.280 We're going to use the chain rule again. 00:07:01.280 --> 00:07:02.750 What's the derivative of the second term? 00:07:02.750 --> 00:07:04.890 I'll switch back to the light blue. 00:07:04.890 --> 00:07:07.840 Light blue means the derivative of one of the terms. 00:07:07.840 --> 00:07:11.320 So we take the derivative of the inside, the derivative of 00:07:11.320 --> 00:07:18.710 inside is minus 3 times 3 is minus 9, x go down one to 00:07:18.710 --> 00:07:23.210 the minus 4, plus 2x. 00:07:23.210 --> 00:07:26.180 And now we take the derivative of the whole thing. 00:07:26.180 --> 00:07:34.090 Times minus 7 times something to the minus 8, and that 00:07:34.090 --> 00:07:36.360 something is this inside. 00:07:36.360 --> 00:07:40.280 3x to the minus 3 plus x squared. 00:07:40.280 --> 00:07:42.770 And then we multiply this thing, this whole thing which 00:07:42.770 --> 00:07:44.870 is the derivative of the second term times the first term. 00:07:47.730 --> 00:07:53.010 Times, and I'm just going to keep going, times x squared 00:07:53.010 --> 00:07:57.190 plus 2x to the fifth. 00:07:57.190 --> 00:07:59.710 So this is a really, I mean you might want to 00:07:59.710 --> 00:08:00.730 simplify at this point. 00:08:00.730 --> 00:08:02.480 You can take this minus 7 and multiply it 00:08:02.480 --> 00:08:03.490 out and all of that. 00:08:03.490 --> 00:08:05.350 But I think this gives you the idea. 00:08:05.350 --> 00:08:08.850 And if you have to multiply this out and then do the 00:08:08.850 --> 00:08:10.220 derivative if it's just a polynomial, this would 00:08:10.220 --> 00:08:11.080 take you forever. 00:08:11.080 --> 00:08:14.110 But using the chain rule, you're actually able to, even 00:08:14.110 --> 00:08:16.250 though we ended up with a pretty complicated answer, 00:08:16.250 --> 00:08:17.110 we got the right answer. 00:08:17.110 --> 00:08:20.230 And now we could actually evaluate the slope of this very 00:08:20.230 --> 00:08:22.990 complicated function at any point just by substituting the 00:08:22.990 --> 00:08:25.190 point into this fairly complicated expression. 00:08:25.190 --> 00:08:27.830 But at least we could do it. 00:08:27.830 --> 00:08:29.970 I think you're going to find that the chain and the product 00:08:29.970 --> 00:08:33.230 rules become even more useful once we start doing derivatives 00:08:33.230 --> 00:08:35.970 of expressions other than polynomials. 00:08:35.970 --> 00:08:38.620 I'm going to teach you about trigonometric functions and 00:08:38.620 --> 00:08:42.710 natural log and logarithm and exponential functions. 00:08:42.710 --> 00:08:44.460 Actually, I'll probably do that in the next presentation. 00:08:44.460 --> 00:08:47.240 So I will see you soon.

Even More Chain Rule

https://www.youtube.com/watch?v=DYb-AN-lK94

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WEBVTT Kind: captions Language: en 00:00:01.260 --> 00:00:03.796 Now that you've seen some examples of the chain rule 00:00:03.796 --> 00:00:06.930 in use, I think the actual definition of the chain rule 00:00:06.930 --> 00:00:08.770 might be more digestible now. 00:00:08.770 --> 00:00:11.550 So let me give you the actual definition of the chain rule. 00:00:11.550 --> 00:00:27.340 Let's say I have a function f of x and it equals h of g of x. 00:00:27.340 --> 00:00:29.640 And you remember all this from composite functions. 00:00:29.640 --> 00:00:33.350 So the chain rule just says that the derivative of f of x 00:00:33.350 --> 00:00:39.500 or f prime of x is equal to the derivative of this inner 00:00:39.500 --> 00:00:48.370 function, g prime of x times the derivative of this h 00:00:48.370 --> 00:00:52.600 function, h prime of x. 00:00:52.600 --> 00:00:54.105 But it's not going to be just h prime of x. 00:00:54.105 --> 00:00:59.070 It's going to be h prime of g of x. 00:00:59.070 --> 00:01:01.750 So let's apply that to some examples like we were 00:01:01.750 --> 00:01:04.290 doing before, and I think it'll make some sense. 00:01:04.290 --> 00:01:17.870 So let's say we had f of x is equal to x squared plus 5x 00:01:17.870 --> 00:01:24.080 plus 3, all of this to the fifth power. 00:01:24.080 --> 00:01:27.580 So in this example, what's h of x, what's g of x, and 00:01:27.580 --> 00:01:28.440 you know what f of x is. 00:01:28.440 --> 00:01:33.240 Well let's say g of x would be this inner function. 00:01:33.240 --> 00:01:40.010 So we would say-- let me pick a different color-- g of x here 00:01:40.010 --> 00:01:47.260 is x squared plus 5x plus 3. 00:01:47.260 --> 00:01:49.480 It's the stuff, f g of x. 00:01:52.260 --> 00:01:55.715 Well h of g of x is this whole thing, so what would h of x be? 00:02:01.750 --> 00:02:04.840 This is h of g of x, but h of x would just be x 00:02:04.840 --> 00:02:06.940 to the fifth, right? 00:02:06.940 --> 00:02:10.530 Because this expression as you took this entire g of x and you 00:02:10.530 --> 00:02:12.370 put it in for x right here. 00:02:12.370 --> 00:02:15.630 I think that make sense if you take this entire expression and 00:02:15.630 --> 00:02:17.720 you substitute x here for this entire expression you 00:02:17.720 --> 00:02:19.810 get this expression. 00:02:19.810 --> 00:02:30.280 And this shows that this is equal to h of g of x. 00:02:30.280 --> 00:02:33.450 If you just take this blue part and substitute it for x, you 00:02:33.450 --> 00:02:35.410 get this entire expression. 00:02:35.410 --> 00:02:38.860 So the chain rule just tells us that the derivative of this, 00:02:38.860 --> 00:02:40.980 that f prime of x-- and I have a feeling I'm going to run out 00:02:40.980 --> 00:02:45.570 of space-- f prime of x-- well actually before I do anything, 00:02:45.570 --> 00:02:46.950 let's figure out the derivatives of g 00:02:46.950 --> 00:02:48.470 of x and h of x. 00:02:48.470 --> 00:02:52.700 g of x of g prime of x-- let me draw a little line here to 00:02:52.700 --> 00:02:55.530 divide it out, I know I'm running out of space. 00:02:55.530 --> 00:03:03.530 So g prime of x is equal to 2x plus 5. 00:03:06.200 --> 00:03:09.290 2x plus 5, and then derivative 3 is just 0, right? 00:03:09.290 --> 00:03:17.150 And the derivative of h of x? h prime of x is equal 00:03:17.150 --> 00:03:22.410 to 5 x to the fourth. 00:03:22.410 --> 00:03:25.270 So the chain rule just says that the derivative of this 00:03:25.270 --> 00:03:30.530 entire composite function is just-- let me just 00:03:30.530 --> 00:03:31.240 write it down here. 00:03:31.240 --> 00:03:34.850 I'm doing this to optimally confuse you. 00:03:34.850 --> 00:03:40.050 The derivative of this entire function is just g prime of x. 00:03:40.050 --> 00:03:42.450 Well we figured out with g prime of x is here, it's 2x 00:03:42.450 --> 00:03:55.260 plus 5 times h times h prime of g of x. 00:03:55.260 --> 00:03:56.310 So what's h prime of g of x? 00:03:56.310 --> 00:03:59.750 Well h prime of x is 5x to the fourth, but we want 00:03:59.750 --> 00:04:01.800 h prime of g of x. 00:04:01.800 --> 00:04:11.330 So h prime of g of x would equal 5 times g 00:04:11.330 --> 00:04:14.190 of x to the fourth. 00:04:14.190 --> 00:04:17.310 And we know what g of x is, it's this whole thing. 00:04:17.310 --> 00:04:21.970 So it would be times 5, and this whole thing x squared 00:04:21.970 --> 00:04:26.990 plus 5x plus 3, all that to the fourth power. 00:04:26.990 --> 00:04:29.815 I think I have truly, truly confused you, so I'm going 00:04:29.815 --> 00:04:33.630 to try to do a couple of more examples. 00:04:33.630 --> 00:04:36.310 Clear this. 00:04:36.310 --> 00:04:37.560 OK. 00:04:37.560 --> 00:04:39.320 Let me write it up here again. 00:04:39.320 --> 00:04:55.870 So if we say that f of x is equal to h of g of x, then f 00:04:55.870 --> 00:05:07.055 prime of x is equal to g prime of x times h prime of g of x. 00:05:09.790 --> 00:05:10.955 So I'll do another example. 00:05:15.270 --> 00:05:28.750 Let's say that g of x is equal to x to the seventh minus 00:05:28.750 --> 00:05:33.150 3x to the ninth is 3. 00:05:33.150 --> 00:05:44.870 And let's say that h of x is equal to-- let's do something 00:05:44.870 --> 00:05:46.260 reasonably straightforward. 00:05:46.260 --> 00:05:55.570 Let's say h of x is x to the minus 10. 00:05:55.570 --> 00:06:00.430 So what is f of x? f of x is just h of g of x, and this 00:06:00.430 --> 00:06:03.660 should be a bit of a reminder from composite functions. 00:06:03.660 --> 00:06:04.310 So let's see. 00:06:04.310 --> 00:06:15.995 h of g of x would just be-- you take g of x and you substitute 00:06:15.995 --> 00:06:25.090 it for x here, so it would just be this expression, x to the 00:06:25.090 --> 00:06:31.670 seventh minus 3x to the minus third, and then all of that 00:06:31.670 --> 00:06:33.810 to the minus 10th power. 00:06:33.810 --> 00:06:35.720 So this is our f of x. 00:06:35.720 --> 00:06:38.700 And this is of course equal to f of x, right, because f of 00:06:38.700 --> 00:06:41.350 x is equal to h of g of x. 00:06:41.350 --> 00:06:44.050 I know this very confusing, but bear with me. 00:06:44.050 --> 00:06:45.650 Maybe you have to watch the video twice and it'll 00:06:45.650 --> 00:06:47.130 start making more sense. 00:06:47.130 --> 00:06:49.770 Well we want to now figure out what f prime of x is. 00:06:54.760 --> 00:06:58.570 Well the chain rule tells us all it is, is we take the 00:06:58.570 --> 00:07:00.930 derivative of g of x, right? 00:07:00.930 --> 00:07:03.240 So the derivative of g of x is what? 00:07:03.240 --> 00:07:04.450 That's easy. 00:07:04.450 --> 00:07:06.190 Or hopefully it's easy by now. 00:07:06.190 --> 00:07:12.280 Derivative of g of x is 7x to the sixth, and minus 3 times 00:07:12.280 --> 00:07:16.720 minus 3 is plus 9x to the minus 4. 00:07:16.720 --> 00:07:20.790 I just took minus 3 and went down 1, so that's g prime of x. 00:07:23.350 --> 00:07:31.130 And then times h prime of g of x. 00:07:31.130 --> 00:07:34.550 Well what's h prime of x? 00:07:34.550 --> 00:07:35.010 That's easy. 00:07:35.010 --> 00:07:39.890 That's just minus 10 times x to the minus 11. 00:07:39.890 --> 00:07:43.050 But we want to do h prime of g of x. 00:07:43.050 --> 00:07:45.590 So instead of having an x here, we're going to substitute that 00:07:45.590 --> 00:07:48.320 x with the entire g of x expression. 00:07:48.320 --> 00:07:55.760 So this is just times 10 time something to the minus eleven, 00:07:55.760 --> 00:07:57.600 and that something is just g of x. 00:08:00.310 --> 00:08:05.780 x to the seventh, minus 3x access to the minus 3. 00:08:05.780 --> 00:08:10.780 And there's our answer. f prime of x is the derivative of kind 00:08:10.780 --> 00:08:15.300 of the inner function, g of x, times the derivative of the 00:08:15.300 --> 00:08:19.420 outer function, but instead of it just being applied to x it'd 00:08:19.420 --> 00:08:23.960 be applied to the entire g of x instead of an x being here. 00:08:23.960 --> 00:08:25.470 Maybe I've confused you more. 00:08:25.470 --> 00:08:27.680 Let me do one quick example just to show you that you 00:08:27.680 --> 00:08:30.940 don't have to kind of do this whole h of g of every time. 00:08:33.790 --> 00:08:48.980 So if I have f of x is equal to 5 times minus x to the eighth, 00:08:48.980 --> 00:08:54.180 plus x to the minus eighth, all of that over to 00:08:54.180 --> 00:08:57.630 the fifth power. 00:08:57.630 --> 00:09:00.610 If I want to figure out f prime of x I just take the derivative 00:09:00.610 --> 00:09:05.480 of this inner function I guess I could call it, so that's 00:09:05.480 --> 00:09:12.480 minus 8x to the seventh minus 8x-- because it's just take the 00:09:12.480 --> 00:09:18.580 negative 8-- to the minus ninth, times the derivative 00:09:18.580 --> 00:09:19.920 of this larger function. 00:09:19.920 --> 00:09:27.740 So 5 times 5 is 25 times something to the fourth. 00:09:27.740 --> 00:09:30.440 And that something is just going to be this expression 00:09:30.440 --> 00:09:35.730 minus x to the eighth plus x to the minus eighth. 00:09:35.730 --> 00:09:36.290 And we're done. 00:09:36.290 --> 00:09:37.100 You could simplify it. 00:09:37.100 --> 00:09:40.080 You could multiply this 25 out and do et cetera, et cetera. 00:09:40.080 --> 00:09:42.540 Hopefully this gives you more of an intuition of what the 00:09:42.540 --> 00:09:45.730 chain rule is all about, and I'm going to do a lot more 00:09:45.730 --> 00:09:47.910 examples in the next couple of presentations as well. 00:09:47.910 --> 00:09:48.250 See you soon.

Chain Rule Examples

https://www.youtube.com/watch?v=6_lmiPDedsY

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https://www.youtube.com/api/timedtext?v=6_lmiPDedsY&ei=eWeUZdPQO7G1xN8P2Z-5uAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249834&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=876DB98125DB1CCAF1B21E618F8E1E4DB3CE2B31.4F5CDBD02866CA0DC803CD27F311C9F98EEC64D5&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.960 --> 00:00:02.790 I'm now going to do a bunch more examples 00:00:02.790 --> 00:00:04.120 using the chain rule. 00:00:04.120 --> 00:00:05.320 So let's see. 00:00:05.320 --> 00:00:05.920 Once again. 00:00:05.920 --> 00:00:13.930 If I had f of x is equal to, let's see, I don't like this 00:00:13.930 --> 00:00:16.100 tool that I'm using now, let's have one of these. 00:00:16.100 --> 00:00:28.390 f of x is equal to, say, x to the third plus 2x squared 00:00:28.390 --> 00:00:32.600 minus, let's say, minus x to the negative 2. 00:00:32.600 --> 00:00:34.960 We haven't put any negative exponents in yet, but I 00:00:34.960 --> 00:00:38.060 think you'll see that the same patterns apply. 00:00:38.060 --> 00:00:43.520 And all of that to, let's say, the minus seven. 00:00:43.520 --> 00:00:47.077 We want to figure out what f prime of x, what the 00:00:47.077 --> 00:00:48.260 derivative of f of x is. 00:00:48.260 --> 00:00:50.540 So this might seem very complicated and daunting to 00:00:50.540 --> 00:00:53.220 you, and obviously to take this entire polynomial to 00:00:53.220 --> 00:00:55.470 the negative seventh power would take you forever. 00:00:55.470 --> 00:00:57.590 But using the chain rule, we can do it quite quickly. 00:00:57.590 --> 00:01:00.610 So the first thing we want to do, is we want to take the 00:01:00.610 --> 00:01:03.460 derivative of the inner function, I guess 00:01:03.460 --> 00:01:04.420 you could call it. 00:01:04.420 --> 00:01:06.630 We want to take the derivative of this. 00:01:06.630 --> 00:01:08.990 And what's the derivative of x to the third plus 2x squared 00:01:08.990 --> 00:01:10.770 minus x to the negative 2? 00:01:10.770 --> 00:01:11.600 Well, we know how to do that. 00:01:11.600 --> 00:01:13.790 That was the first type of derivatives we 00:01:13.790 --> 00:01:14.210 learned how to do. 00:01:14.210 --> 00:01:22.210 It's 3x squared and 2 times 2, plus 4x to the first, or just 00:01:22.210 --> 00:01:27.270 4x, and then here, with a negative exponent, we do 00:01:27.270 --> 00:01:28.120 the exact same thing. 00:01:28.120 --> 00:01:31.320 We say negative 2 times negative 1, right, there's a 1 00:01:31.320 --> 00:01:33.450 here, we don't write it down. 00:01:33.450 --> 00:01:40.360 So negative 2 times negative 1 is plus 2 x to the, and then we 00:01:40.360 --> 00:01:43.210 decrease the exponent by 1, so it's x to the 00:01:43.210 --> 00:01:45.530 negative 3, right? 00:01:45.530 --> 00:01:47.980 So we figured out what the derivative of the inside is, 00:01:47.980 --> 00:01:53.990 and then we just multiply that, that whole thing, times 00:01:53.990 --> 00:01:56.750 the derivative of kind of the entire expression. 00:01:56.750 --> 00:02:01.210 So then that'll be, we take the minus 7, let me 00:02:01.210 --> 00:02:02.270 do a different color. 00:02:02.270 --> 00:02:05.800 So this is the entire thing. 00:02:05.800 --> 00:02:13.620 So then we take minus 7, so it's times minus 7, this 00:02:13.620 --> 00:02:16.130 whole expression, I'm going to run out of space. 00:02:16.130 --> 00:02:22.370 x to the third plus 2x squared minus x to the minus 2. 00:02:22.370 --> 00:02:24.400 That's minus x to the minus 2. 00:02:24.400 --> 00:02:26.900 And all of that, we just decrease this exponent 00:02:26.900 --> 00:02:29.980 by 1, to the minus 8. 00:02:29.980 --> 00:02:33.040 So let me write it all down a little bit neater now. 00:02:33.040 --> 00:02:38.230 So we get f prime of x as the derivative of f of x is equal 00:02:38.230 --> 00:02:52.500 to 3x squared plus 4x plus 2x to the minus third power, I 00:02:52.500 --> 00:02:53.310 don't know why did that. 00:02:53.310 --> 00:02:55.190 That's minus 3. 00:02:55.190 --> 00:03:04.450 Times minus seven times x to the third plus 2x squared minus 00:03:04.450 --> 00:03:09.710 x to the minus two, all of that to the negative eight power. 00:03:09.710 --> 00:03:10.970 And we could simplify it little bit. 00:03:10.970 --> 00:03:14.640 Maybe we could just multiply this minus 7, times, we could 00:03:14.640 --> 00:03:16.005 distribute it across this expression. 00:03:16.005 --> 00:03:23.510 So we'd say, that equals minus 7, so this equals minus 21 x 00:03:23.510 --> 00:03:34.260 squared, minus 28 x minus 14x to the negative 3. 00:03:34.260 --> 00:03:43.980 All of that times x to the third plus 2 x squared minus x 00:03:43.980 --> 00:03:48.950 to the minus 2 to the minus 8. 00:03:48.950 --> 00:03:49.950 So there we did it. 00:03:49.950 --> 00:03:53.990 We took this, what I would say is a very complicated function, 00:03:53.990 --> 00:03:57.290 and using the chain rule and just the basic rules we had 00:03:57.290 --> 00:03:59.260 introduced a couple of presentations ago, we were able 00:03:59.260 --> 00:04:01.030 to find the derivative of it. 00:04:01.030 --> 00:04:03.650 And now, if we wanted to, for whatever application, we could 00:04:03.650 --> 00:04:07.320 find the slope of this function at any point x by just 00:04:07.320 --> 00:04:13.910 substituting that point into this equation, and we'll get 00:04:13.910 --> 00:04:15.120 the slope at that point. 00:04:15.120 --> 00:04:17.400 Let me do a slightly harder one, to show you that the chain 00:04:17.400 --> 00:04:20.400 rule, you can kind of go arbitrarily deep in 00:04:20.400 --> 00:04:20.969 the chain rule. 00:04:30.010 --> 00:04:30.610 OK. 00:04:30.610 --> 00:04:34.010 So let's say I had, let me see if I can write it 00:04:34.010 --> 00:04:35.000 a little bit thinner. 00:04:35.000 --> 00:04:39.340 If I had f of x, I don't know if you can see that, I'm going 00:04:39.340 --> 00:04:40.560 to do it a little fatter. 00:04:40.560 --> 00:04:46.480 f of x is equal to, I want to make it a little bit more 00:04:46.480 --> 00:04:48.520 complicated this time. 00:04:48.520 --> 00:05:03.420 3x to the minus 2 plus 5 x to the third minus 7x, all of that 00:05:03.420 --> 00:05:07.620 to the fifth, and then this whole expression to 00:05:07.620 --> 00:05:08.900 the third power. 00:05:08.900 --> 00:05:11.560 So I imagine you saying, Sal, you're starting to go nuts, 00:05:11.560 --> 00:05:13.010 this is going to take us forever. 00:05:13.010 --> 00:05:14.630 Well, I'll show you, using the chain rule, it will 00:05:14.630 --> 00:05:16.500 not take that long. 00:05:16.500 --> 00:05:20.200 So the way I think about it, so-- f prime of x, 00:05:20.200 --> 00:05:22.480 f prime of x equals. 00:05:22.480 --> 00:05:25.360 I start off kind with the innermost function. 00:05:25.360 --> 00:05:28.210 So let me see if I can use colors to make it 00:05:28.210 --> 00:05:28.820 a little bit simpler. 00:05:28.820 --> 00:05:34.430 Let's take the derivative of this innermost function first. 00:05:34.430 --> 00:05:35.940 Actually, let me give you the big picture. 00:05:35.940 --> 00:05:40.820 We want to find the derivative of the innermost function, and 00:05:40.820 --> 00:05:42.860 then a little bit bigger, and then a little bit 00:05:42.860 --> 00:05:43.890 more big than that. 00:05:43.890 --> 00:05:46.980 I know that's not precise mathematical terms, but 00:05:46.980 --> 00:05:49.310 you'll get the point when I show you this example. 00:05:49.310 --> 00:05:51.860 So first we'll do this inner function, this 00:05:51.860 --> 00:05:52.860 inner expression. 00:05:52.860 --> 00:05:54.520 And the derivative of that's pretty easy, right? 00:05:54.520 --> 00:06:00.270 It's 15x squared minus 7, right? 00:06:00.270 --> 00:06:02.020 that was pretty straightforward. 00:06:02.020 --> 00:06:06.150 And now we're going to want to multiply that times this 00:06:06.150 --> 00:06:08.010 entire derivative here. 00:06:08.010 --> 00:06:10.280 So let me circle that in a different-- so then 00:06:10.280 --> 00:06:11.750 we want to do this. 00:06:11.750 --> 00:06:15.240 We're going to multiply that times this entire derivative. 00:06:15.240 --> 00:06:20.140 Well, that's just times 5. 00:06:20.140 --> 00:06:24.420 And we just pretend like this is just an x here, right? 00:06:24.420 --> 00:06:26.795 Because the derivative of x to the fifth is 5x 00:06:26.795 --> 00:06:28.150 to the fourth, right? 00:06:28.150 --> 00:06:30.590 But instead of an x, we have this whole expression, 5x 00:06:30.590 --> 00:06:31.680 to the third minus 7x. 00:06:31.680 --> 00:06:33.130 So we'll write that. 00:06:33.130 --> 00:06:37.360 5x to the third minus 7x. 00:06:37.360 --> 00:06:41.520 Now the exponent here goes down by one. 00:06:41.520 --> 00:06:44.540 So it's 5 times 5x to the third minus 7x, all that 00:06:44.540 --> 00:06:46.070 to the fourth power. 00:06:46.070 --> 00:06:48.330 So we figured out the derivative of this so far, and 00:06:48.330 --> 00:06:51.646 then we want to figure out the derivative of this, so we'll 00:06:51.646 --> 00:06:53.000 add it, right, because we're trying to figure the derivative 00:06:53.000 --> 00:06:54.880 of this entire expression. 00:06:54.880 --> 00:06:56.120 So this is an easy one. 00:06:56.120 --> 00:07:00.020 Let me draw that in a different color. 00:07:00.020 --> 00:07:02.870 So we want the derivative of this. 00:07:02.870 --> 00:07:06.450 So that's negative 2 times 3, so that's negative 00:07:06.450 --> 00:07:10.430 6x to the minus three. 00:07:10.430 --> 00:07:11.610 So what have we done so far? 00:07:11.610 --> 00:07:16.270 We've so far figured out the derivative of this entire 00:07:16.270 --> 00:07:18.540 expression, right? 00:07:18.540 --> 00:07:20.850 The derivative of that entire expression using 00:07:20.850 --> 00:07:25.170 the chain rule is this. 00:07:25.170 --> 00:07:26.910 And now, we're almost done. 00:07:26.910 --> 00:07:28.290 We just have to multiply that. 00:07:28.290 --> 00:07:30.630 So I'm going to just, I've run out of space on that line, but 00:07:30.630 --> 00:07:32.170 let's just assume that the line continues. 00:07:32.170 --> 00:07:33.770 So that's times. 00:07:33.770 --> 00:07:35.360 And now we just take the derivative of kind of 00:07:35.360 --> 00:07:37.410 this whole big thing. 00:07:37.410 --> 00:07:40.346 And now it's going to be the derivative of, I'm going 00:07:40.346 --> 00:07:42.580 to use this brown color. 00:07:42.580 --> 00:07:45.950 So it's a whole big expression to the third power, right? 00:07:45.950 --> 00:07:50.520 So that becomes times 3 times the whole expression, right? 00:07:50.520 --> 00:07:55.270 That's 3 times, now I'm going to write the whole thing, 3x to 00:07:55.270 --> 00:08:06.670 the minus 2 plus 5 x the third minus 7x, that to fifth, and 00:08:06.670 --> 00:08:10.220 then you decrement this by 1, to the second power. 00:08:10.220 --> 00:08:12.600 That was an ultraconfusing example, and this is probably 00:08:12.600 --> 00:08:15.850 the hardest chain rule problem you'll see in a lot of 00:08:15.850 --> 00:08:17.340 the questions you'll have on your test. 00:08:17.340 --> 00:08:18.450 You see, it wasn't that difficult. 00:08:18.450 --> 00:08:21.200 We just kind of went to the smallest possible function, and 00:08:21.200 --> 00:08:22.760 actually the smallest possible function would have been one of 00:08:22.760 --> 00:08:25.710 these terms, but we just found the derivative of this, which 00:08:25.710 --> 00:08:30.690 was 15 x squared minus 7, and then we just used the principle 00:08:30.690 --> 00:08:33.420 that the derivative of kind of a function is just the 00:08:33.420 --> 00:08:36.960 derivative of each of its parts-- well, actually, the 00:08:36.960 --> 00:08:40.400 derivative of-- we figured out the derivative of this inner 00:08:40.400 --> 00:08:43.464 piece, which was 15x squared minus 7, and then we multiplied 00:08:43.464 --> 00:08:47.350 it times the derivative of this slightly larger piece, which is 00:08:47.350 --> 00:08:52.210 5 times this entire expression to the fourth, then we added it 00:08:52.210 --> 00:08:54.980 to the derivative of 3x to the minus 2. 00:08:54.980 --> 00:08:57.470 And then that whole thing, and actually I should put a big 00:08:57.470 --> 00:09:00.810 parentheses around here, that whole thing, we multiply it 00:09:00.810 --> 00:09:04.180 times the derivative of this larger expression. 00:09:04.180 --> 00:09:07.630 I think I might have confused you, so I apologize if I have, 00:09:07.630 --> 00:09:09.810 and in the next presentation I'm going to just do a bunch 00:09:09.810 --> 00:09:13.000 more chain rule problems, and at some point, it should 00:09:13.000 --> 00:09:14.670 start to make sense to you. 00:09:14.670 --> 00:09:18.210 I think it's just a matter of seeing example, after 00:09:18.210 --> 00:09:19.940 example, after example. 00:09:19.940 --> 00:09:22.380 I'll see you into the next presentation, and I apologize 00:09:22.380 --> 00:09:24.440 if I have confused you.

The Chain Rule

https://www.youtube.com/watch?v=XIQ-KnsAsbg

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.960 --> 00:00:01.930 Welcome back. 00:00:01.930 --> 00:00:05.460 I'm now going to do some more examples of a bit of a review 00:00:05.460 --> 00:00:07.300 of some of the derivatives that we've been seeing. 00:00:07.300 --> 00:00:09.450 And then I'll introduce you to something called the chain rule 00:00:09.450 --> 00:00:13.080 which expands the universe of the types of functions we can 00:00:13.080 --> 00:00:14.290 take the derivatives of. 00:00:14.290 --> 00:00:18.540 So in the last presentation, I showed you how to function if I 00:00:18.540 --> 00:00:30.270 had f of x is equal to 10x to the seventh plus 6x to the 00:00:30.270 --> 00:00:39.330 third plus 15x minus x to the 16th. 00:00:39.330 --> 00:00:42.160 To take the derivative of this entire function, we take just 00:00:42.160 --> 00:00:43.840 the derivatives of each of the pieces, right? 00:00:43.840 --> 00:00:45.640 Because you can add them up. 00:00:45.640 --> 00:00:51.560 So f prime of x in this example, is equal to-- and 00:00:51.560 --> 00:00:52.755 I think you get the hang of it at this point. 00:00:52.755 --> 00:00:54.150 It's actually fairly straightforward. 00:00:54.150 --> 00:00:56.480 We take the 7, multiply it by the 10. 00:00:56.480 --> 00:01:01.620 So we get 70x, then 1 degree less. 00:01:01.620 --> 00:01:10.940 So 70x to the sixth plus 18x squared plus 15. 00:01:10.940 --> 00:01:13.010 We can kind of view this as x to the 1, right? 00:01:13.010 --> 00:01:16.920 So it's 1 times 15 times x to the 0. 00:01:16.920 --> 00:01:17.550 Which is 1. 00:01:17.550 --> 00:01:23.840 So that's just 15 minus 16x to the 15th. 00:01:23.840 --> 00:01:25.330 And I don't want you to lose sight of what we're 00:01:25.330 --> 00:01:26.090 actually doing here. 00:01:26.090 --> 00:01:27.870 What is f prime of x? 00:01:27.870 --> 00:01:32.770 This is the function the tells us the slope of any point 00:01:32.770 --> 00:01:35.450 x, along the curve f of x. 00:01:35.450 --> 00:01:37.860 It's a pretty interesting thing. 00:01:37.860 --> 00:01:40.795 Let me just draw to maybe give you a little bit of intuition. 00:01:45.300 --> 00:01:47.490 I don't know what the slope of f of x really looks like. 00:01:47.490 --> 00:01:49.560 And actually, let's pretend like this isn't f of x. 00:01:49.560 --> 00:01:51.360 Let's pretend like this is just some arbitrary 00:01:51.360 --> 00:01:53.270 function I'm drawing. 00:01:53.270 --> 00:01:56.210 If this is f of x, just some curve that does all sorts of 00:01:56.210 --> 00:02:01.430 crazy things, f prime of x tells me the slope at any 00:02:01.430 --> 00:02:02.810 point along that line. 00:02:02.810 --> 00:02:09.950 So if I wanted to know the slope at this point right here, 00:02:09.950 --> 00:02:13.060 I could use the derivative function to figure out the 00:02:13.060 --> 00:02:15.120 slope of the tangent line. 00:02:15.120 --> 00:02:18.740 The tangent line is something like that right there. 00:02:18.740 --> 00:02:21.930 Or if I wanted to figure out the slope at this point, 00:02:21.930 --> 00:02:23.890 once again I'd use the derivative function. 00:02:23.890 --> 00:02:26.340 And it would tell me the slope of the tangent 00:02:26.340 --> 00:02:27.100 line at that point. 00:02:27.100 --> 00:02:29.150 Which would be something like that. 00:02:29.150 --> 00:02:30.960 So it's a pretty useful thing. 00:02:30.960 --> 00:02:34.900 And once I give you all the tools to analytically solve a 00:02:34.900 --> 00:02:36.560 whole host of derivatives, then we'll actually do a bunch 00:02:36.560 --> 00:02:39.140 of word problems and applications of derivatives. 00:02:39.140 --> 00:02:41.270 And I think you'll see that it's a really, really, 00:02:41.270 --> 00:02:42.700 really useful concept. 00:02:42.700 --> 00:02:43.600 So let's move on. 00:02:43.600 --> 00:02:45.960 I think you get the idea of how to do these derivatives 00:02:45.960 --> 00:02:48.610 of polynomials. 00:02:48.610 --> 00:02:49.760 Let me erase this. 00:02:49.760 --> 00:02:52.650 I'm actually using a different tool now. 00:02:52.650 --> 00:02:56.110 So I think it might be a bit easier. 00:02:56.110 --> 00:02:57.630 Let's see, someone was calling me. 00:02:57.630 --> 00:03:00.520 But you're more important so I will not answer the phone. 00:03:08.440 --> 00:03:11.640 I'm going to introduce you-- this tool doesn't have, I 00:03:11.640 --> 00:03:14.330 don't think it has a straight up eraser. 00:03:14.330 --> 00:03:15.230 Actually, maybe let's see. 00:03:15.230 --> 00:03:16.350 If I do it like this. 00:03:27.480 --> 00:03:28.400 Oh let me see. 00:03:30.900 --> 00:03:32.150 No that doesn't work. 00:03:32.150 --> 00:03:36.080 Let me just erase like this, the old-fashioned way. 00:03:39.530 --> 00:03:41.000 You just have to bear with me. 00:03:41.000 --> 00:03:45.280 And then once I finish erasing, I will show you the chain rule. 00:03:45.280 --> 00:03:45.900 This is good. 00:03:45.900 --> 00:03:48.410 It feels like I'm a real teacher with a real chalkboard 00:03:48.410 --> 00:03:49.490 and a real eraser now. 00:03:53.530 --> 00:03:56.750 This is a lot cleaner than a normal chalkboard as well. 00:03:56.750 --> 00:04:00.960 Bear with me, almost there. 00:04:00.960 --> 00:04:02.630 I'll figure out a faster way to do this over the 00:04:02.630 --> 00:04:04.590 next couple of videos. 00:04:04.590 --> 00:04:05.210 It's pretty sad. 00:04:05.210 --> 00:04:07.670 I'm showing you how to do derivatives in calculus, but 00:04:07.670 --> 00:04:10.910 I don't know how to erase a faster way than this. 00:04:14.270 --> 00:04:15.560 There, we're done. 00:04:15.560 --> 00:04:15.900 OK. 00:04:15.900 --> 00:04:18.290 So now I'm going show you how to solve the derivatives of a 00:04:18.290 --> 00:04:21.360 slightly more complicated type of a function. 00:04:21.360 --> 00:04:23.080 It's actually not more complicated. 00:04:23.080 --> 00:04:24.450 It's just different. 00:04:24.450 --> 00:04:42.890 So let's say f of x is equal to 2x plus 3 to the fifth power. 00:04:42.890 --> 00:04:45.220 And I want to figure out the derivative of this. 00:04:45.220 --> 00:04:46.320 We're going to use something called the chain rule. 00:04:46.320 --> 00:04:49.020 Because one thing we could do, we could just multiply out 2x 00:04:49.020 --> 00:04:50.600 plus 3 to the fifth power. 00:04:50.600 --> 00:04:54.420 And if you've ever done that, you know it's a pain. 00:04:54.420 --> 00:04:55.540 So that's not something we'd want to do. 00:04:55.540 --> 00:04:57.280 So we're going to use something called the chain rule. 00:04:57.280 --> 00:04:59.390 And I'm just going to give you a bunch of examples before I 00:04:59.390 --> 00:05:01.060 even show you the definition of the chain rule. 00:05:01.060 --> 00:05:02.330 Because I think this is something that you just 00:05:02.330 --> 00:05:04.260 have to learn by example. 00:05:04.260 --> 00:05:07.110 So the chain rule just tells us that the derivative of let's 00:05:07.110 --> 00:05:10.820 say this function right here. 00:05:10.820 --> 00:05:14.100 You take the derivative of the subfunctions, and then you can 00:05:14.100 --> 00:05:16.000 take a derivative of the entire function. 00:05:16.000 --> 00:05:17.290 I'll tell you that formally. 00:05:17.290 --> 00:05:19.010 But I think when you introduce it formally, it gets 00:05:19.010 --> 00:05:19.890 more confusing. 00:05:19.890 --> 00:05:23.455 So what I do, I just take the derivative of 2x plus 3 first. 00:05:23.455 --> 00:05:25.000 And actually let me use colors. 00:05:25.000 --> 00:05:26.730 I think that might simplify it. 00:05:26.730 --> 00:05:29.510 So I take the derivative of 2x plus 3. 00:05:29.510 --> 00:05:31.350 What's the derivative of 2x plus 3? 00:05:31.350 --> 00:05:32.130 Well you know that. 00:05:32.130 --> 00:05:36.470 It's just the derivative of 2x, which is 2. 00:05:36.470 --> 00:05:37.860 And then the derivative of 3 is 0. 00:05:37.860 --> 00:05:41.300 So the derivative of 2x plus 3 is just 2. 00:05:41.300 --> 00:05:45.030 And then I'm going to multiply that times the derivative 00:05:45.030 --> 00:05:46.150 of the whole thing. 00:05:46.150 --> 00:05:48.930 And I just pretend like 2x plus 3 is just like 00:05:48.930 --> 00:05:51.060 a variable by itself. 00:05:51.060 --> 00:05:54.620 So then what's the derivative of x to the fifth? 00:05:54.620 --> 00:05:56.630 Well the derivative of x to the fifth-- I'm going to do that in 00:05:56.630 --> 00:06:00.366 a different color-- the derivative of x to the 00:06:00.366 --> 00:06:05.590 fifth is 5x to the fourth. 00:06:05.590 --> 00:06:13.270 So it'll be 5 times something to the fourth. 00:06:13.270 --> 00:06:15.760 But here we didn't take the derivative of x the fifth. 00:06:15.760 --> 00:06:18.010 We took the derivative of 2x plus 3 to the fifth. 00:06:18.010 --> 00:06:20.280 So we just put the 2x plus 3 there instead. 00:06:24.230 --> 00:06:26.190 So what did we do here? 00:06:26.190 --> 00:06:29.290 We went in the inside of the function, and we took 00:06:29.290 --> 00:06:29.860 the derivative here. 00:06:29.860 --> 00:06:32.570 And the derivative of 2x plus 3 was just 2. 00:06:32.570 --> 00:06:35.570 And then we multiplied it by the derivative of 00:06:35.570 --> 00:06:36.460 the greater function. 00:06:36.460 --> 00:06:38.980 And we just pretended like the 2x plus 3 was a variable. 00:06:38.980 --> 00:06:40.120 It was like x. 00:06:40.120 --> 00:06:44.410 So instead of 5x to the fourth, we got 5 times 00:06:44.410 --> 00:06:46.170 2x plus 3 to the fourth. 00:06:46.170 --> 00:06:51.540 And if we just simplify that, f prime of x is equal to 00:06:51.540 --> 00:06:58.220 2 times 5 is 10; 10 times 2x plus 3 to the fourth. 00:06:58.220 --> 00:07:01.650 That was a lot simpler than multiplying out 2x plus 3 to 00:07:01.650 --> 00:07:04.260 the fifth power, and then doing the derivatives the old way. 00:07:04.260 --> 00:07:06.080 I know this was probably a little confusing to you, 00:07:06.080 --> 00:07:09.270 so I'm going to try to do a couple more examples. 00:07:09.270 --> 00:07:25.570 Let's say I had g of x is equal to x-squared plus 2x plus 00:07:25.570 --> 00:07:30.620 3 to the eighth power. 00:07:30.620 --> 00:07:36.430 So g prime of x is going to equal-- well what did we say? 00:07:36.430 --> 00:07:38.090 We take the derivative of the inside. 00:07:38.090 --> 00:07:40.260 This is called the chain rule. 00:07:40.260 --> 00:07:41.410 What's the derivative of the inside? 00:07:41.410 --> 00:07:48.250 It's 2x plus 2 plus 0, right? 00:07:48.250 --> 00:07:50.190 And then we take the derivative of the whole thing. 00:07:50.190 --> 00:07:53.100 And we just pretend like this whole expression, x-squared 00:07:53.100 --> 00:07:57.540 plus 2x plus 3 is just kind of like the variable x. 00:07:57.540 --> 00:07:59.590 We know that the derivative of x to the eighth is 00:07:59.590 --> 00:08:01.520 8x to the seventh. 00:08:01.520 --> 00:08:07.410 So it'll be 8 times something to the seventh. 00:08:07.410 --> 00:08:10.580 And that something is this entire expression here, 8 times 00:08:10.580 --> 00:08:15.660 x-squared plus 2x plus 3. 00:08:15.660 --> 00:08:17.040 I hope I didn't confuse you too much. 00:08:17.040 --> 00:08:18.790 And you can simplify this more in any way. 00:08:18.790 --> 00:08:21.900 Because it's 2x plus 2 times 8 times x-squared plus 2x 00:08:21.900 --> 00:08:23.035 plus 3 to the seventh. 00:08:23.035 --> 00:08:26.300 To multiply this out, or to multiply this out is 00:08:26.300 --> 00:08:27.800 a huge pain as you know. 00:08:27.800 --> 00:08:30.840 But we could simplify a little bit. 00:08:30.840 --> 00:08:33.630 Let me draw a divider here. 00:08:33.630 --> 00:08:44.526 We could say that that equals 8 times 2x, 16x plus 16 times-- 00:08:44.526 --> 00:08:48.300 I'm making it really messy-- x-squared plus 2x plus 3 00:08:48.300 --> 00:08:50.610 to the seventh power. 00:08:50.610 --> 00:08:51.870 I hope I didn't confuse you too much. 00:08:51.870 --> 00:08:53.920 In the next presentation, I'm just going to do a ton of 00:08:53.920 --> 00:08:55.500 examples using the chain rule. 00:08:55.500 --> 00:08:57.850 And I think the more examples you see, it's going to 00:08:57.850 --> 00:08:58.470 hit the point home. 00:08:58.470 --> 00:09:00.440 And then after I've done a bunch of examples, then I'm 00:09:00.440 --> 00:09:01.960 going to give you a formal definition. 00:09:01.960 --> 00:09:04.190 I think that's actually an easier way to digest the chain 00:09:04.190 --> 00:09:06.970 rule than giving you the formal definition first, and then 00:09:06.970 --> 00:09:08.440 showing you a bunch of examples. 00:09:08.440 --> 00:09:10.520 So I'll see you in the next presentation.

Power rule introduction (old)

https://www.youtube.com/watch?v=z1lwai-lIzY

vtt

https://www.youtube.com/api/timedtext?v=z1lwai-lIzY&ei=eWeUZYr8N9edmLAPxoyyiAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249833&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=9554E66D3C84695FD20D1D6F2B6A571B25D91417.D99E33D3A7FB638EFC35D70D1A502E3E6251F27A&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.710 --> 00:00:01.820 Welcome back. 00:00:01.820 --> 00:00:04.820 In the last presentation I showed you that if I had the 00:00:04.820 --> 00:00:13.530 function f of x is equal to x squared, that the derivative of 00:00:13.530 --> 00:00:17.980 this function, which is denoted by f-- look at that, my pen 00:00:17.980 --> 00:00:19.835 is already malfunctioning. 00:00:19.835 --> 00:00:26.190 The derivative of that function, f prime of 00:00:26.190 --> 00:00:28.770 x, is equal to 2x. 00:00:28.770 --> 00:00:32.050 And I used the limit definition of a derivative. 00:00:32.050 --> 00:00:34.375 I used, let me write it down here. 00:00:38.020 --> 00:00:39.540 This pen is horrible. 00:00:39.540 --> 00:00:42.520 I need to really figure out some other tool to use. 00:00:42.520 --> 00:00:47.160 The limit as h approaches 0 -- sometimes you'll see delta x 00:00:47.160 --> 00:00:51.440 instead of h, but it's the same thing-- of f of x plus 00:00:51.440 --> 00:00:57.900 h minus f of x over h. 00:00:57.900 --> 00:01:00.890 And I used this definition of a derivative, which is really 00:01:00.890 --> 00:01:03.860 just the slope at any given point along the curve, 00:01:03.860 --> 00:01:04.670 to figure this out. 00:01:04.670 --> 00:01:06.940 That if f of x is equal to x squared, that 00:01:06.940 --> 00:01:08.970 the derivative is 2x. 00:01:08.970 --> 00:01:11.360 And you could actually use this to do others. 00:01:11.360 --> 00:01:12.940 And I won't do it now, maybe I'll do it in a 00:01:12.940 --> 00:01:13.710 future presentation. 00:01:13.710 --> 00:01:21.176 But it turns out that if you have f of x is equal to x to 00:01:21.176 --> 00:01:26.630 the third, that the derivative is f prime of x is 00:01:26.630 --> 00:01:32.700 equal to 3x squared. 00:01:32.700 --> 00:01:39.310 If f of x is equal to x to the fourth, well then the 00:01:39.310 --> 00:01:46.550 derivative is equal to 4x to the third. 00:01:46.550 --> 00:01:48.840 I think you're starting to see a pattern here. 00:01:48.840 --> 00:01:51.812 If I actually wrote up here that if f of x -- let me see 00:01:51.812 --> 00:01:55.080 if I have space to write it neatly. 00:01:55.080 --> 00:01:57.510 If I wrote f of x -- I hope you can see this -- f 00:01:57.510 --> 00:01:59.340 of x is equal to x. 00:01:59.340 --> 00:02:00.000 Well you know this. 00:02:00.000 --> 00:02:02.270 I mean, y equals x, what's the slope of y equals x? 00:02:02.270 --> 00:02:05.600 That's just 1, right? 00:02:05.600 --> 00:02:07.330 y equals x, that's a slope of 1. 00:02:07.330 --> 00:02:09.820 You didn't need to know calculus to know that. 00:02:09.820 --> 00:02:13.870 f prime of x is just equal to 1. 00:02:13.870 --> 00:02:16.270 And then you can probably guess what the next one is. 00:02:16.270 --> 00:02:23.760 If f of x is equal to x to the fifth, then the derivative is-- 00:02:23.760 --> 00:02:29.230 I think you could guess-- 5 x to the fourth. 00:02:29.230 --> 00:02:34.730 So in general, for any expression within a polynomial, 00:02:34.730 --> 00:02:42.090 or any degree x to whatever power-- let's say f of x is 00:02:42.090 --> 00:02:47.000 equal to-- this pen drives me nuts. 00:02:47.000 --> 00:02:50.380 f of x is equal to x to the n, right? 00:02:50.380 --> 00:02:52.650 Where n could be any exponent. 00:02:52.650 --> 00:03:02.430 Then f prime of x is equal to nx to the n minus 1. 00:03:02.430 --> 00:03:04.160 And you see this is what the case was in all 00:03:04.160 --> 00:03:05.700 these situations. 00:03:05.700 --> 00:03:07.620 That 1 didn't show up. 00:03:07.620 --> 00:03:08.730 n minus 1. 00:03:08.730 --> 00:03:13.110 So if n was 25, x to the 25th power, the derivative 00:03:13.110 --> 00:03:16.110 would be 25 x to the 24th. 00:03:16.110 --> 00:03:18.826 So I'm going to use this rule and then I'm going to show 00:03:18.826 --> 00:03:19.690 you a couple of other ones. 00:03:19.690 --> 00:03:21.920 And then now we can figure out the derivative of pretty much 00:03:21.920 --> 00:03:23.270 any polynomial function. 00:03:26.400 --> 00:03:28.690 So just another couple of rules. 00:03:28.690 --> 00:03:32.180 This might be a little intuitive for you, and if you 00:03:32.180 --> 00:03:33.830 use that limit definition of a derivative, you could 00:03:33.830 --> 00:03:36.630 actually prove it. 00:03:36.630 --> 00:03:42.170 But if I want to figure out the derivative of, let's say, the 00:03:42.170 --> 00:03:51.200 derivative of-- So another way of-- this is kind of, what is 00:03:51.200 --> 00:03:52.730 the change with respect to x? 00:03:52.730 --> 00:03:54.620 This is another notation. 00:03:54.620 --> 00:03:57.530 I think this is what Leibniz uses to figure out the 00:03:57.530 --> 00:03:58.400 derivative operator. 00:03:58.400 --> 00:04:08.855 So if I wanted to find the derivative of A f of x, where A 00:04:08.855 --> 00:04:09.910 is just some constant number. 00:04:09.910 --> 00:04:11.660 It could be 5 times f of x. 00:04:11.660 --> 00:04:15.850 This is the same thing as saying A times the 00:04:15.850 --> 00:04:21.460 derivative of f of x. 00:04:21.460 --> 00:04:22.400 And what does that tell us? 00:04:22.400 --> 00:04:27.880 Well, this tells us that, let's say I had f of x. 00:04:27.880 --> 00:04:31.180 f of x is equal to-- and this only works with the constants-- 00:04:31.180 --> 00:04:35.680 f of x is equal to 5x squared. 00:04:35.680 --> 00:04:36.460 Right? 00:04:36.460 --> 00:04:40.590 Well this is the same thing as 5 times x squared. 00:04:40.590 --> 00:04:42.070 I know I'm stating the obvious. 00:04:42.070 --> 00:04:44.510 So we can just say that the derivative of this is just 5 00:04:44.510 --> 00:04:46.970 times the derivative of x squared. 00:04:46.970 --> 00:04:53.690 So f prime of x is equal to 5 times, and what's the 00:04:53.690 --> 00:04:55.170 derivative of x squared? 00:04:55.170 --> 00:04:57.260 Right, it's 2x. 00:04:57.260 --> 00:04:59.300 So it equals 10x. 00:04:59.300 --> 00:04:59.970 Right? 00:04:59.970 --> 00:05:02.700 Similarly, let's say I had g of x, just using 00:05:02.700 --> 00:05:03.910 a different letter. 00:05:03.910 --> 00:05:08.000 g of x is equal to-- and my pen keeps malfunctioning. 00:05:08.000 --> 00:05:18.090 g of x is equal to, let's say, 3x to the 12th. 00:05:18.090 --> 00:05:23.530 Then g prime of x, or the derivative of g, is equal 00:05:23.530 --> 00:05:27.490 to 3 times the derivative of x to the 12th. 00:05:27.490 --> 00:05:28.450 Well we know what that is. 00:05:28.450 --> 00:05:33.540 It's 12 x to the 11th. 00:05:33.540 --> 00:05:34.800 Which you would have seen. 00:05:34.800 --> 00:05:36.680 12x to the 11th. 00:05:36.680 --> 00:05:41.180 This equals 36x to the 11th. 00:05:41.180 --> 00:05:42.190 Pretty straightforward, right? 00:05:42.190 --> 00:05:44.540 You just multiply the constant times whatever the 00:05:44.540 --> 00:05:45.250 derivative would have been. 00:05:45.250 --> 00:05:48.450 I think you get that. 00:05:48.450 --> 00:05:51.050 Now one other thing. 00:05:51.050 --> 00:05:55.930 If I wanted to apply the derivative operator-- let me 00:05:55.930 --> 00:05:58.330 change colors just to mix things up a little bit. 00:05:58.330 --> 00:06:02.450 Let's say if I wanted to apply the derivative of operator-- I 00:06:02.450 --> 00:06:04.650 think this is called the addition rule. 00:06:04.650 --> 00:06:06.650 It might be a little bit obvious. 00:06:06.650 --> 00:06:13.020 f of x plus g of x. 00:06:16.910 --> 00:06:26.060 This is the same thing as the derivative of f of x plus 00:06:26.060 --> 00:06:29.110 the derivative3 of g of x. 00:06:29.110 --> 00:06:30.640 That might seem a little complicated to you, but all 00:06:30.640 --> 00:06:33.290 it's saying is that you can find the derivative of each of 00:06:33.290 --> 00:06:35.920 the parts when you're adding up, and then that's the 00:06:35.920 --> 00:06:37.860 derivative of the whole thing. 00:06:37.860 --> 00:06:40.570 I'll do a couple of examples. 00:06:40.570 --> 00:06:42.230 So what does this tell us? 00:06:42.230 --> 00:06:43.910 This is also the same thing, of course. 00:06:43.910 --> 00:06:46.630 This is, I believe, Leibniz's notation. 00:06:46.630 --> 00:06:50.000 And then Lagrange's notation is-- of course these were the 00:06:50.000 --> 00:06:53.530 founding fathers of calculus. 00:06:53.530 --> 00:06:57.335 That's the same thing as f prime of x plus g prime of x. 00:06:57.335 --> 00:06:59.910 And let me apply this, because whenever you apply it, I think 00:06:59.910 --> 00:07:01.140 it starts to seem a lot more obvious. 00:07:01.140 --> 00:07:19.500 So let's say f of x is equal to 3x squared plus 5x plus 3. 00:07:19.500 --> 00:07:22.080 Well, if we just want to figure out the derivative, we say f 00:07:22.080 --> 00:07:25.445 prime of x, we just find the derivative of each 00:07:25.445 --> 00:07:26.080 of these terms. 00:07:26.080 --> 00:07:29.070 Well, this is 3 times the derivative of x squared. 00:07:29.070 --> 00:07:30.200 The derivative of x squared, we already figured 00:07:30.200 --> 00:07:31.930 out, is 2x, right? 00:07:31.930 --> 00:07:34.230 So this becomes 6x. 00:07:34.230 --> 00:07:36.140 Really you just take the 2, multiply it by the 3, and 00:07:36.140 --> 00:07:38.580 then decrement the 2 by 1. 00:07:38.580 --> 00:07:42.120 So it's really 6x to the first, which is the same thing as 6x. 00:07:42.120 --> 00:07:45.182 Plus the derivative of 5x is 5. 00:07:45.182 --> 00:07:48.440 And you know that because if I just had a line that's y equals 00:07:48.440 --> 00:07:50.920 5x, the slope is 5, right? 00:07:50.920 --> 00:07:53.950 Plus, what's the derivative of a constant function? 00:07:53.950 --> 00:07:55.410 What's the derivative of 3? 00:07:55.410 --> 00:07:57.410 Well, I'll give you a hint. 00:07:57.410 --> 00:08:00.950 Graph y equals 3 and tell me what the slope is. 00:08:00.950 --> 00:08:05.240 Right, the derivative of a constant is 0. 00:08:05.240 --> 00:08:08.230 I'll show other times why that might be more intuitive. 00:08:08.230 --> 00:08:09.500 Plus 0. 00:08:09.500 --> 00:08:10.720 You can just ignore that. 00:08:10.720 --> 00:08:13.770 f prime of x is equal to 6x plus 5. 00:08:13.770 --> 00:08:14.480 Let's do some more. 00:08:17.260 --> 00:08:19.510 I think the more examples we do, the better. 00:08:24.470 --> 00:08:27.790 And I want to keep switching notations, so you don't get 00:08:27.790 --> 00:08:29.340 daunted whenever you see it in a different way. 00:08:29.340 --> 00:08:39.540 Let's say y equals 10x to the fifth minus 7x to the 00:08:39.540 --> 00:08:46.350 third plus 4x plus 1. 00:08:46.350 --> 00:08:48.920 So here we're going to apply the derivative operator. 00:08:48.920 --> 00:08:55.760 So we say dy-- this is I think Leibniz's 00:08:55.760 --> 00:08:59.040 notation-- dy over dx. 00:08:59.040 --> 00:09:01.170 And that's just the change in y over the change in x, 00:09:01.170 --> 00:09:02.960 over very small changes. 00:09:02.960 --> 00:09:06.040 That's kind of how I view this d, like a very small delta. 00:09:06.040 --> 00:09:17.500 Is equal to 5 times 10 is 50 x to the fourth minus 21 -- 00:09:17.500 --> 00:09:24.530 right, 3 times 7-- x squared plus 4. 00:09:24.530 --> 00:09:27.160 And then the 1, the derivative of 1 is just 0. 00:09:27.160 --> 00:09:27.650 So there it is. 00:09:27.650 --> 00:09:29.340 We figured out the derivative of this very 00:09:29.340 --> 00:09:30.210 complicated function. 00:09:30.210 --> 00:09:31.510 And it was pretty straightforward. 00:09:31.510 --> 00:09:34.050 I think you'll find that derivatives of polynomials are 00:09:34.050 --> 00:09:36.570 actually more straightforward than a lot of concepts that you 00:09:36.570 --> 00:09:39.170 learned a lot earlier in mathematics. 00:09:39.170 --> 00:09:41.160 That's all the time I have now for this presentation. 00:09:41.160 --> 00:09:43.820 In the next couple I'll just do a bunch of more examples, and 00:09:43.820 --> 00:09:46.100 I'll show you some more rules for solving even more 00:09:46.100 --> 00:09:47.175 complicated derivatives. 00:09:47.175 --> 00:09:49.300 See you in the next presentation.

Calculus: Derivatives 2

https://www.youtube.com/watch?v=ay8838UZ4nM

vtt

https://www.youtube.com/api/timedtext?v=ay8838UZ4nM&ei=fGeUZYDrI4qhp-oP7paXIA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249836&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=6F3A6F7EA9E5E3D198B1DCD0CE0C20F5B5578FDC.796CD612A63A0BA4E25EEA86172903FF84BB1AD5&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.790 --> 00:00:02.730 In the last presentation, I hopefully gave you a little 00:00:02.730 --> 00:00:05.290 bit of an intuition of what a derivative is. 00:00:05.290 --> 00:00:07.920 It's really just a way to find the slope at a given 00:00:07.920 --> 00:00:10.100 point along the curve. 00:00:10.100 --> 00:00:12.210 Now we'll actually apply it to some functions. 00:00:12.210 --> 00:00:18.200 So let's say I had the function f of x. 00:00:18.200 --> 00:00:22.150 f of x is equal to x squared. 00:00:22.150 --> 00:00:27.325 And I want to know what is the slope of this curve. 00:00:40.410 --> 00:00:47.220 What is the slope at x is equal to-- let's say at x equals 3. 00:00:47.220 --> 00:00:48.520 What is the slope of x? 00:00:48.520 --> 00:00:51.720 Let's draw out what I'm asking. 00:00:51.720 --> 00:00:53.300 Coordinate axis. 00:00:53.300 --> 00:00:57.310 x-coordinate, that's the y-coordinate. 00:00:57.310 --> 00:00:59.260 And then if I were to draw-- let me pick a different color. 00:01:05.910 --> 00:01:10.690 So we want to say what is the slope when x is equal to 3. 00:01:16.310 --> 00:01:19.590 This is x equals 3. 00:01:19.590 --> 00:01:23.240 And of course when x equals 3, f of x is equal to 9. 00:01:23.240 --> 00:01:25.010 We know that, right? 00:01:28.830 --> 00:01:32.000 So what we do is we take a point, maybe a little bit 00:01:32.000 --> 00:01:33.120 further along the curve. 00:01:33.120 --> 00:01:37.880 Let's say this point right here is 3 plus h. 00:01:37.880 --> 00:01:39.780 And I keep it abstract as h because as you know we're 00:01:39.780 --> 00:01:41.720 going to take the limit as h approaches 0. 00:01:41.720 --> 00:01:45.230 And at this point right here is what? 00:01:45.230 --> 00:01:48.390 It's 3 plus h squared, right? 00:01:48.390 --> 00:01:52.000 Because the function is f of x is equal to x squared. 00:01:52.000 --> 00:02:08.700 So this point right here is 3 plus h, 3 plus h squared. 00:02:08.700 --> 00:02:10.909 Because we just take the 3 plus h and put it into x squared 00:02:10.909 --> 00:02:12.360 and we get 3 plus h squared. 00:02:12.360 --> 00:02:17.000 And this point here is of course 3, 9. 00:02:17.000 --> 00:02:19.340 What we want to do is we want to find the slope 00:02:19.340 --> 00:02:20.540 between these two point. 00:02:23.600 --> 00:02:25.310 I really have to find a better tool. 00:02:25.310 --> 00:02:28.890 This one keeps freezing, I think it's too CPU intensive. 00:02:28.890 --> 00:02:29.500 But anyway. 00:02:29.500 --> 00:02:31.480 So we want to find the slope between these two points. 00:02:31.480 --> 00:02:34.500 So what's the slope? so it's a change in y, so it's 3 plus h 00:02:34.500 --> 00:02:50.240 squared minus this y minus 9 over the change in x. 00:02:50.240 --> 00:03:03.000 Well that's 3 plus h minus 3. 00:03:03.000 --> 00:03:05.780 So if we simplify this top part or we multiply it out, 00:03:05.780 --> 00:03:06.720 what's 3 plus h squared? 00:03:06.720 --> 00:03:19.260 That's 9 plus 6h plus h squared, and then get the minus 00:03:19.260 --> 00:03:26.520 9, and all of that is over-- well this 3 and this minus 00:03:26.520 --> 00:03:29.730 3 cancel out, so all you're left is with h. 00:03:29.730 --> 00:03:31.890 And even if we simplify this, this 9 minus 00:03:31.890 --> 00:03:34.892 9, they cancel out. 00:03:34.892 --> 00:03:37.005 So let me go up here. 00:03:40.700 --> 00:03:49.210 We're left with-- this pen keeps freezing-- it's 6h 00:03:49.210 --> 00:03:53.040 plus h squared over h. 00:03:53.040 --> 00:03:55.500 And now we would simplify this, right, because we can divide 00:03:55.500 --> 00:03:57.160 the top and the bottom, that numerator and the 00:03:57.160 --> 00:03:58.260 denominator by h. 00:03:58.260 --> 00:04:05.200 And you get 6 plus h squared. 00:04:05.200 --> 00:04:08.230 So that's the slope between these two points. 00:04:08.230 --> 00:04:10.240 It's 6 plus h squared. 00:04:10.240 --> 00:04:12.980 So if we want to find the instantaneous slope at the 00:04:12.980 --> 00:04:17.560 point x equals 3, f of x is equal to 9, or the point 3,9, 00:04:17.560 --> 00:04:20.450 we just have to find the limit as h approaches 0 here. 00:04:20.450 --> 00:04:27.670 So we'll just take the limit as h approaches 0. 00:04:27.670 --> 00:04:29.240 Well this is an easy limit problem, right? 00:04:29.240 --> 00:04:32.620 What's the limit of 6 plus h squared as h approaches 0? 00:04:32.620 --> 00:04:34.340 Well it equals 6. 00:04:34.340 --> 00:04:38.300 So we now know that the slope of this curve at the 00:04:38.300 --> 00:04:41.970 point x equals 3 is 6. 00:04:45.860 --> 00:04:49.600 So if you actually did a traditional rise over run, 00:04:49.600 --> 00:04:52.390 the slope, this change in y over change in x is 6. 00:04:52.390 --> 00:04:54.830 So we have the instantaneous slope at exactly the 00:04:54.830 --> 00:04:57.140 point x is equal to 3. 00:04:57.140 --> 00:05:00.070 So that's useful. 00:05:00.070 --> 00:05:05.960 You know if this was a graph of someone's position, we would 00:05:05.960 --> 00:05:08.570 then know kind of the instantaneous velocity, which 00:05:08.570 --> 00:05:09.820 is-- well I won't go into that. 00:05:09.820 --> 00:05:11.710 I'll do a separate module on physics. 00:05:11.710 --> 00:05:13.620 But this was useful, but let's see if we can do more 00:05:13.620 --> 00:05:16.440 generalized version where we don't have to know ahead of 00:05:16.440 --> 00:05:18.550 time what point we want to find the slope at. 00:05:18.550 --> 00:05:22.290 If we can get a generalized formula for the slope at any 00:05:22.290 --> 00:05:26.870 point along the graph f of x is equal to x squared. 00:05:26.870 --> 00:05:30.140 So let me clear this. 00:05:30.140 --> 00:05:42.470 So we're going to stick with f of x is equal to x squared. 00:05:42.470 --> 00:05:46.180 And we know that the slope at any point of this is just going 00:05:46.180 --> 00:06:03.050 to be the limit as h approaches 0 of f of x plus h 00:06:03.050 --> 00:06:07.530 minus f of access. 00:06:07.530 --> 00:06:11.650 All of that over h. 00:06:11.650 --> 00:06:14.590 This part right here, this is just the slope formula that 00:06:14.590 --> 00:06:16.080 you learned years ago. 00:06:16.080 --> 00:06:18.610 It's just change in y over change in x. 00:06:18.610 --> 00:06:21.450 And all we're doing is we're seeing what happens as the 00:06:21.450 --> 00:06:23.860 change in x gets smaller and smaller and smaller as it 00:06:23.860 --> 00:06:24.860 actually approaches 0. 00:06:24.860 --> 00:06:27.090 And that's why we can get the instantaneous change at 00:06:27.090 --> 00:06:28.250 that point in the curve. 00:06:28.250 --> 00:06:30.610 So let's apply this definition of a derivative 00:06:30.610 --> 00:06:33.490 to this function. 00:06:33.490 --> 00:06:37.040 And actually if you want to know the notation, I think 00:06:37.040 --> 00:06:39.190 this is the notation Lagrange came up with. 00:06:39.190 --> 00:06:43.950 This is equal to f prime of x. 00:06:43.950 --> 00:06:45.210 Don't take my word on it on Lagrange. 00:06:45.210 --> 00:06:46.970 You might want to look it up on Wikipedia. 00:06:46.970 --> 00:06:48.250 But this [UNINTELLIGIBLE] 00:06:48.250 --> 00:06:51.980 derivative of f of x is f prime of x. 00:06:51.980 --> 00:06:53.630 Let's apply it to x squared. 00:06:53.630 --> 00:06:59.490 So we're going to say the limit as h approaches 00:06:59.490 --> 00:07:02.790 0 of f of x plus h. 00:07:02.790 --> 00:07:08.210 Well, f of x plus h is just-- this pen driving me 00:07:08.210 --> 00:07:13.580 crazy-- x plus h squared. 00:07:13.580 --> 00:07:16.690 I just took the x plus h and put it into f of x. 00:07:16.690 --> 00:07:24.730 Minus f of x--well that's just x squared-- over h. 00:07:24.730 --> 00:07:30.490 And this is equal to the limit as h approaches 0. 00:07:30.490 --> 00:07:32.240 Just multiply this out of. 00:07:32.240 --> 00:07:43.710 x squared plus 2xh plus h squared minus x squared-- 00:07:43.710 --> 00:07:48.350 running out of space-- all of that over h. 00:07:48.350 --> 00:07:49.310 Let's simplify this. 00:07:49.310 --> 00:07:53.160 This x squared cancels out with this minus x squared. 00:07:53.160 --> 00:07:56.260 And then we can divide the numerator and the denominator 00:07:56.260 --> 00:08:07.005 by h, and we're left with the limit as h approaches 0. 00:08:07.005 --> 00:08:14.180 Numerator and denominator by h of 2x plus h. 00:08:14.180 --> 00:08:16.470 Well this is easy. 00:08:16.470 --> 00:08:19.520 This goes to 0, this is just equal to 2x. 00:08:19.520 --> 00:08:20.640 So there we have it. 00:08:20.640 --> 00:08:24.520 The limit as h approaches 0 is equal to 2x. 00:08:24.520 --> 00:08:27.650 And this is equal to f prime of x, so the derivative of f of 00:08:27.650 --> 00:08:32.415 x, which is the denoted by f prime of x is equal to 2x. 00:08:32.415 --> 00:08:33.660 Well what does it tell us? 00:08:33.660 --> 00:08:35.270 What have we done for ourselves? 00:08:35.270 --> 00:08:38.930 Well now I can give you any point along the curve. 00:08:38.930 --> 00:08:43.464 Let's say we want to know the slope at the point 16, right. 00:08:46.260 --> 00:08:51.200 When at the point 16,256. 00:08:51.200 --> 00:08:53.870 That's a point along f of x equals x squared. 00:08:53.870 --> 00:08:55.870 It's just 16 and then 16 squared. 00:08:55.870 --> 00:08:57.250 What's the slope at that point? 00:08:57.250 --> 00:08:59.746 Well we now know the slope is 2 times 16. 00:09:04.640 --> 00:09:09.340 So the slope is equal to 32. 00:09:09.340 --> 00:09:12.545 Whatever the x value is you just put into this f prime of 00:09:12.545 --> 00:09:15.750 x function or the derivative function, and you'll get 00:09:15.750 --> 00:09:17.060 the slope at that point. 00:09:17.060 --> 00:09:19.270 I think that's pretty neat and I'll show you how in future 00:09:19.270 --> 00:09:22.100 presentations how we can apply this to physics and 00:09:22.100 --> 00:09:24.090 optimization problems and a whole other set of things. 00:09:24.090 --> 00:09:26.290 And I'm also going to show you how to find the derivatives for 00:09:26.290 --> 00:09:28.180 a whole set of other functions. 00:09:28.180 --> 00:09:29.030 I'll see in the next presentation.

Calculus: Derivatives 1

https://www.youtube.com/watch?v=rAof9Ld5sOg

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WEBVTT Kind: captions Language: en 00:00:01.080 --> 00:00:03.790 Welcome to the presentation on derivatives. 00:00:03.790 --> 00:00:06.290 I think you're going to find that this is when math starts 00:00:06.290 --> 00:00:10.530 to become a lot more fun than it was just a few topics ago. 00:00:10.530 --> 00:00:11.990 Well let's get started with our derivatives. 00:00:11.990 --> 00:00:13.150 I know it sounds very complicated. 00:00:13.150 --> 00:00:16.560 Well, in general, if I have a straight line-- let me see if I 00:00:16.560 --> 00:00:20.990 can draw a straight line properly-- if I had a straight 00:00:20.990 --> 00:00:28.020 line-- that's my coordinate axes, which aren't straight-- 00:00:28.020 --> 00:00:29.120 this is a straight line. 00:00:31.700 --> 00:00:34.630 But when I have a straight line like that, and I ask you to 00:00:34.630 --> 00:00:37.150 find the slope-- I think you already know how to do this-- 00:00:37.150 --> 00:00:39.540 it's just the change in y divided by the change in x. 00:00:39.540 --> 00:00:43.800 If I wanted to find the slope-- really I mean the slope is the 00:00:43.800 --> 00:00:46.140 same, because it is a straight line, the slope is the same 00:00:46.140 --> 00:00:50.040 across the whole line, but if I want to find the slope at any 00:00:50.040 --> 00:00:51.840 point in this line, what I would do is I would pick a 00:00:51.840 --> 00:00:56.000 point x-- say I'd pick this point. 00:00:56.000 --> 00:01:00.280 We'd pick a different color-- I'd take this point, I'd pick 00:01:00.280 --> 00:01:02.840 this point-- it's pretty arbitrary, I could pick any two 00:01:02.840 --> 00:01:05.990 points, and I would figure out what the change in y is-- this 00:01:05.990 --> 00:01:09.860 is the change in y, delta y, that's just another way of 00:01:09.860 --> 00:01:15.860 saying change in y-- and this is the change in x. 00:01:15.860 --> 00:01:16.920 delta x. 00:01:16.920 --> 00:01:22.140 And we figured out that the slope is defined really as 00:01:22.140 --> 00:01:30.250 change in y divided by change in x. 00:01:33.540 --> 00:01:37.640 And another way of saying that is delta-- it's that triangle-- 00:01:37.640 --> 00:01:41.180 delta y divided by delta x. 00:01:41.180 --> 00:01:42.660 Very straightforward. 00:01:42.660 --> 00:01:45.100 Now what happens, though, if we're not dealing 00:01:45.100 --> 00:01:45.860 with a straight line? 00:01:45.860 --> 00:01:49.780 Let me see if I have space to draw that. 00:01:49.780 --> 00:01:52.920 Another coordinate axes. 00:01:52.920 --> 00:01:55.780 Still pretty messy, but I think you'll get the point. 00:02:00.073 --> 00:02:02.960 Now let's say, instead of just a regular line like this, this 00:02:02.960 --> 00:02:04.690 follows the standard y equals mx plus b. 00:02:04.690 --> 00:02:09.640 Let's just say I had the curve y equals x squared. 00:02:09.640 --> 00:02:12.150 Let me draw it in a different color. 00:02:12.150 --> 00:02:15.940 So y equals x squared looks something like this. 00:02:15.940 --> 00:02:19.320 It's a curve, you're probably pretty familiar with it by now. 00:02:19.320 --> 00:02:20.650 And what I'm going to ask you is, what is the 00:02:20.650 --> 00:02:23.180 slope of this curve? 00:02:23.180 --> 00:02:24.440 And think about that. 00:02:24.440 --> 00:02:26.930 What does it mean to take the slope of a curve now? 00:02:26.930 --> 00:02:29.150 Well, in this line, the slope was the same throughout 00:02:29.150 --> 00:02:30.400 the whole line. 00:02:30.400 --> 00:02:31.780 But if you look at this curve, doesn't the 00:02:31.780 --> 00:02:32.960 slope change, right? 00:02:32.960 --> 00:02:36.560 Here it's almost flat, and it gets steeper steeper steeper 00:02:36.560 --> 00:02:38.865 steeper steeper until gets pretty steep. 00:02:38.865 --> 00:02:41.000 And if you go really far out, it gets extremely steep. 00:02:41.000 --> 00:02:42.630 So you're probably saying, well, how do you figure out 00:02:42.630 --> 00:02:45.910 the slope of a curve whose slope keeps changing? 00:02:45.910 --> 00:02:48.170 Well there is no slope for the entire curve. 00:02:48.170 --> 00:02:50.900 For a line, there is a slope for the entire line, because 00:02:50.900 --> 00:02:52.250 the slope never changes. 00:02:52.250 --> 00:02:54.380 But what we could try to do is figure out what the 00:02:54.380 --> 00:02:56.720 slope is at a given point. 00:02:56.720 --> 00:02:59.540 And the slope at a given point would be the same as the 00:02:59.540 --> 00:03:00.890 slope of a tangent line. 00:03:00.890 --> 00:03:08.260 For example-- let me pick a green-- the slope at this point 00:03:08.260 --> 00:03:17.970 right here would be the same as the slope of this line. 00:03:17.970 --> 00:03:19.180 Right? 00:03:19.180 --> 00:03:20.550 Because this line is tangent to it. 00:03:20.550 --> 00:03:24.350 So it just touches that curve, and at that exact point, they 00:03:24.350 --> 00:03:27.820 would have-- this blue curve, y equals x squared, would have 00:03:27.820 --> 00:03:30.650 the same slope as this green line. 00:03:30.650 --> 00:03:33.050 But if we go to a point back here, even though this is a 00:03:33.050 --> 00:03:36.940 really badly drawn graph, the slope would be 00:03:36.940 --> 00:03:38.700 something like this. 00:03:38.700 --> 00:03:40.015 The tangent slope. 00:03:40.015 --> 00:03:42.520 The slope would be a negative slope, and here it's a positive 00:03:42.520 --> 00:03:47.940 slope, but if we took a point here, the slope would 00:03:47.940 --> 00:03:50.630 be even more positive. 00:03:50.630 --> 00:03:52.030 So how are we going to figure this out? 00:03:52.030 --> 00:03:55.900 How are we going to figure out what the slope is at any point 00:03:55.900 --> 00:03:58.850 along the curve y equals x squared? 00:03:58.850 --> 00:04:01.590 That's where the derivative comes into use, and now for the 00:04:01.590 --> 00:04:03.290 first time you'll actually see why a limit is actually 00:04:03.290 --> 00:04:06.010 a useful concept. 00:04:06.010 --> 00:04:09.130 So let me try to redraw the curve. 00:04:09.130 --> 00:04:15.750 OK, I'll draw my axes, that's the y-axis-- I'll just do it in 00:04:15.750 --> 00:04:22.630 the first quadrant-- and this is-- I really have to find a 00:04:22.630 --> 00:04:29.470 better tool to do my-- this is x coordinate, and then let 00:04:29.470 --> 00:04:31.620 me draw my curve in yellow. 00:04:34.135 --> 00:04:37.790 So y equals x squared looks something like this. 00:04:37.790 --> 00:04:40.520 I'm really concentrating to draw this at 00:04:40.520 --> 00:04:41.700 least decently good. 00:04:41.700 --> 00:04:42.800 OK. 00:04:42.800 --> 00:04:46.890 So let's say we want to find the slope at this point. 00:04:53.560 --> 00:05:00.040 Let's call this point a. 00:05:00.040 --> 00:05:02.070 At this point, x equals a. 00:05:02.070 --> 00:05:06.845 And of course this is f of a. 00:05:11.340 --> 00:05:13.190 So what we could try to do is, we could try to find 00:05:13.190 --> 00:05:15.410 the slope of a secant line. 00:05:15.410 --> 00:05:19.920 A line between-- we take another point, say, somewhat 00:05:19.920 --> 00:05:26.550 close, to this point on the graph, let's say here, and if 00:05:26.550 --> 00:05:29.530 we could figure out the slope of this line, it would be a 00:05:29.530 --> 00:05:34.000 bit of an approximation of the slope of the curve 00:05:34.000 --> 00:05:35.090 exactly at this point. 00:05:35.090 --> 00:05:37.740 So let me draw that secant line. 00:05:44.460 --> 00:05:45.100 Something like that. 00:05:45.100 --> 00:05:46.690 Secant line looks something like that. 00:05:46.690 --> 00:05:56.050 And let's say that this point right here is a plus h, where 00:05:56.050 --> 00:05:59.860 this distance is just h, this is a plus h, we're just going 00:05:59.860 --> 00:06:05.160 h away from a, and then this point right here 00:06:05.160 --> 00:06:09.060 is f of a plus h. 00:06:11.640 --> 00:06:13.103 My pen is malfunctioning. 00:06:17.730 --> 00:06:19.550 So this would be an approximation for what the 00:06:19.550 --> 00:06:21.220 slope is at this point. 00:06:21.220 --> 00:06:24.980 And the closer that h gets, the closer this point gets to 00:06:24.980 --> 00:06:27.390 this point, the better our approximation is going to be, 00:06:27.390 --> 00:06:30.520 all the way to the point that if we could actually get the 00:06:30.520 --> 00:06:34.140 slope where h equals 0, that would actually be the slope, 00:06:34.140 --> 00:06:37.050 the instantaneous slope, at that point in the curve. 00:06:37.050 --> 00:06:41.440 But how can we figure out what the slope is when h equals 0? 00:06:44.800 --> 00:06:46.670 So right now, we're saying that the slope between these two 00:06:46.670 --> 00:06:49.900 points, it would be the change in y, so what's 00:06:49.900 --> 00:06:51.040 the change in y? 00:06:51.040 --> 00:06:57.170 It's this, so that this point right here is-- the x 00:06:57.170 --> 00:07:00.710 coordinate is-- my thing just keeps messing up-- the x 00:07:00.710 --> 00:07:11.330 coordinate is a plus h, and the y coordinate is f of a plus h. 00:07:15.180 --> 00:07:22.050 And this point right here, the coordinate is a and f of a. 00:07:22.050 --> 00:07:25.370 So if we just use the standard slope formula, like before, we 00:07:25.370 --> 00:07:27.610 would say change in y over change in x. 00:07:27.610 --> 00:07:29.100 Well, what's the change in y? 00:07:29.100 --> 00:07:37.680 It's f of a plus h-- this y coordinate minus this y 00:07:37.680 --> 00:07:46.850 coordinate-- minus f of a over the change in x. 00:07:46.850 --> 00:07:53.010 Well that change in x is this x coordinate, a plus h, minus 00:07:53.010 --> 00:07:55.720 this x coordinate, minus a. 00:07:55.720 --> 00:07:58.480 And of course this a and this a cancel out. 00:07:58.480 --> 00:08:01.490 So it's f of a plus h, minus f of a, all over h. 00:08:01.490 --> 00:08:05.400 This is just the slope of this secant line. 00:08:05.400 --> 00:08:08.810 And if we want to get the slope of the tangent line, we would 00:08:08.810 --> 00:08:11.980 just have to find what happens as h gets smaller and 00:08:11.980 --> 00:08:12.780 smaller and smaller. 00:08:12.780 --> 00:08:14.470 And I think you know where I'm going. 00:08:14.470 --> 00:08:16.840 Really, we just want to, if we want to find the slope of this 00:08:16.840 --> 00:08:19.140 tangent line, we just have to find the limit of this 00:08:19.140 --> 00:08:28.780 value as h approaches 0. 00:08:28.780 --> 00:08:32.700 And then, as h approaches 0, this secant line is going to 00:08:32.700 --> 00:08:36.710 get closer and closer to the slope of the tangent line. 00:08:36.710 --> 00:08:40.590 And then we'll know the exact slope at the instantaneous 00:08:40.590 --> 00:08:41.900 point along the curve. 00:08:41.900 --> 00:08:44.150 And actually, it turns out that this is the definition 00:08:44.150 --> 00:08:46.800 of the derivative. 00:08:46.800 --> 00:08:50.780 And the derivative is nothing more than the slope of a 00:08:50.780 --> 00:08:53.010 curve at an exact point. 00:08:53.010 --> 00:08:56.310 And this is super useful, because for the first time, 00:08:56.310 --> 00:08:58.570 everything we've talked about to this point is 00:08:58.570 --> 00:08:59.560 the slope of a line. 00:08:59.560 --> 00:09:03.050 But now we can take any continuous curve, or most 00:09:03.050 --> 00:09:06.800 continuous curves, and find the slope of that curve 00:09:06.800 --> 00:09:08.340 at an exact point. 00:09:08.340 --> 00:09:11.940 So now that I've given you the definition of what a derivative 00:09:11.940 --> 00:09:13.690 is, and maybe hopefully a little bit of intuition, in the 00:09:13.690 --> 00:09:17.070 next presentation I'm going to use this definition to actually 00:09:17.070 --> 00:09:20.040 apply it to some functions, like x squared and others, and 00:09:20.040 --> 00:09:21.930 give you some more problems. 00:09:21.930 --> 00:09:24.070 I'll see you in the next presentation

Limit examples (part 3)

https://www.youtube.com/watch?v=gWSDDopD9sk

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https://www.youtube.com/api/timedtext?v=gWSDDopD9sk&ei=eWeUZZr6FeaBxs0Psf-RwAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249833&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=837C781BD26BED58C019D9CD77FDF5F3F188FCBA.CF62ADE804F04795484A7EB8C64280ECA8F49BFF&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.800 --> 00:00:03.330 Let's do some more limit examples. 00:00:03.330 --> 00:00:05.250 So let's get another problem. 00:00:05.250 --> 00:00:19.860 If I had the limit as x approaches 3 of, let's say, 00:00:19.860 --> 00:00:31.190 x squared minus 6x plus 9 over x squared minus 9. 00:00:31.190 --> 00:00:33.450 So the first thing I like to do whenever I see any of these 00:00:33.450 --> 00:00:35.150 limits problems is just substitute the number in and 00:00:35.150 --> 00:00:36.413 see if I get something that makes sense, and 00:00:36.413 --> 00:00:37.670 then we'd be done. 00:00:37.670 --> 00:00:39.170 Well, usually we'd be done. 00:00:39.170 --> 00:00:41.490 I don't want to make these sweeping statements. 00:00:41.490 --> 00:00:43.880 If the function is continuous, we'd be done. 00:00:43.880 --> 00:00:47.250 But if we put the 3 in the numerator, we get 3 squared, 00:00:47.250 --> 00:00:51.040 which is 9, minus 18 plus 9. 00:00:51.040 --> 00:00:52.660 So that equals 0. 00:00:52.660 --> 00:00:55.510 And the denominator also-- let's see, 3 squared minus 00:00:55.510 --> 00:00:56.400 9, that also equals 0. 00:00:56.400 --> 00:01:00.970 So we don't like having 0/0. 00:01:00.970 --> 00:01:03.840 My pen tool is malfunctioning again. 00:01:03.840 --> 00:01:07.290 So we don't like getting 0, 0, 0, so is there any way we can 00:01:07.290 --> 00:01:12.010 simplify this expression to maybe get it to an expression 00:01:12.010 --> 00:01:15.110 that, when we evaluate it at x equals 3, we actually get 00:01:15.110 --> 00:01:16.640 something that makes sense? 00:01:16.640 --> 00:01:19.180 Well, whenever I see two of these polynomials here, and 00:01:19.180 --> 00:01:22.220 they look, just by inspecting them, relatively easy to 00:01:22.220 --> 00:01:25.220 factor, I like to factor them out because maybe there's the 00:01:25.220 --> 00:01:27.250 same factor in the numerator and the denominator, and 00:01:27.250 --> 00:01:28.610 then we can simplify it. 00:01:28.610 --> 00:01:32.110 So let's say that this is the same thing as-- that looks 00:01:32.110 --> 00:01:39.690 like it's x plus 3-- no, no, no, x minus 3. 00:01:39.690 --> 00:01:42.450 This is x minus 3. 00:01:42.450 --> 00:01:44.280 It actually looks like it's x minus 3 squared, but we're 00:01:44.280 --> 00:01:46.840 just going to write x minus 3 times x minus 3, which is, of 00:01:46.840 --> 00:01:48.820 course, x minus 3 squared. 00:01:48.820 --> 00:01:51.940 And then in the denominator, you know how to factor these, 00:01:51.940 --> 00:02:01.310 this is x plus 3 times x minus 3, all right? 00:02:01.310 --> 00:02:03.990 So the limit as x approaches 3 of this expression is the same 00:02:03.990 --> 00:02:07.960 thing as the limit as x approaches 3 of 00:02:07.960 --> 00:02:08.760 this expression. 00:02:08.760 --> 00:02:12.660 And, of course, there's nothing we can do to change the fact 00:02:12.660 --> 00:02:16.150 that this function, or this expression, is undefined 00:02:16.150 --> 00:02:17.360 at x equals 3. 00:02:17.360 --> 00:02:19.500 But if we can simplify it, we can figure out 00:02:19.500 --> 00:02:21.320 what it approaches. 00:02:21.320 --> 00:02:26.010 Well, if we assume that x is any number but 3, we can cross 00:02:26.010 --> 00:02:28.330 out these two terms because then they wouldn't be 0, right? 00:02:28.330 --> 00:02:31.020 It only is 0 when x is equal to 3 because-- so in the numerator 00:02:31.020 --> 00:02:32.990 and the denominator, we can cross this out. 00:02:32.990 --> 00:02:36.250 And we can say-- and I'm not being very rigorous here, but 00:02:36.250 --> 00:02:38.610 this is kind of how it's taught, and I think you get the 00:02:38.610 --> 00:02:43.180 intuition-- that this is the same thing as the limit as x 00:02:43.180 --> 00:02:49.200 approaches 3 of x minus 3 over x plus 3. 00:02:49.200 --> 00:02:52.330 Now let's just try to stick the x in and see what we get. 00:02:52.330 --> 00:02:54.010 Well, in the numerator, we get 3 minus 3. 00:02:54.010 --> 00:02:56.030 We still get 0. 00:02:56.030 --> 00:02:58.190 But in the denominator here, we get 6, right? 00:02:58.190 --> 00:02:59.520 3 plus 3 is 6. 00:02:59.520 --> 00:03:00.770 So now we get a good number. 00:03:00.770 --> 00:03:04.680 0 or 6, well, that's a real number, so it's 0. 00:03:04.680 --> 00:03:05.710 0/6 is 0. 00:03:05.710 --> 00:03:07.170 So that was interesting. 00:03:07.170 --> 00:03:09.660 The first time we did it, we got the answer 0/0. 00:03:09.660 --> 00:03:12.760 And now we get the answer 0 by simplifying. 00:03:12.760 --> 00:03:15.590 But, of course, it's very important to remember that 00:03:15.590 --> 00:03:18.810 this expression is not defined at x equals 3. 00:03:18.810 --> 00:03:21.280 It's defined everywhere but, but if we were to graph it, and 00:03:21.280 --> 00:03:24.070 I encourage you to do so, you would see that as you get 00:03:24.070 --> 00:03:26.980 closer and closer to x equals 3, the value of this 00:03:26.980 --> 00:03:28.960 expression will equal 0. 00:03:28.960 --> 00:03:30.060 And I know what you're thinking. 00:03:30.060 --> 00:03:31.350 Well, this was 0/0. 00:03:31.350 --> 00:03:38.190 Is every time I get 0/0 going to end up just becoming 0 when 00:03:38.190 --> 00:03:39.300 I evaluate the expression? 00:03:39.300 --> 00:03:42.240 Well, let's explore that. 00:03:42.240 --> 00:03:45.570 Let me clear this. 00:03:45.570 --> 00:03:56.080 Let's say what is-- pen is not working-- the limit as x 00:03:56.080 --> 00:04:17.860 approaches 1 of x squared minus x minus 2. 00:04:21.650 --> 00:04:23.640 No, let's say x squared plus x minus 2. 00:04:23.640 --> 00:04:26.110 As you can see, I do all this in my head, and 00:04:26.110 --> 00:04:27.450 I'm prone to mistakes. 00:04:27.450 --> 00:04:32.370 And all of that over x minus 1. 00:04:32.370 --> 00:04:33.940 Well, once again, if we just evaluate it, let's see what 00:04:33.940 --> 00:04:34.930 happens when x equals 1. 00:04:34.930 --> 00:04:38.200 You get 1 squared plus 1, so it's 2 minus 2. 00:04:38.200 --> 00:04:40.010 You get 0/0. 00:04:40.010 --> 00:04:44.640 So once again, we get 0/0, and we have to do something to 00:04:44.640 --> 00:04:46.960 this maybe to simplify it. 00:04:46.960 --> 00:04:48.620 Well, let's factor the top. 00:04:48.620 --> 00:04:54.530 So that's the same thing as the limit as x approaches 1. 00:04:54.530 --> 00:05:01.450 Well, that's x minus 1 times x plus 2, right? 00:05:07.020 --> 00:05:09.840 And I think you'll often discover when you see a lot of 00:05:09.840 --> 00:05:13.720 limit problems that even if this top factor, if this top 00:05:13.720 --> 00:05:16.710 expression, is hard to factor, chances are, one of the things 00:05:16.710 --> 00:05:19.390 in the denominator that are making this expression 00:05:19.390 --> 00:05:21.770 undefined is probably a factor up here. 00:05:21.770 --> 00:05:24.390 So sometimes you might get a more complex thing that isn't 00:05:24.390 --> 00:05:27.510 as easy to factor as this, but a good starting point is to 00:05:27.510 --> 00:05:30.840 guess that one of the factors is going to be in the bottom 00:05:30.840 --> 00:05:33.360 expression because that's kind of the trick of these problems, 00:05:33.360 --> 00:05:35.750 to just simplify the expression. 00:05:35.750 --> 00:05:39.390 So once again, if we assume that x does not equal 1, and 00:05:39.390 --> 00:05:43.440 this expression would not be 0 and this would not be 0, 00:05:43.440 --> 00:05:47.230 then these two could be canceled out. 00:05:47.230 --> 00:05:51.010 And we get that this is just the same thing as the limit as 00:05:51.010 --> 00:05:55.070 x approaches 1 of x plus 2. 00:05:55.070 --> 00:05:56.030 Well, now this is pretty easy. 00:05:56.030 --> 00:05:58.850 What's the limit as x approaches 1 of x plus 2? 00:05:58.850 --> 00:06:03.180 Well, you just stick 1 in there, and you get 3. 00:06:03.180 --> 00:06:04.010 So it's interesting. 00:06:04.010 --> 00:06:07.840 When we just tried to evaluate the expression at 00:06:07.840 --> 00:06:10.580 x equals 1, we got 0/0. 00:06:10.580 --> 00:06:14.840 And in the previous example, we saw that it evaluated out when 00:06:14.840 --> 00:06:17.870 you simplified it to 0, and in this example, it came out to 3. 00:06:17.870 --> 00:06:19.760 And I really encourage you, if you have a graphing calculator, 00:06:19.760 --> 00:06:23.110 graph these functions that we're doing and see and show 00:06:23.110 --> 00:06:26.080 yourself visually that it's true, that the limit as you 00:06:26.080 --> 00:06:30.190 approach, say, x equals 1 actually does approach the 00:06:30.190 --> 00:06:31.950 limits that were solving for. 00:06:31.950 --> 00:06:34.600 And make up your own problems. 00:06:34.600 --> 00:06:36.940 Hell, that's what I'm doing. 00:06:36.940 --> 00:06:38.650 So you could prove it to yourself. 00:06:38.650 --> 00:06:40.320 So let's do another. 00:06:40.320 --> 00:06:42.280 Let's do one that I think is pretty interesting. 00:06:48.020 --> 00:06:57.770 Let's say what's the limit as x approaches infinity? 00:06:57.770 --> 00:07:04.300 The limit as x approaches infinity of, let's say, x 00:07:04.300 --> 00:07:17.650 squared plus 3 over x to the third. 00:07:17.650 --> 00:07:19.960 So the way I think about these problems as they approach 00:07:19.960 --> 00:07:21.910 infinity, just think about what happens when you get 00:07:21.910 --> 00:07:24.960 really, really, really large values of x. 00:07:24.960 --> 00:07:28.030 And kind of a cheating way of doing this is, if you have a 00:07:28.030 --> 00:07:29.840 calculator, even if you don't have a calculator, put 00:07:29.840 --> 00:07:31.120 in huge numbers here. 00:07:31.120 --> 00:07:34.610 See what happens when x is a million, see what happens when 00:07:34.610 --> 00:07:36.900 x is a billion, see what happens when x is a trillion, 00:07:36.900 --> 00:07:38.030 and I think you'll get the point. 00:07:38.030 --> 00:07:40.270 You'll see what-- if there is a limit here, you'll 00:07:40.270 --> 00:07:41.500 see what it's going to. 00:07:41.500 --> 00:07:44.090 But the way I think about it is, in the numerator, kind of 00:07:44.090 --> 00:07:48.180 the fastest-growing term here is the x squared term, right? 00:07:48.180 --> 00:07:50.970 This is the fastest-growing term here. 00:07:50.970 --> 00:07:52.820 In the denominator, what's the fastest-growing term? 00:07:52.820 --> 00:07:54.475 Well, in the denominator, the fastest-growing term 00:07:54.475 --> 00:07:56.440 is this x to the third. 00:07:56.440 --> 00:07:58.160 Well, what's going to grow faster, x to the 00:07:58.160 --> 00:08:00.000 third or x squared? 00:08:00.000 --> 00:08:01.840 Well, yeah, x to the third's going to grow a lot 00:08:01.840 --> 00:08:03.130 faster than x squared. 00:08:03.130 --> 00:08:06.310 So this denominator, as you get larger and larger and larger 00:08:06.310 --> 00:08:10.230 values of x, is going to grow a lot faster than that numerator. 00:08:10.230 --> 00:08:12.700 So you could imagine if the denominator's growing much, 00:08:12.700 --> 00:08:14.905 much, much faster than the numerator, as you get larger 00:08:14.905 --> 00:08:17.130 and larger numbers, you're going to get a smaller and 00:08:17.130 --> 00:08:18.630 smaller and smaller fraction, right? 00:08:18.630 --> 00:08:20.280 It's going to approach 0. 00:08:20.280 --> 00:08:26.590 And so as you go to infinity, it approaches 0. 00:08:26.590 --> 00:08:29.530 I know that I kind of just hand waved, but that's really 00:08:29.530 --> 00:08:30.320 how you think about it. 00:08:30.320 --> 00:08:32.830 Another way you could do it is you could actually 00:08:32.830 --> 00:08:34.940 divide this fraction. 00:08:34.940 --> 00:08:37.470 You could actually divide this rational expression, and you'll 00:08:37.470 --> 00:08:39.750 get something like 1/x plus something, something, 00:08:39.750 --> 00:08:42.620 something, and then you'd also see, oh, well, the limit as x 00:08:42.620 --> 00:08:45.270 approaches infinity of 1/x is also 0. 00:08:45.270 --> 00:08:46.900 Let's do one more. 00:08:46.900 --> 00:08:49.190 I'll do this fast so I can confuse you. 00:08:49.190 --> 00:09:00.490 The limit as x approaches infinity of 3x squared plus 00:09:00.490 --> 00:09:05.670 x over 4x squared minus 5. 00:09:08.170 --> 00:09:10.680 These problems kind of look confusing sometimes, but 00:09:10.680 --> 00:09:11.480 they're really easy. 00:09:11.480 --> 00:09:13.330 You just have to think about what happens as you get 00:09:13.330 --> 00:09:14.700 really large values of x. 00:09:14.700 --> 00:09:19.320 Well, as you get really large values of x, these small terms, 00:09:19.320 --> 00:09:21.670 these ones that don't grow as fast as these large terms, 00:09:21.670 --> 00:09:23.640 kind of don't matter anymore, right, because you're getting 00:09:23.640 --> 00:09:25.270 really large values of x. 00:09:25.270 --> 00:09:28.210 And this case, these don't matter anymore, and then 00:09:28.210 --> 00:09:32.210 these two x terms grow at the same pace, right? 00:09:32.210 --> 00:09:34.070 And they'll always be kind of growing in 00:09:34.070 --> 00:09:35.000 this ratio of 3 to 4. 00:09:35.000 --> 00:09:37.870 So the limit here is actually that easy. 00:09:37.870 --> 00:09:39.880 It's 3/4. 00:09:39.880 --> 00:09:41.320 So what you do is you just figure out what's the 00:09:41.320 --> 00:09:43.940 fastest-growing term on the top, what's the fastest-growing 00:09:43.940 --> 00:09:47.000 term on the bottom, and then figure out what it approaches. 00:09:47.000 --> 00:09:49.680 If they're the same term, then they kind of cancel out, and 00:09:49.680 --> 00:09:51.700 you say the limit approaches 3/4. 00:09:51.700 --> 00:09:54.410 It's a very nonrigorous way of doing it, but it gets 00:09:54.410 --> 00:09:55.570 you the right answer. 00:09:55.570 --> 00:09:57.370 See you in the next presentation.

Limit examples (part 1)

https://www.youtube.com/watch?v=GGQngIp0YGI

vtt

https://www.youtube.com/api/timedtext?v=GGQngIp0YGI&ei=eWeUZfD-FoCCp-oPrPuFoAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249833&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B719D672878E1A5C7EDFCA5B2C61850F7CFEE1F1.18729DAEA7E4D59ACF0302C4F25551079E5B5082&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.280 --> 00:00:02.440 Welcome back. 00:00:02.440 --> 00:00:04.410 Now that we hopefully have a little bit of an intuition of 00:00:04.410 --> 00:00:07.360 what a limit is, or finding the limit of a function is, 00:00:07.360 --> 00:00:08.150 let's do some problems. 00:00:08.150 --> 00:00:12.150 Some of these you might actually see on your exams or 00:00:12.150 --> 00:00:14.610 when you're actually trying to solve a general limit problem. 00:00:14.610 --> 00:00:20.440 So let's say what is the limit-- once again, my 00:00:20.440 --> 00:00:23.020 pen is not working. 00:00:23.020 --> 00:00:31.740 What is the limit as x approaches-- let's 00:00:31.740 --> 00:00:34.960 say negative 1. 00:00:34.960 --> 00:00:40.170 And let me see, what's a good-- let's say my expression is-- 00:00:40.170 --> 00:00:42.480 I'll put it in parentheses so it's cleaner. 00:00:42.480 --> 00:00:53.490 It's 2x plus 2 over x plus 1. 00:00:53.490 --> 00:00:56.070 So the first thing I would always try to do is just say 00:00:56.070 --> 00:00:58.930 what happens if I just stick x straight into this expression? 00:00:58.930 --> 00:00:59.810 What happens? 00:00:59.810 --> 00:01:03.360 Well, what's 2x plus 2 when x is equal to negative 1? 00:01:03.360 --> 00:01:04.790 2 times negative 1. 00:01:04.790 --> 00:01:12.090 2 times negative 1 plus 2 over negative 1 plus 1. 00:01:12.090 --> 00:01:17.030 Well, the numerator is negative 2 plus 2-- that equals 0-- 00:01:17.030 --> 00:01:18.540 over-- what's the denominator? 00:01:18.540 --> 00:01:19.930 Negative 1 plus 1. 00:01:19.930 --> 00:01:21.150 Over 0. 00:01:21.150 --> 00:01:23.510 And do we know what 0 over 0 is? 00:01:23.510 --> 00:01:24.020 Well, no. 00:01:24.020 --> 00:01:26.250 It's undefined, right? 00:01:26.250 --> 00:01:29.320 So here's a case, just like what we saw in that first 00:01:29.320 --> 00:01:33.940 video, where the limit actually can't equal what the expression 00:01:33.940 --> 00:01:36.770 equals when you substitute x for the number you're trying to 00:01:36.770 --> 00:01:40.560 find the limit of because you get an undefined answer. 00:01:40.560 --> 00:01:42.750 So let's see if, using the limit, we can come up 00:01:42.750 --> 00:01:45.080 with a better answer for what it's approaching. 00:01:45.080 --> 00:01:47.120 Well, since we're just starting with these limit problems, 00:01:47.120 --> 00:01:48.580 let me draw a graph. 00:01:48.580 --> 00:01:50.700 And I think this is going to give you the intuition 00:01:50.700 --> 00:01:52.090 for what we're doing. 00:01:52.090 --> 00:01:53.560 It'll probably give you the answer. 00:01:53.560 --> 00:01:57.180 But then I'll show you how to solve this analytical. 00:01:57.180 --> 00:02:06.120 So if I draw a graph, these are the axes. 00:02:06.120 --> 00:02:09.710 Actually, I'll do the graphical and the analytical 00:02:09.710 --> 00:02:11.130 at the same time. 00:02:11.130 --> 00:02:13.450 So I want to rewrite this expression in a way that 00:02:13.450 --> 00:02:15.020 maybe I can simplify it. 00:02:15.020 --> 00:02:16.840 So 2x plus 2. 00:02:16.840 --> 00:02:22.805 Isn't that the same thing as 2 times x plus 1? 00:02:25.390 --> 00:02:27.960 2 times x plus 1, right? 00:02:27.960 --> 00:02:31.250 2x plus 2 is the same thing as 2 times x plus one, and then 00:02:31.250 --> 00:02:39.190 all of that is over x plus 1. 00:02:39.190 --> 00:02:44.440 So as long as this expression and this expression don't equal 00:02:44.440 --> 00:02:48.040 0, it actually turns out that this function-- let's say 00:02:48.040 --> 00:02:51.260 this is f of x, right? 00:02:51.260 --> 00:02:52.940 This function. 00:02:52.940 --> 00:02:56.660 Well, for every value other than x is equal to negative 00:02:56.660 --> 00:02:59.770 1, you could actually cancel this and this out. 00:02:59.770 --> 00:03:06.360 And so really, we see that f of x is equal to-- I need to find 00:03:06.360 --> 00:03:15.750 a better tool-- f of x is equal to 2 when x does not 00:03:15.750 --> 00:03:18.010 equal negative 1. 00:03:18.010 --> 00:03:22.510 And we saw when x is equal to negative 1, it's undefined. 00:03:22.510 --> 00:03:32.050 So undefined when equals negative 1. 00:03:32.050 --> 00:03:33.470 So how would we graph that? 00:03:33.470 --> 00:03:36.090 We showed that f of x is equal to 2 when x does not equal 00:03:36.090 --> 00:03:38.240 negative 1 and f of x is undefined when x 00:03:38.240 --> 00:03:39.630 equals negative 1. 00:03:39.630 --> 00:03:43.240 And once again, all I did is kind of rewrite this exact 00:03:43.240 --> 00:03:44.520 same function, right? 00:03:44.520 --> 00:03:47.380 I showed that I could simplify and I could divide the 00:03:47.380 --> 00:03:50.930 numerator and denominator by x plus 1 as long as x does not 00:03:50.930 --> 00:03:53.200 equal negative 1, and that otherwise, it's undefined. 00:03:53.200 --> 00:03:54.310 So let me graph this. 00:03:54.310 --> 00:03:56.200 I'm going to get a different color. 00:03:56.200 --> 00:03:58.890 Maybe I'll go with red. 00:03:58.890 --> 00:04:00.280 So this is 2. 00:04:00.280 --> 00:04:03.660 So we see that x is-- and let me say this is negative 1. 00:04:06.190 --> 00:04:10.080 So for every other value other than negative 1, the value of 00:04:10.080 --> 00:04:13.910 this, of f of x, is equal to 2. 00:04:13.910 --> 00:04:22.170 This is 1, this is 2, this is 3, and so on. 00:04:22.170 --> 00:04:24.950 At negative 1, the graph is undefined. 00:04:24.950 --> 00:04:26.160 So there's a hole there. 00:04:26.160 --> 00:04:29.940 And then we keep going to the left-hand side. 00:04:29.940 --> 00:04:33.480 So if we're going to do the limit, we can just visually 00:04:33.480 --> 00:04:37.560 say, well, as x-- let me do another color now. 00:04:37.560 --> 00:04:44.400 As x comes from the left-hand side, what does f of x equal? 00:04:44.400 --> 00:04:47.660 Well, f of x is 2, 2, 2, 2, 2. f of x is equal to 2 00:04:47.660 --> 00:04:51.230 until we get to exactly negative 1, right? 00:04:51.230 --> 00:04:53.700 And similarly, when we go from the other hand, it's 00:04:53.700 --> 00:04:54.940 the exact same thing. 00:04:54.940 --> 00:04:59.040 f of x is 2, 2, 2 until we get to negative 1. 00:04:59.040 --> 00:05:02.310 So you'll see, and I'll make sure you see it visually here, 00:05:02.310 --> 00:05:09.180 that the limit as approaches negative 1 of 2x plus 2 over 00:05:09.180 --> 00:05:11.900 x plus 1, it equals 2. 00:05:11.900 --> 00:05:13.500 Let me draw a line here so you don't get messed 00:05:13.500 --> 00:05:15.520 up with all of it. 00:05:15.520 --> 00:05:21.090 And I'm not formally, I guess, proving here that the limit is 00:05:21.090 --> 00:05:23.460 2, but I'm showing you kind of an analytical way, and this is 00:05:23.460 --> 00:05:26.040 actually how it tends to be done in algebra class, is that 00:05:26.040 --> 00:05:29.730 you tend to simplify the expression so that you say, oh, 00:05:29.730 --> 00:05:32.630 if there wasn't a hole here, what would the f of 00:05:32.630 --> 00:05:33.760 x equal, right? 00:05:33.760 --> 00:05:35.910 And then you'd just evaluate it at that point. 00:05:35.910 --> 00:05:39.790 I think this might give you a little intuition, but this 00:05:39.790 --> 00:05:42.050 isn't a formal solution. 00:05:42.050 --> 00:05:46.710 But unless you're asked to, you tend not to be asked 00:05:46.710 --> 00:05:47.800 for a formal solution. 00:05:47.800 --> 00:05:49.800 You actually just tend to ask what the limit is, and this is 00:05:49.800 --> 00:05:50.690 the way you could solve it. 00:05:50.690 --> 00:05:54.430 And actually another way that you could-- I mean, I often 00:05:54.430 --> 00:05:57.310 used to check my answers when I used to do it is you could take 00:05:57.310 --> 00:06:01.620 a calculator and try in-- what happens when: what is f 00:06:01.620 --> 00:06:08.035 of minus 1.001, right? 00:06:08.035 --> 00:06:15.730 And you can also try what is f of negative 0.999, right? 00:06:15.730 --> 00:06:17.210 Because what you want to do is you want to say, well, what 00:06:17.210 --> 00:06:20.010 does the function equal when I get really close to negative 1? 00:06:20.010 --> 00:06:23.410 And then you could keep going closer and closer to negative 00:06:23.410 --> 00:06:25.490 1 and see what the function approaches, and in this 00:06:25.490 --> 00:06:27.750 case, you'll see that it approaches 2. 00:06:27.750 --> 00:06:28.740 So let's do another problem. 00:06:32.240 --> 00:06:46.160 Well, let's say, what is the limit as x approaches 00:06:46.160 --> 00:06:51.570 0 of 1 over x? 00:06:51.570 --> 00:06:55.360 I think here it might be useful to draw this graph because 00:06:55.360 --> 00:06:58.520 it'll give you a visual reason, a visual represent-- actually, 00:06:58.520 --> 00:06:59.730 let's do it both ways. 00:06:59.730 --> 00:07:01.970 Let's say-- let's do it the picking-numbers method because 00:07:01.970 --> 00:07:03.580 I think that'll give you an intuition and maybe it'll 00:07:03.580 --> 00:07:04.500 help us draw the graph. 00:07:04.500 --> 00:07:09.410 So let's say that this is f of x. 00:07:09.410 --> 00:07:12.580 f of x-- you can tell my presentation is very 00:07:12.580 --> 00:07:17.690 unplanned-- f of x is equal to 1 over x. 00:07:17.690 --> 00:07:19.520 And we want to find the limit as x approaches 0. 00:07:19.520 --> 00:07:26.870 So what is f of-- actually, let's make a table. 00:07:26.870 --> 00:07:27.760 f of x. 00:07:36.670 --> 00:07:41.000 So clearly when x is equal to 0, we don't know. 00:07:41.000 --> 00:07:42.310 It's undefined. 00:07:42.310 --> 00:07:43.340 1 over 0 is undefined. 00:07:51.960 --> 00:07:59.170 But what happens when x equals minus 0.01? 00:07:59.170 --> 00:08:04.050 Well, with minus 0.01, 1 over minus 0.01, that is equal 00:08:04.050 --> 00:08:07.750 to negative 100, right? 00:08:07.750 --> 00:08:13.070 What happens when x is equal to minus 0.001, right? 00:08:13.070 --> 00:08:14.880 So we're getting closer and closer to 0 from 00:08:14.880 --> 00:08:16.240 the negative direction. 00:08:16.240 --> 00:08:20.430 Well, here it equals-- make sure my pen is 00:08:20.430 --> 00:08:24.900 working, color right. 00:08:24.900 --> 00:08:28.540 Something's wrong with my tool. 00:08:28.540 --> 00:08:30.250 Now my computer's breaking down. 00:08:30.250 --> 00:08:31.740 Let's see what's going on. 00:08:38.060 --> 00:08:40.260 I think my computer just froze. 00:08:40.260 --> 00:08:43.330 Well, I'm going to try to solve this, and in the very next 00:08:43.330 --> 00:08:46.730 video, I'm going to continue on with this problem. 00:08:46.730 --> 00:08:48.650 So I'll actually see you in the next presentation while I 00:08:48.650 --> 00:08:52.770 figure out why my pen isn't working, and then we'll 00:08:52.770 --> 00:08:53.780 continue with this problem. 00:08:53.780 --> 00:08:56.290 See you very soon.

Limit examples (part 2)

https://www.youtube.com/watch?v=YRw8udexH4o

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https://www.youtube.com/api/timedtext?v=YRw8udexH4o&ei=emeUZeTiAp65mLAP4Ya0wAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249834&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D5E74F0F24BA8CE3C734412BDA6BE96F0D1F7252.5578585BE83EF6D2B71527C22D45B035017753A7&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.880 --> 00:00:03.240 OK, hopefully, my tool is working now. 00:00:03.240 --> 00:00:07.490 But anyway, so we were saying when x is equal to minus 0.001, 00:00:07.490 --> 00:00:09.590 so we're getting closer and closer to 0 from the negative 00:00:09.590 --> 00:00:13.140 side, f of x is equal to minus 1,000, right? 00:00:13.140 --> 00:00:14.750 You can just evaluate it yourself, right? 00:00:14.750 --> 00:00:18.800 And as you see, as x approaches 0 from the negative direction, 00:00:18.800 --> 00:00:22.590 we get larger and larger-- or I guess you could say smaller and 00:00:22.590 --> 00:00:24.110 smaller negative numbers, right? 00:00:24.110 --> 00:00:29.690 You get-- you know, if it's minus 0.0001, you'd get minus 00:00:29.690 --> 00:00:32.730 10,000, and then minus 100,000, and then minus 1 million, you 00:00:32.730 --> 00:00:35.040 could imagine the closer and closer you get to zero. 00:00:35.040 --> 00:00:38.440 Similarly, when you go from the other direction, when you say 00:00:38.440 --> 00:00:45.650 what is-- when x is 0.01, there you get positive 100, right? 00:00:45.650 --> 00:00:54.890 When x is point-- the thing is frozen again-- when it's 0.001, 00:00:54.890 --> 00:00:58.420 you get positive 1,000. 00:00:58.420 --> 00:01:03.270 So as you see, as you approach 0 from the negative direction, 00:01:03.270 --> 00:01:06.210 you get larger and larger negative values, or I guess 00:01:06.210 --> 00:01:07.420 smaller and smaller negative values. 00:01:07.420 --> 00:01:11.440 And as you go from the positive direction, you get larger 00:01:11.440 --> 00:01:12.190 and larger values. 00:01:12.190 --> 00:01:14.700 Let me graph this just to give you a sense of what this graph 00:01:14.700 --> 00:01:16.190 looks like because this is actually a good graph to know 00:01:16.190 --> 00:01:17.320 what it looks like just generally. 00:01:20.040 --> 00:01:25.510 So let's say I have the x-axis. 00:01:25.510 --> 00:01:26.530 This is the y-axis. 00:01:29.200 --> 00:01:31.570 Change my color. 00:01:31.570 --> 00:01:37.090 So when x is a negative number, as x gets really, really, 00:01:37.090 --> 00:01:40.320 really negative, as x is like negative infinity, this 00:01:40.320 --> 00:01:41.890 is approaching zero, but it's still going to be a 00:01:41.890 --> 00:01:43.330 slightly negative number. 00:01:43.330 --> 00:01:47.040 And then as we see from what we drew, as we approach x is equal 00:01:47.040 --> 00:01:50.040 to 0, we asymptote, and we approach negative 00:01:50.040 --> 00:01:53.090 infinity, right? 00:01:53.090 --> 00:01:56.850 And similarly, from positive numbers, if you go out to 00:01:56.850 --> 00:01:59.210 the right really far, it approaches 0, but it's 00:01:59.210 --> 00:02:00.980 still going to be positive. 00:02:00.980 --> 00:02:04.400 And as we gets closer and closer to 0, it spikes up, and 00:02:04.400 --> 00:02:05.450 it goes to positive infinity. 00:02:05.450 --> 00:02:08.770 You never quite get x is equal to 0. 00:02:08.770 --> 00:02:13.260 So in this situation, you actually have as x approaches-- 00:02:13.260 --> 00:02:16.320 so let me give you a different notation, which you'll 00:02:16.320 --> 00:02:17.380 probably see eventually. 00:02:17.380 --> 00:02:19.800 I might actually do a separate presentation on this. 00:02:19.800 --> 00:02:28.940 The limit as x approaches 0 from the positive direction, 00:02:28.940 --> 00:02:35.180 that's this notation here, of 1/x, right? 00:02:35.180 --> 00:02:38.260 So this is as x approaches 0 from the positive direction, 00:02:38.260 --> 00:02:43.585 from the right-hand side, well, this is equal to infinity. 00:02:46.550 --> 00:02:56.120 And then the limit as x-- this pen, this pen-- the limit as x 00:02:56.120 --> 00:03:01.340 approaches 0 from the negative side of 1/x. 00:03:01.340 --> 00:03:03.460 This notation just says the limit as I approach 00:03:03.460 --> 00:03:04.420 from the negative side. 00:03:04.420 --> 00:03:09.120 So as I approach x equal 0 from this direction, right, from 00:03:09.120 --> 00:03:10.560 this direction, what happens? 00:03:10.560 --> 00:03:13.530 Well, that is equal to minus infinity. 00:03:16.550 --> 00:03:19.110 So since I'm approaching a different value when I 00:03:19.110 --> 00:03:21.500 approach from one side or the other, this limit 00:03:21.500 --> 00:03:23.130 is actually undefined. 00:03:23.130 --> 00:03:25.625 I mean, we could say that from the positive side, it's 00:03:25.625 --> 00:03:27.660 positive infinity, or from the negative side, it's negative 00:03:27.660 --> 00:03:30.310 infinity, but they have to equal the same thing for 00:03:30.310 --> 00:03:31.800 this limit to be defined. 00:03:31.800 --> 00:03:34.390 So this is equal to undefined. 00:03:39.740 --> 00:03:43.590 So let's do another problem, and I think this should 00:03:43.590 --> 00:03:44.810 be interesting now. 00:03:44.810 --> 00:03:48.350 So let's say, just keeping that last problem we had in mind, 00:03:48.350 --> 00:04:03.160 what's the limit as x approaches 0 of 1/x squared? 00:04:03.160 --> 00:04:06.270 So in this situation, I'll draw the graph. 00:04:09.080 --> 00:04:12.620 That's my x-axis. 00:04:12.620 --> 00:04:14.140 That's my y-axis. 00:04:14.140 --> 00:04:17.400 So here, no matter what value we put into x, we get a 00:04:17.400 --> 00:04:18.560 positive value, right? 00:04:18.560 --> 00:04:19.370 Because you're going to square it. 00:04:19.370 --> 00:04:25.360 If you put minus-- you could actually-- oh, let me do it. 00:04:25.360 --> 00:04:28.830 It'll be instructive, I think. 00:04:28.830 --> 00:04:30.940 Once again, obviously you can't just put x equal to 0. 00:04:30.940 --> 00:04:33.330 You'll get 1/0, which is undefined. 00:04:33.330 --> 00:04:35.220 But let's say 1 over x squared. 00:04:35.220 --> 00:04:37.400 What does 1 over x squared evaluate to? 00:04:37.400 --> 00:04:46.250 So when x is 0.1, 0.1 squared is 0.01, so 1/x is 100. 00:04:46.250 --> 00:04:52.600 Similarly, if I do minus 0.1, minus 0.1 squared is positive 00:04:52.600 --> 00:04:56.440 0.01, so then 1 over that is still 100, right? 00:04:56.440 --> 00:04:58.630 So regardless of whether we put a negative or positive number 00:04:58.630 --> 00:05:01.380 here, we get a positive value. 00:05:01.380 --> 00:05:07.070 And similarly, if I put-- if we say x is 0.01, if you evaluate 00:05:07.070 --> 00:05:14.540 it, you'll get 10,000, and if we put minus 0.01, you'll get 00:05:14.540 --> 00:05:15.920 positive 10,000 as well, right? 00:05:15.920 --> 00:05:17.420 Because we square it. 00:05:17.420 --> 00:05:19.240 So in this graph, if you were to draw it, and if you have a 00:05:19.240 --> 00:05:22.060 graphing calculator, you should experiment, it 00:05:22.060 --> 00:05:24.720 looks something like this. 00:05:24.720 --> 00:05:26.480 I can see this dark blue. 00:05:26.480 --> 00:05:29.680 So from the negative side, it approaches infinity, right? 00:05:29.680 --> 00:05:30.180 You can see that. 00:05:30.180 --> 00:05:33.000 As we get to smaller and smaller-- as we get closer and 00:05:33.000 --> 00:05:35.860 closer to 0 from the negative side, it approaches infinity. 00:05:35.860 --> 00:05:43.280 As we go from the positive side-- these are actually 00:05:43.280 --> 00:05:45.050 symmetric, although I didn't draw it that symmetric-- it 00:05:45.050 --> 00:05:46.390 also approaches infinity. 00:05:46.390 --> 00:05:51.780 So this is a case in which the limit-- oh, that's 00:05:51.780 --> 00:05:52.380 not too bright. 00:05:52.380 --> 00:05:58.720 I don't know if you can see -- the limit as x approaches 0 00:05:58.720 --> 00:06:03.850 from the negative side of 1 over x squared is equal to 00:06:03.850 --> 00:06:11.240 infinity, and the limit as x approaches 0 from the positive 00:06:11.240 --> 00:06:16.070 side of 1 over x squared is also equal to infinity. 00:06:16.070 --> 00:06:18.550 So when you go from the left-hand side, it 00:06:18.550 --> 00:06:19.560 equals infinity, right? 00:06:19.560 --> 00:06:21.860 It goes to infinity as you approach 0. 00:06:21.860 --> 00:06:23.360 And as you go from the right-hand side, it 00:06:23.360 --> 00:06:26.030 also goes to infinity. 00:06:26.030 --> 00:06:29.800 And so the limit in general is equal to infinity. 00:06:29.800 --> 00:06:34.520 And this is why I got excited when I first started 00:06:34.520 --> 00:06:35.100 learning limits. 00:06:35.100 --> 00:06:38.770 Because for the first time, infinity is a legitimate answer 00:06:38.770 --> 00:06:41.950 to your problem, which, I don't know, on some metaphysical 00:06:41.950 --> 00:06:43.660 level got me kind of excited. 00:06:43.660 --> 00:06:47.750 But anyway, I will do more problems in the next 00:06:47.750 --> 00:06:49.950 presentation because you can never do enough limit problems. 00:06:49.950 --> 00:06:51.610 And in a couple of presentations, I actually give 00:06:51.610 --> 00:06:54.670 you the formal, kind of rigorous mathematical 00:06:54.670 --> 00:06:56.890 definition of the limits.

Introduction to limits 2

https://www.youtube.com/watch?v=W0VWO4asgmk

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https://www.youtube.com/api/timedtext?v=W0VWO4asgmk&ei=emeUZYeSArLoxN8Pi8uj2A8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249834&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1F5ED88069403A2C2DA2C21D7B0C632769DFDCB0.A09CEBF60FF09C41E244D2F728E46F179133031F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.900 --> 00:00:03.610 Welcome to the presentation on limits. 00:00:03.610 --> 00:00:06.290 Let's get started with some-- well, first an explanation 00:00:06.290 --> 00:00:07.530 before I do any problems. 00:00:07.530 --> 00:00:11.110 So let's say I had-- let me make sure I have the right 00:00:11.110 --> 00:00:12.070 color and my pen works. 00:00:12.070 --> 00:00:17.480 OK, let's say I had the limit, and I'll explain what a 00:00:17.480 --> 00:00:18.840 limit is in a second. 00:00:18.840 --> 00:00:22.780 But the way you write it is you say the limit-- oh, my color is 00:00:22.780 --> 00:00:26.860 on the wrong-- OK, let me use the pen and yellow. 00:00:26.860 --> 00:00:38.181 OK, the limit as x approaches 2 of x squared. 00:00:42.550 --> 00:00:46.340 Now, all this is saying is what value does the expression x 00:00:46.340 --> 00:00:49.680 squared approach as x approaches 2? 00:00:49.680 --> 00:00:51.130 Well, this is pretty easy. 00:00:51.130 --> 00:00:53.065 If we look at-- let me at least draw a graph. 00:00:53.065 --> 00:00:57.010 I'll stay in this yellow color. 00:00:57.010 --> 00:00:59.420 So let me draw. 00:00:59.420 --> 00:01:03.813 x squared looks something like-- let me use 00:01:03.813 --> 00:01:05.610 a different color. 00:01:05.610 --> 00:01:07.900 x square looks something like this, right? 00:01:10.730 --> 00:01:20.040 And when x is equal to 2, y, or the expression-- because 00:01:20.040 --> 00:01:21.320 we don't say what this is equal to. 00:01:21.320 --> 00:01:23.570 It's just the expression-- x squared is equal to 4, right? 00:01:27.650 --> 00:01:33.100 So a limit is saying, as x approaches 2, as x approaches 2 00:01:33.100 --> 00:01:37.920 from both sides, from numbers left than 2 and from numbers 00:01:37.920 --> 00:01:41.940 right than 2, what does the expression approach? 00:01:41.940 --> 00:01:44.630 And you might, I think, already see where this is going and be 00:01:44.630 --> 00:01:46.920 wondering why we're even going to the trouble of learning this 00:01:46.920 --> 00:01:50.040 new concept because it seems pretty obvious, but as x-- as 00:01:50.040 --> 00:01:53.320 we get to x closer and closer to 2 from this direction, and 00:01:53.320 --> 00:01:55.530 as we get to x closer and closer to 2 to this 00:01:55.530 --> 00:01:58.940 direction, what does this expression equal? 00:01:58.940 --> 00:02:01.670 Well, it essentially equals 4, right? 00:02:01.670 --> 00:02:04.690 The expression is equal to 4. 00:02:04.690 --> 00:02:08.090 The way I think about it is as you move on the curve closer 00:02:08.090 --> 00:02:12.390 and closer to the expression's value, what does the 00:02:12.390 --> 00:02:13.500 expression equal? 00:02:13.500 --> 00:02:14.660 In this case, it equals 4. 00:02:14.660 --> 00:02:17.850 You're probably saying, Sal, this seems like a useless 00:02:17.850 --> 00:02:20.690 concept because I could have just stuck 2 in there, and I 00:02:20.690 --> 00:02:23.850 know that if this is-- say this is f of x, that if f of x is 00:02:23.850 --> 00:02:27.590 equal to x squared, that f of 2 is equal to 4, and that would 00:02:27.590 --> 00:02:29.050 have been a no-brainer. 00:02:29.050 --> 00:02:33.840 Well, let me maybe give you one wrinkle on that, and hopefully 00:02:33.840 --> 00:02:36.970 now you'll start to see what the use of a limit is. 00:02:36.970 --> 00:02:51.620 Let me to define-- let me say f of x is equal to x squared 00:02:51.620 --> 00:02:59.390 when, if x does not equal 2, and let's say it equals 00:02:59.390 --> 00:03:04.270 3 when x equals 2. 00:03:04.270 --> 00:03:04.560 Interesting. 00:03:04.560 --> 00:03:08.600 So it's a slight variation on this expression right here. 00:03:08.600 --> 00:03:09.490 So this is our new f of x. 00:03:09.490 --> 00:03:10.670 So let me ask you a question. 00:03:10.670 --> 00:03:16.300 What is-- my pen still works-- what is the limit-- I used 00:03:16.300 --> 00:03:23.180 cursive this time-- what is the limit as x-- that's an x-- 00:03:23.180 --> 00:03:25.580 as x approaches 2 of f of x? 00:03:29.590 --> 00:03:30.210 That's an x. 00:03:30.210 --> 00:03:31.120 It says x approaches 2. 00:03:31.120 --> 00:03:31.800 It's just like that. 00:03:31.800 --> 00:03:33.000 I just-- I don't know. 00:03:33.000 --> 00:03:35.460 For some reason, my brain is working functionally. 00:03:35.460 --> 00:03:39.250 OK, so let me graph this now. 00:03:39.250 --> 00:03:43.360 So that's an equally neat-looking graph as 00:03:43.360 --> 00:03:46.100 the one I just drew. 00:03:46.100 --> 00:03:46.790 Let me draw. 00:03:46.790 --> 00:03:50.100 So now it's almost the same as this curve, except something 00:03:50.100 --> 00:03:52.330 interesting happens at x equals 2. 00:03:52.330 --> 00:03:53.715 So it's just like this. 00:03:53.715 --> 00:03:58.480 It's like an x squared curve like that. 00:03:58.480 --> 00:04:02.710 But at x equals 2 and f of x equals 4, we 00:04:02.710 --> 00:04:04.890 draw a little hole. 00:04:04.890 --> 00:04:08.050 We draw a hole because it's not defined at x equals 2. 00:04:08.050 --> 00:04:10.320 This is x equals 2. 00:04:10.320 --> 00:04:11.960 This is 2. 00:04:11.960 --> 00:04:12.680 This is 4. 00:04:12.680 --> 00:04:14.690 This is the f of x axis, of course. 00:04:14.690 --> 00:04:19.820 And when x is equal to 2-- let's say this is 3. 00:04:19.820 --> 00:04:23.860 When x is equal to 2, f of x is equal to 3. 00:04:23.860 --> 00:04:25.240 This is actually right below this. 00:04:25.240 --> 00:04:27.610 I should-- it doesn't look completely right below it, 00:04:27.610 --> 00:04:29.730 but I think you got to get the picture. 00:04:29.730 --> 00:04:31.400 See, this graph is x squared. 00:04:31.400 --> 00:04:35.983 It's exactly x squared until we get to x equals 2. 00:04:35.983 --> 00:04:40.030 At x equals 2, We have a grap-- No, not a grap. 00:04:40.030 --> 00:04:42.770 We have a gap in the graph, which maybe 00:04:42.770 --> 00:04:44.720 could be called a grap. 00:04:44.720 --> 00:04:49.540 We have a gap in the graph, and then we keep-- and then after x 00:04:49.540 --> 00:04:51.350 equals 2, we keep moving on. 00:04:51.350 --> 00:04:54.300 And that gap, and that gap is defined right here, what 00:04:54.300 --> 00:04:55.350 happens when x equals 2? 00:04:55.350 --> 00:04:57.300 Well, then f of x is equal to 3. 00:04:57.300 --> 00:05:02.960 So this graph kind of goes-- it's just like x squared, but 00:05:02.960 --> 00:05:08.470 instead of f of 2 being 4, f of 2 drops down to 3, but 00:05:08.470 --> 00:05:10.260 then we keep on going. 00:05:10.260 --> 00:05:12.400 So going back to the limit problem, what is the 00:05:12.400 --> 00:05:14.880 limit as x approaches 2? 00:05:14.880 --> 00:05:17.050 Now, well, let's think about the same thing. 00:05:17.050 --> 00:05:19.200 We're going to go-- this is how I visualize it. 00:05:19.200 --> 00:05:21.000 I go along the curve. 00:05:21.000 --> 00:05:23.130 Let me pick a different color. 00:05:23.130 --> 00:05:28.050 So as x approaches 2 from this side, from the left-hand side 00:05:28.050 --> 00:05:35.200 or from numbers less than 2, f of x is approaching values 00:05:35.200 --> 00:05:40.520 approaching 4, right? f of x is approaching 4 as x 00:05:40.520 --> 00:05:41.540 approaches 2, right? 00:05:41.540 --> 00:05:42.200 I think you see that. 00:05:42.200 --> 00:05:46.820 If you just follow along the curve, as you approach f of 2, 00:05:46.820 --> 00:05:48.420 you get closer and closer to 4. 00:05:48.420 --> 00:05:53.010 Similarly, as you go from the right-hand side-- make sure 00:05:53.010 --> 00:05:54.370 my thing's still working. 00:05:54.370 --> 00:05:57.780 As you go from the right-hand side, you go along the 00:05:57.780 --> 00:06:05.200 curve, and f of x is also slowly approaching 4. 00:06:05.200 --> 00:06:07.120 So, as you can see, as we go closer and closer and 00:06:07.120 --> 00:06:11.610 closer to x equals 2, f of whatever number that is 00:06:11.610 --> 00:06:13.540 approaches 4, right? 00:06:13.540 --> 00:06:15.860 So, in this case, the limit as x approaches 00:06:15.860 --> 00:06:21.130 2 is also equal to 4. 00:06:21.130 --> 00:06:24.120 Well, this is interesting because, in this case, the 00:06:24.120 --> 00:06:35.980 limit as x approaches 2 of f of x does not equal f of 2. 00:06:35.980 --> 00:06:38.170 Now, normally, this would be on this line. 00:06:38.170 --> 00:06:41.300 In this case, the limit as you approach the expression is 00:06:41.300 --> 00:06:44.030 equal to evaluating the expression of that value. 00:06:44.030 --> 00:06:46.450 In this case, the limit isn't. 00:06:46.450 --> 00:06:49.220 I think now you're starting to see why the limit is a slightly 00:06:49.220 --> 00:06:51.260 different concept than just evaluating the function at 00:06:51.260 --> 00:06:53.770 that point because you have functions where, for whatever 00:06:53.770 --> 00:06:57.530 reason at a certain point, either the function might not 00:06:57.530 --> 00:07:01.640 be defined or the function kind of jumps up or down, but as you 00:07:01.640 --> 00:07:05.370 approach that point, you still approach a value different than 00:07:05.370 --> 00:07:06.700 the function at that point. 00:07:06.700 --> 00:07:08.110 Now, that's my introduction. 00:07:08.110 --> 00:07:12.020 I think this will give you intuition for what a limit is. 00:07:12.020 --> 00:07:14.110 In another presentation, I'll give you the more formal 00:07:14.110 --> 00:07:16.460 mathematical, you know, the delta-epsilon 00:07:16.460 --> 00:07:17.890 definition of a limit. 00:07:17.890 --> 00:07:20.190 And actually, in the very next module, I'm now going to 00:07:20.190 --> 00:07:23.210 do a bunch of problems involving the limit. 00:07:23.210 --> 00:07:25.510 I think as you do more and more problems, you'll get more and 00:07:25.510 --> 00:07:28.380 more of an intuition as to what a limit is. 00:07:28.380 --> 00:07:30.490 And then as we go into drill derivatives and integrals, 00:07:30.490 --> 00:07:33.350 you'll actually understand why people probably even invented 00:07:33.350 --> 00:07:34.910 limits to begin with. 00:07:34.910 --> 00:07:36.930 We'll see you in the next presentation.

Domain of a function

https://www.youtube.com/watch?v=U-k5N1WPk4g

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https://www.youtube.com/api/timedtext?v=U-k5N1WPk4g&ei=fGeUZavPIsizvdIPxY60sAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249836&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=32D70B845FB7544022796E18AD5ADC5265681556.ECBDAC399CE1F2DE01E55913CF8FF8BC739C11C6&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:04.070 Welcome to my presentation on domain of a function. 00:00:04.070 --> 00:00:05.060 So what's is the domain? 00:00:05.060 --> 00:00:07.720 The domain of a function, you'll often hear it combined 00:00:07.720 --> 00:00:09.090 with domain and range. 00:00:09.090 --> 00:00:12.830 But the domain of a function is just what values can I put into 00:00:12.830 --> 00:00:15.150 a function and get a valid output. 00:00:15.150 --> 00:00:16.360 So let's start with something examples. 00:00:16.360 --> 00:00:32.880 Let's say I had f of x is equal to, let's say, x squared. 00:00:35.410 --> 00:00:36.270 So let me ask you a question. 00:00:36.270 --> 00:00:39.440 What values of x can I put in here so I get a valid 00:00:39.440 --> 00:00:41.460 answer for x squared? 00:00:41.460 --> 00:00:44.820 Well, I can really put anything in here, any real number. 00:00:44.820 --> 00:00:53.980 So here I'll say that the domain is the set of x's 00:00:53.980 --> 00:00:58.620 such that x is a member of the real numbers. 00:00:58.620 --> 00:01:02.020 So this is just a fancy way of saying that OK, this r with 00:01:02.020 --> 00:01:04.370 this kind of double backbone here, that just means real 00:01:04.370 --> 00:01:06.210 numbers, and I think you're familiar with real numbers now. 00:01:06.210 --> 00:01:10.420 That's pretty much every number outside of the complex numbers. 00:01:10.420 --> 00:01:11.640 And if you don't know what complex numbers 00:01:11.640 --> 00:01:12.910 are, that's fine. 00:01:12.910 --> 00:01:14.980 You probably won't need to know it right now. 00:01:14.980 --> 00:01:18.210 The real numbers are every number that most people are 00:01:18.210 --> 00:01:20.290 familiar with, including irrational numbers, including 00:01:20.290 --> 00:01:23.180 transcendental numbers, including fractions -- every 00:01:23.180 --> 00:01:24.470 number is a real number. 00:01:24.470 --> 00:01:27.820 So the domain here is x -- x just has to be a member 00:01:27.820 --> 00:01:28.730 of the real numbers. 00:01:28.730 --> 00:01:31.840 And this little backwards looking e or something, this 00:01:31.840 --> 00:01:35.150 just means x is a member of the real numbers. 00:01:35.150 --> 00:01:37.200 So let's do another one in a slight variation. 00:01:42.445 --> 00:01:51.310 So let's say I had f of x is equal to 1 over x squared. 00:01:51.310 --> 00:01:52.620 So is this same thing now? 00:01:52.620 --> 00:01:54.890 Can I still put any x value in here and get 00:01:54.890 --> 00:01:56.970 a reasonable answer? 00:01:56.970 --> 00:01:57.950 Well what's f of 0? 00:02:08.370 --> 00:02:12.880 f of zero is equal to 1 over 0. 00:02:12.880 --> 00:02:14.870 And what's 1 over 0? 00:02:14.870 --> 00:02:17.860 I don't know what it is, so this is undefined. 00:02:22.470 --> 00:02:25.780 No one ever took the trouble to define what 1 over 0 should be. 00:02:25.780 --> 00:02:27.810 And they probably didn't do, so some people probably thought 00:02:27.810 --> 00:02:31.170 about what should be, but they probably couldn't find out with 00:02:31.170 --> 00:02:33.450 a good definition for 1 over 0 that's consistent with 00:02:33.450 --> 00:02:34.250 the rest of mathematics. 00:02:34.250 --> 00:02:35.710 So 1 over 0 stays undefined. 00:02:35.710 --> 00:02:37.930 So f of 0 is undefined. 00:02:37.930 --> 00:02:42.700 So we can't put 0 in and get a valid answer for f of 0. 00:02:42.700 --> 00:02:50.630 So here we say the domain is equal to -- do little brackets, 00:02:50.630 --> 00:02:52.900 that shows kind of the set of what x's apply. 00:02:52.900 --> 00:02:54.303 That's those little curly brackets, I didn't 00:02:54.303 --> 00:02:56.400 draw it that well. 00:02:56.400 --> 00:03:02.030 x is a member of the real numbers still, such that 00:03:02.030 --> 00:03:05.560 x does not equal 0. 00:03:05.560 --> 00:03:10.230 So here I just made a slight variation on what I had before. 00:03:10.230 --> 00:03:12.925 Before we said when f of x is equal to x squared that x 00:03:12.925 --> 00:03:15.050 is just any real number. 00:03:15.050 --> 00:03:20.250 Now we're saying that x is any real number except for 0. 00:03:20.250 --> 00:03:22.400 This is just a fancy way of saying it, and then these curly 00:03:22.400 --> 00:03:24.290 brackets just mean a set. 00:03:24.290 --> 00:03:26.390 Let's do a couple more ones. 00:03:26.390 --> 00:03:44.360 Let's say f of x is equal to the square root of x minus 3. 00:03:44.360 --> 00:03:48.170 So up here we said, well this function isn't defined when we 00:03:48.170 --> 00:03:49.680 get a 0 in the denominator. 00:03:49.680 --> 00:03:52.090 But what's interesting about this function? 00:03:52.090 --> 00:03:55.600 Can we take a square root of a negative number? 00:03:55.600 --> 00:03:58.000 Well until we learn about imaginary and complex 00:03:58.000 --> 00:03:59.030 numbers we can't. 00:03:59.030 --> 00:04:03.280 So here we say well, any x is valid here except for the x's 00:04:03.280 --> 00:04:07.220 that make this expression under the radical sign negative. 00:04:07.220 --> 00:04:11.510 So we have to say that x minus 3 has to be greater than or 00:04:11.510 --> 00:04:13.690 equal to 0, right, because you could have the square to 0, 00:04:13.690 --> 00:04:15.235 that's fine, it's just 0. 00:04:15.235 --> 00:04:20.070 So x minus 3 has to be greater than or equal to 0, so x has to 00:04:20.070 --> 00:04:22.700 be greater than or equal to 3. 00:04:22.700 --> 00:04:35.130 So here our domain is x is a member of the real numbers, 00:04:35.130 --> 00:04:41.165 such that x is greater than or equal to 3. 00:04:45.840 --> 00:04:50.140 Let's do a slightly more difficult one. 00:04:50.140 --> 00:05:01.260 What if I said f of x is equal to the square root of the 00:05:01.260 --> 00:05:06.200 absolute value of x minus 3. 00:05:06.200 --> 00:05:08.320 So now it's getting a little bit more complicated. 00:05:08.320 --> 00:05:11.020 Well just like this time around, this expression of 00:05:11.020 --> 00:05:13.180 the radical still has to be greater than or equal to 0. 00:05:13.180 --> 00:05:19.400 So you can just say that the absolute value of x minus 3 is 00:05:19.400 --> 00:05:21.910 greater than or equal to 0. 00:05:21.910 --> 00:05:25.580 So we have the absolute value of x has to be greater 00:05:25.580 --> 00:05:27.830 than or equal to 3. 00:05:27.830 --> 00:05:30.480 And if order for the absolute value of something to be 00:05:30.480 --> 00:05:35.440 greater than or equal to something, then that means that 00:05:35.440 --> 00:05:46.290 x has to be less than or equal to negative 3, or x has to be 00:05:46.290 --> 00:05:48.530 greater than or equal to 3. 00:05:48.530 --> 00:05:52.460 It makes sense because x can't be negative 2, right? 00:05:52.460 --> 00:05:55.240 Because negative 2 has an absolute value less than 3. 00:05:55.240 --> 00:05:57.700 So x has to be less than negative 3. 00:05:57.700 --> 00:06:00.930 It has to be further in the negative direction than 00:06:00.930 --> 00:06:03.220 negative 3, or it has to be further in the positive 00:06:03.220 --> 00:06:04.790 direction than positive 3. 00:06:04.790 --> 00:06:10.490 So, once again, x has to be less than negative 3 or x 00:06:10.490 --> 00:06:13.000 has to be greater than 3, so we have our domain. 00:06:13.000 --> 00:06:15.920 So we have it as x is a member of the reals 00:06:15.920 --> 00:06:19.960 -- I always forget. 00:06:19.960 --> 00:06:21.460 Is that the line? 00:06:21.460 --> 00:06:23.080 I forget, it's either a colon or a line. 00:06:23.080 --> 00:06:25.510 I'm rusty, it's been years since I've done 00:06:25.510 --> 00:06:26.400 this kind of stuff. 00:06:26.400 --> 00:06:29.050 But anyway, I think you get the point. 00:06:29.050 --> 00:06:32.800 It could be any real number here, as long as x is less 00:06:32.800 --> 00:06:37.650 than negative 3, less than or equal to negative 3, or x is 00:06:37.650 --> 00:06:39.980 greater than or equal to 3. 00:06:39.980 --> 00:06:41.650 Let me ask a question now. 00:06:41.650 --> 00:06:50.970 What if instead of this it was -- that was the denominator, 00:06:50.970 --> 00:06:53.270 this is all a separate problem up here. 00:06:53.270 --> 00:06:56.500 So now we have 1 over the square root of the absolute 00:06:56.500 --> 00:06:59.160 value of x minus 3. 00:06:59.160 --> 00:07:00.730 So now how does this change the situation? 00:07:00.730 --> 00:07:03.650 So not only does this expression in the denominator, 00:07:03.650 --> 00:07:06.170 not only does this have to be greater than or equal to 00:07:06.170 --> 00:07:08.360 0, can it be 0 anymore? 00:07:08.360 --> 00:07:10.960 Well no, because then you would get the square root of 0, which 00:07:10.960 --> 00:07:13.560 is 0 and you would get a 0 in the denominator. 00:07:13.560 --> 00:07:15.310 So it's kind of like this problem plus this 00:07:15.310 --> 00:07:16.600 problem combined. 00:07:16.600 --> 00:07:20.210 So now when you have 1 over the square root of the absolute 00:07:20.210 --> 00:07:25.430 value of x minus 3, now it's no longer greater than or equal to 00:07:25.430 --> 00:07:28.690 0, it's just a greater than 0, right? 00:07:28.690 --> 00:07:30.420 it's just greater than 0. 00:07:30.420 --> 00:07:32.040 Because we can't have a 0 here in the denominator. 00:07:32.040 --> 00:07:37.080 So if it's greater than 0, then we just say greater than 3. 00:07:37.080 --> 00:07:40.510 And essentially just get rid of the equal signs right here. 00:07:40.510 --> 00:07:41.790 Let me erase it properly. 00:07:44.310 --> 00:07:45.750 It's a slightly different color, but maybe 00:07:45.750 --> 00:07:47.640 you won't notice. 00:07:47.640 --> 00:07:50.130 So there you go. 00:07:50.130 --> 00:07:52.450 Actually, we should do another example since we have time. 00:07:58.470 --> 00:08:01.180 Let me erase this. 00:08:01.180 --> 00:08:01.810 OK. 00:08:01.810 --> 00:08:35.140 Now let's say that f of x is equal to 2, if x is even, 00:08:35.140 --> 00:08:53.970 and 1 over x minus 2 times x minus 1, if x is odd. 00:08:53.970 --> 00:08:55.970 So what's the domain here? 00:08:55.970 --> 00:08:57.620 What is a valid x I can put in here. 00:08:57.620 --> 00:08:59.820 So immediately we have two clauses. 00:08:59.820 --> 00:09:13.270 If x is even we use this clause, so f of 4 -- well, 00:09:13.270 --> 00:09:16.030 that's just equal to 2 because we used this clause here. 00:09:16.030 --> 00:09:19.300 But this clause applies when x is odd. 00:09:19.300 --> 00:09:21.370 Just like we did in the last example, what are the 00:09:21.370 --> 00:09:24.270 situations where this kind of breaks down? 00:09:24.270 --> 00:09:26.200 Well, when the denominator is 0. 00:09:26.200 --> 00:09:29.850 Well the denominator is 0 when x is equal to 2, or 00:09:29.850 --> 00:09:32.870 x is equal to 1, right? 00:09:32.870 --> 00:09:34.970 But this clause only applies when x is odd. 00:09:34.970 --> 00:09:37.350 So x is equal to 2 won't apply to this clause. 00:09:37.350 --> 00:09:41.370 So only x is equal to 1 would apply to this clause. 00:09:41.370 --> 00:09:49.010 So the domain is x is a member of the reals, such that 00:09:49.010 --> 00:09:52.320 x does not equal 1. 00:09:52.320 --> 00:09:55.160 I think that's all the time I have for now. 00:09:55.160 --> 00:09:58.210 Have fun practicing these domain problems.

Collateralized debt obligation (CDO)

https://www.youtube.com/watch?v=XjoJ9UF2hqg

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en

WEBVTT Kind: captions Language: en 00:00:01.230 --> 00:00:02.280 Welcome back. 00:00:02.280 --> 00:00:03.860 Well, in the last presentation, we described a 00:00:03.860 --> 00:00:06.200 situation where you had a bunch of borrowers. 00:00:06.200 --> 00:00:08.160 They needed $1 billion collectively, because there's 00:00:08.160 --> 00:00:10.760 1000 of them and they each needed $1 million 00:00:10.760 --> 00:00:12.810 to buy their house. 00:00:12.810 --> 00:00:14.790 And they borrowed the money essentially from a special 00:00:14.790 --> 00:00:15.530 purpose entity. 00:00:15.530 --> 00:00:17.580 They borrowed it from their local mortgage broker, who 00:00:17.580 --> 00:00:20.250 then sold it to a bank, or to an investment bank, who 00:00:20.250 --> 00:00:22.770 created the special purpose entity, and then they IPO the 00:00:22.770 --> 00:00:25.840 special purpose entity and raise the money from people 00:00:25.840 --> 00:00:27.700 who bought the mortgage-backed securities. 00:00:27.700 --> 00:00:29.580 But essentially what happened is the investors in the 00:00:29.580 --> 00:00:31.940 mortgage-backed securities provided the money to the 00:00:31.940 --> 00:00:33.160 special purpose entity to 00:00:33.160 --> 00:00:34.830 essentially loan to the borrowers. 00:00:34.830 --> 00:00:37.120 And then the reason why we call it a security is because, 00:00:37.120 --> 00:00:40.040 not only are these people getting this 10% a year, but 00:00:40.040 --> 00:00:42.030 if they want to -- let's say that you had one of these 00:00:42.030 --> 00:00:44.220 mortgage-backed securities and you paid $1000 for it. 00:00:44.220 --> 00:00:45.700 And you're getting this 10% a year, but then all of a 00:00:45.700 --> 00:00:49.390 sudden, you think that the whole mortgage industry is 00:00:49.390 --> 00:00:50.950 about to collapse, a bunch of people are going to default, 00:00:50.950 --> 00:00:52.380 and you want out. 00:00:52.380 --> 00:00:53.860 If you just gave someone a loan, there'd be 00:00:53.860 --> 00:00:54.720 no way to get out. 00:00:54.720 --> 00:00:56.570 You'd have to sell that loan to someone else. 00:00:56.570 --> 00:00:58.380 But if you have a mortgage-backed security, you 00:00:58.380 --> 00:01:00.070 can actually trade the security with someone else. 00:01:00.070 --> 00:01:01.980 And they might pay you, who knows, they might pay more 00:01:01.980 --> 00:01:02.780 than $1000. 00:01:02.780 --> 00:01:03.460 They might pay you less. 00:01:03.460 --> 00:01:05.880 But there will be at least some type of a market in the 00:01:05.880 --> 00:01:08.240 security, so you could have what you could call liquidity. 00:01:08.240 --> 00:01:10.960 Liquidity just means that I have the security 00:01:10.960 --> 00:01:11.740 and I can sell it. 00:01:11.740 --> 00:01:14.340 I could trade it just like I could trade a share of IBM or 00:01:14.340 --> 00:01:17.600 I could trade a share of Microsoft. 00:01:17.600 --> 00:01:20.800 But like we said before, this security, in order to place a 00:01:20.800 --> 00:01:24.020 value on it, you have to do some type of analysis of what 00:01:24.020 --> 00:01:25.240 you think it's worth. 00:01:25.240 --> 00:01:31.180 Or what you think the real interest will be after you 00:01:31.180 --> 00:01:33.520 take into account people pre-paying their mortgage, 00:01:33.520 --> 00:01:35.340 people defaulting on their mortgage, and other things 00:01:35.340 --> 00:01:37.770 like short-term interest rates, et cetera, et cetera. 00:01:37.770 --> 00:01:40.280 And there is only maybe a small group of people who are 00:01:40.280 --> 00:01:42.870 sophisticated enough to be able to figure that out to 00:01:42.870 --> 00:01:45.470 make some type of models and who knows if even they're 00:01:45.470 --> 00:01:46.720 sophisticated enough. 00:01:55.350 --> 00:01:58.440 There might be a whole other class of investors 00:01:58.440 --> 00:02:00.340 here, say this guy. 00:02:00.340 --> 00:02:02.800 He would love to kind of invest in insecurities, but he 00:02:02.800 --> 00:02:03.800 thinks this is too risky. 00:02:03.800 --> 00:02:08.940 He'd be willing to take a lower return as long as he was 00:02:08.940 --> 00:02:10.979 allowed to invest in less risky investments. 00:02:10.979 --> 00:02:13.530 Maybe by law, maybe he's a pension fund or he's some type 00:02:13.530 --> 00:02:16.140 of a mutual fund, that's forced to invest in something 00:02:16.140 --> 00:02:17.860 of a certain grade. 00:02:17.860 --> 00:02:26.230 And say that there's another investor here, and he thinks 00:02:26.230 --> 00:02:27.180 that this is boring. 00:02:27.180 --> 00:02:28.350 You know, 9%, 10%. 00:02:28.350 --> 00:02:29.160 Who cares about that? 00:02:29.160 --> 00:02:31.370 He wants to see bigger and bigger returns. 00:02:31.370 --> 00:02:34.750 So there's no way for him to invest in this security and to 00:02:34.750 --> 00:02:35.810 get better returns. 00:02:35.810 --> 00:02:39.510 So now we're going to take this mortgage-backed security 00:02:39.510 --> 00:02:44.650 and introduce one step further kind of permutation or 00:02:44.650 --> 00:02:45.910 derivative of what this is. 00:02:45.910 --> 00:02:46.930 That's all derivatives are. 00:02:46.930 --> 00:02:48.730 You've probably heard the term derivatives and people do a 00:02:48.730 --> 00:02:52.320 lot of hand-waving saying, oh, it's a more complicated form 00:02:52.320 --> 00:02:53.030 of security. 00:02:53.030 --> 00:02:56.770 All derivative means is you take one type of asset and you 00:02:56.770 --> 00:03:00.120 slice and dice it in a way to spread the risk, or whatever. 00:03:00.120 --> 00:03:01.950 And so you create a derivative asset. 00:03:01.950 --> 00:03:04.450 It's derived from the original asset. 00:03:04.450 --> 00:03:09.000 So let's see how we could use this same asset pool, the same 00:03:09.000 --> 00:03:12.810 pool of loans, and satisfy all of these people. 00:03:12.810 --> 00:03:15.630 Satisfy this guy, who wants maybe a lower return but lower 00:03:15.630 --> 00:03:18.680 risk, and this guy, who's willing to take a little bit 00:03:18.680 --> 00:03:21.410 higher risk in exchange for higher return. 00:03:24.380 --> 00:03:27.060 So now in this situation, we have the same borrowers. 00:03:27.060 --> 00:03:29.390 They borrowed $1 billion collectively, right, because 00:03:29.390 --> 00:03:31.310 there's 1000 of them, et cetera, et cetera. 00:03:31.310 --> 00:03:34.410 And they're still a special purpose entity, but now, 00:03:34.410 --> 00:03:38.780 instead of just slicing up the special purpose entity a 00:03:38.780 --> 00:03:42.680 million ways, what we're going to do is we're going to split 00:03:42.680 --> 00:03:46.810 it up first into three, what we could call, tranches. 00:03:46.810 --> 00:03:51.190 A tranche is just a bucket, if you will, of the asset. 00:03:51.190 --> 00:03:53.370 And we're going to call the three tranches: equity, 00:03:53.370 --> 00:03:56.140 mezzanine, and senior. 00:03:56.140 --> 00:03:57.710 These are the words that are commonly 00:03:57.710 --> 00:03:59.350 used in this industry. 00:03:59.350 --> 00:04:05.010 A senior just means, if this entity were to lose money, 00:04:05.010 --> 00:04:07.810 these people get their money back first. So it's the least 00:04:07.810 --> 00:04:10.220 risk out of all of the tranches. 00:04:10.220 --> 00:04:12.900 Mezzanine, that just means the next level or middle. 00:04:12.900 --> 00:04:14.890 And these guys are some place in between. 00:04:14.890 --> 00:04:18.850 They have a little bit more risk, and they still get a 00:04:18.850 --> 00:04:21.300 little bit more reward than senior, but they have less 00:04:21.300 --> 00:04:23.700 risk than this equity tranche. 00:04:23.700 --> 00:04:24.350 Equity tranche. 00:04:24.350 --> 00:04:27.060 These are the people who first lose money. 00:04:27.060 --> 00:04:29.040 Let's say some of these borrowers start defaulting. 00:04:29.040 --> 00:04:31.190 It all comes out of the equity tranche. 00:04:31.190 --> 00:04:33.770 So that's what protects the senior tranche and the 00:04:33.770 --> 00:04:35.770 mezzanine tranche from defaults. 00:04:35.770 --> 00:04:38.580 So in this situation what we did is we raised -- out of the 00:04:38.580 --> 00:04:42.690 $1 billion we needed -- $400 million from the senior 00:04:42.690 --> 00:04:45.990 tranche, $300 million from the mezzanine tranche, and then 00:04:45.990 --> 00:04:47.810 $300 million from the equity tranche. 00:04:47.810 --> 00:04:52.730 The $400 million senior tranche we raised from soon. 00:04:52.730 --> 00:04:56.880 1000 senior securities, collateralized debt 00:04:56.880 --> 00:04:57.380 obligations. 00:04:57.380 --> 00:04:59.815 These are these, right here. 00:04:59.815 --> 00:05:02.870 Say there were 400,000 of these and these each cost 00:05:02.870 --> 00:05:07.990 $1000, right? 00:05:07.990 --> 00:05:10.030 Let's say these cost $1000. 00:05:10.030 --> 00:05:11.690 And we issued 400,000 of these. 00:05:16.590 --> 00:05:19.780 So we raised $400 million. 00:05:19.780 --> 00:05:22.890 Let's say we give these guys a 6% return. 00:05:22.890 --> 00:05:25.100 And you might say, 6%, that's not much. 00:05:25.100 --> 00:05:29.920 But these guys, it is pretty low risk, because in order for 00:05:29.920 --> 00:05:35.910 them to not get their 6%, the value of this $1 billion asset 00:05:35.910 --> 00:05:38.700 or these $1 billion loans, would have to go down below 00:05:38.700 --> 00:05:40.460 $400 million. 00:05:40.460 --> 00:05:43.440 Maybe I'll do a little bit more math in another example. 00:05:43.440 --> 00:05:45.370 But I think it'll start making sense to you. 00:05:45.370 --> 00:05:47.900 For example, every year we said there's going to be $100 00:05:47.900 --> 00:05:48.930 million in payments, right? 00:05:48.930 --> 00:05:50.170 Because it's 10%. 00:05:50.170 --> 00:05:51.800 $100 million in payments. 00:05:51.800 --> 00:05:54.850 Of that $100 million in payments, 6% on the $400 00:05:54.850 --> 00:05:57.770 million, that's $24 million in payments. 00:05:57.770 --> 00:05:58.550 Right? 00:05:58.550 --> 00:06:01.950 So $24 million in payments will go to the senior tranche. 00:06:01.950 --> 00:06:07.150 Similarly we issued 300,000 shares at $1000 per share on 00:06:07.150 --> 00:06:08.200 the mezzanine tranche. 00:06:08.200 --> 00:06:09.060 This is also 1000. 00:06:09.060 --> 00:06:10.030 This is the mezzanine tranche. 00:06:10.030 --> 00:06:13.660 And let's say they get 7%, a slightly higher return. 00:06:13.660 --> 00:06:16.320 And these percentages are usually determined by some 00:06:16.320 --> 00:06:18.100 type of market or what people are willing to get. 00:06:18.100 --> 00:06:19.700 But let's just say it's fixed for now. 00:06:19.700 --> 00:06:21.350 Let's say it's 7%. 00:06:21.350 --> 00:06:23.950 So 300,000 shares, seven 7%. 00:06:23.950 --> 00:06:26.290 These guys are going to get $21 million. 00:06:26.290 --> 00:06:27.170 Right? 00:06:27.170 --> 00:06:30.080 So out of the $100 million every year, $24 million is 00:06:30.080 --> 00:06:32.570 going to go to these guys, $21 million is going to go to 00:06:32.570 --> 00:06:34.370 these guys, and then whatever's left over is going 00:06:34.370 --> 00:06:37.110 to go to the equity tranche. 00:06:37.110 --> 00:06:39.960 So the $300 million from equity, they're going to get 00:06:39.960 --> 00:06:42.820 $55 million assuming that there are no defaults or 00:06:42.820 --> 00:06:45.400 pre-payments or anything shady happens with the securities. 00:06:49.940 --> 00:06:52.860 But these guys are going to get $55 million. 00:06:52.860 --> 00:06:57.090 Or on $300 million, that's a 16.5% return. 00:06:57.090 --> 00:06:58.170 And I know what you're thinking. 00:06:58.170 --> 00:06:59.880 Boy, Sal, that sounds amazing. 00:06:59.880 --> 00:07:03.280 Why wouldn't everyone want to be an equity investor? 00:07:03.280 --> 00:07:03.590 I don't know. 00:07:03.590 --> 00:07:05.170 My pen has stopped working. 00:07:05.170 --> 00:07:07.900 But anyway, I'll try to move on without my pen. 00:07:07.900 --> 00:07:09.890 So you're saying, why wouldn't everyone want to 00:07:09.890 --> 00:07:10.570 be an equity investor? 00:07:10.570 --> 00:07:11.400 Well, let me ask you a question. 00:07:11.400 --> 00:07:14.850 What happens if -- let's go to that scenario where we talked 00:07:14.850 --> 00:07:20.020 before -- 20% of the borrowers just say, you know what? 00:07:20.020 --> 00:07:21.780 I can't pay this mortgage anymore. 00:07:21.780 --> 00:07:24.326 I'm going to hand you back the keys to these houses. 00:07:24.326 --> 00:07:27.760 And of that 20%, you only get a 50% return. 00:07:27.760 --> 00:07:29.550 So for each of those $1 million houses, you're only 00:07:29.550 --> 00:07:32.060 able to sell it for $500,000. 00:07:32.060 --> 00:07:36.240 So then instead of getting $100 million per year, you're 00:07:36.240 --> 00:07:39.080 only going to get $90 million per year. 00:07:39.080 --> 00:07:40.340 I wish I could use my pen. 00:07:43.050 --> 00:07:46.540 Something about my computer has frozen. 00:07:46.540 --> 00:07:48.960 So instead of $100 million a year, you're now only going to 00:07:48.960 --> 00:07:50.980 get $90 million a year. 00:07:50.980 --> 00:07:51.530 Right? 00:07:51.530 --> 00:07:53.480 And all of a sudden, these guys are not 00:07:53.480 --> 00:07:54.460 going to be cut off. 00:07:54.460 --> 00:07:56.230 This guy is still going to get $24 million, this guy is still 00:07:56.230 --> 00:08:00.320 going to get $21 million, but now this guy is going to get 00:08:00.320 --> 00:08:03.070 $45 million. 00:08:03.070 --> 00:08:04.675 But he's still getting above average yield. 00:08:04.675 --> 00:08:05.830 Now let's say it gets even worse. 00:08:05.830 --> 00:08:07.430 Let's say a bunch of borrowers start 00:08:07.430 --> 00:08:08.720 defaulting on their loans. 00:08:08.720 --> 00:08:14.010 And instead of getting $90 million per year, you start 00:08:14.010 --> 00:08:16.880 only getting $50 million in per year. 00:08:16.880 --> 00:08:18.940 Now you pay this guy $24 million. 00:08:18.940 --> 00:08:21.570 You pay this guy $21 million -- or this group of guys or 00:08:21.570 --> 00:08:22.970 gals -- $21 million. 00:08:22.970 --> 00:08:26.510 And then all you have left is $5 million for this guy. 00:08:26.510 --> 00:08:30.260 And $5 million on $300 million, now he's getting less 00:08:30.260 --> 00:08:31.840 than a 2% return. 00:08:31.840 --> 00:08:35.419 So this guy took on higher risk for higher reward. 00:08:35.419 --> 00:08:38.900 If everyone pays, sure, he gets 16.5%. 00:08:38.900 --> 00:08:42.820 But then if you start having a lot of defaults, if, let's 00:08:42.820 --> 00:08:48.090 say, the return on what you get every month goes in half, 00:08:48.090 --> 00:08:49.470 this guy takes the entire hit. 00:08:49.470 --> 00:08:51.130 So his return goes to 0%. 00:08:51.130 --> 00:08:53.030 So he had higher risk, higher reward, while 00:08:53.030 --> 00:08:54.450 these guys get untouched. 00:08:54.450 --> 00:08:57.130 Of course, if enough people start defaulting, even these 00:08:57.130 --> 00:08:59.440 people start to get hurt. 00:08:59.440 --> 00:09:02.250 So this is a form of a collateralized debt 00:09:02.250 --> 00:09:03.000 obligation. 00:09:03.000 --> 00:09:05.870 This is actually a mortgage-backed collateralized 00:09:05.870 --> 00:09:06.560 debt obligation. 00:09:06.560 --> 00:09:12.100 You can actually do this type of a structure with any type 00:09:12.100 --> 00:09:15.580 of debt obligation that's backed by assets. 00:09:15.580 --> 00:09:18.260 So we did the situation with mortgages, but you could do it 00:09:18.260 --> 00:09:19.110 with a bunch of assets. 00:09:19.110 --> 00:09:21.250 You could do it with corporate debt. 00:09:21.250 --> 00:09:24.450 You could do it with receivables from a company. 00:09:24.450 --> 00:09:26.880 But what you read about the most right now in the 00:09:26.880 --> 00:09:29.910 newspapers is mortgage-backed collateralized debt 00:09:29.910 --> 00:09:30.490 obligations. 00:09:30.490 --> 00:09:33.820 And to some degree, that's what's been getting a lot of 00:09:33.820 --> 00:09:34.890 these hedge funds in trouble. 00:09:34.890 --> 00:09:38.230 And I think I'll do another presentation on exactly how 00:09:38.230 --> 00:09:40.370 and why they have gotten in trouble. 00:09:40.370 --> 00:09:42.760 Look forward to talking to you

Mortgage-backed securities III

https://www.youtube.com/watch?v=q0oSKmC3Mfc

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en

WEBVTT Kind: captions Language: en 00:00:01.110 --> 00:00:03.480 Welcome back to my series of presentations on 00:00:03.480 --> 00:00:05.480 mortgage-backed securities. 00:00:05.480 --> 00:00:08.100 So let's review what we've already gone over. 00:00:08.100 --> 00:00:10.460 So I've already drawn here-- I actually prepared ahead of 00:00:10.460 --> 00:00:13.580 time-- so I've already drawn here kind of what we've 00:00:13.580 --> 00:00:14.570 already talked about. 00:00:14.570 --> 00:00:17.560 So we start with borrowers who need to buy houses. 00:00:17.560 --> 00:00:19.570 Each of them borrowed $1 million. 00:00:19.570 --> 00:00:20.890 Actually let me write that down. 00:00:20.890 --> 00:00:24.240 Let me change the color of my pen. 00:00:24.240 --> 00:00:25.410 Where'd my pen go? 00:00:25.410 --> 00:00:26.110 OK. 00:00:26.110 --> 00:00:27.790 So each of these people borrowed $1 million. 00:00:33.610 --> 00:00:33.920 OK. 00:00:33.920 --> 00:00:36.230 Each of them borrowed $1 million and there were 1,000 00:00:36.230 --> 00:00:37.150 of them, right? 00:00:37.150 --> 00:00:39.170 So $1 million times 1,000. 00:00:39.170 --> 00:00:41.250 That's $1 billion that they needed. 00:00:41.250 --> 00:00:44.460 And they said that they would pay 10% a year on that money 00:00:44.460 --> 00:00:45.100 that they borrowed. 00:00:45.100 --> 00:00:48.940 So that's 10% for each of them is $100,000 and then as we 00:00:48.940 --> 00:00:50.500 said, there's 1,000 borrowers. 00:00:50.500 --> 00:00:53.370 So they're going to put in $100 million, right? 00:00:53.370 --> 00:00:56.030 100,000 times 1,000 is 100 million. 00:00:56.030 --> 00:00:57.070 So just to simplify. 00:00:57.070 --> 00:01:00.000 Keep it in your mind. $1 million goes to a bunch of 00:01:00.000 --> 00:01:03.050 borrowers, goes to 1,000 borrowers, to be specific. 00:01:03.050 --> 00:01:07.130 And then each year, those borrowers are going to give 00:01:07.130 --> 00:01:09.280 the special purpose entity-- this is just a corporation 00:01:09.280 --> 00:01:12.560 designed to kind of structure these mortgage-backed 00:01:12.560 --> 00:01:15.430 securities-- they're going to give 10% of the billion, or 00:01:15.430 --> 00:01:17.690 $100 million back into this. 00:01:17.690 --> 00:01:20.190 And then we said, OK, well where does that money for this 00:01:20.190 --> 00:01:22.960 special purpose entity, or for this corporation, come from? 00:01:22.960 --> 00:01:26.150 Well it comes from the investors in the actual 00:01:26.150 --> 00:01:27.400 mortgage-backed securities. 00:01:31.440 --> 00:01:36.060 And just to be clear, so the asset within this 00:01:36.060 --> 00:01:40.175 entity are the loans. 00:01:44.815 --> 00:01:47.920 The loans are the main asset that's inside of the special 00:01:47.920 --> 00:01:48.470 purpose entity. 00:01:48.470 --> 00:01:51.580 And the loans are just the right on these 10% payments. 00:01:51.580 --> 00:01:54.430 And so money came from when the owners of each of these 00:01:54.430 --> 00:01:57.870 mortgage-backed securities-- each, let's say, paid $1,000 00:01:57.870 --> 00:01:59.140 for the mortgage-backed securities. 00:01:59.140 --> 00:02:01.620 And in return, they're going to get 10% on their money. 00:02:01.620 --> 00:02:03.750 So each security cost $1,000. 00:02:03.750 --> 00:02:05.040 And then they're going to they're going to get 00:02:05.040 --> 00:02:06.640 $100 back per month. 00:02:06.640 --> 00:02:08.669 And we said there are a million of these securities, 00:02:08.669 --> 00:02:12.210 so $1,000 times 1 million, that's where the $1 billion 00:02:12.210 --> 00:02:13.460 comes from. 00:02:16.760 --> 00:02:18.140 My thing's been acting up. 00:02:18.140 --> 00:02:20.550 That's where the $1 billion comes from. 00:02:20.550 --> 00:02:22.800 And that essentially is lent to the borrowers. 00:02:22.800 --> 00:02:24.680 And these guys will get 10%. 00:02:24.680 --> 00:02:28.500 Now one thing I want you to keep in mind is, they get 10% 00:02:28.500 --> 00:02:33.370 only if every one of these borrowers pays their loans, 00:02:33.370 --> 00:02:35.760 never defaults, never pre-pays. 00:02:35.760 --> 00:02:38.300 Pre-paying a mortgage is just saying, I sold the house. 00:02:38.300 --> 00:02:40.470 I don't need the mortgage anymore, so I just pay it off. 00:02:40.470 --> 00:02:44.390 So it's only 10%, indefinitely, if all of the 00:02:44.390 --> 00:02:47.990 borrowers pay all the money and never default or 00:02:47.990 --> 00:02:49.560 anything like that. 00:02:49.560 --> 00:02:51.670 So this 10% is kind of in an ideal world. 00:02:51.670 --> 00:02:54.520 Well everyone knows that it's not going to be exactly 10%. 00:02:54.520 --> 00:02:56.630 Some percentage of these borrowers are going to default 00:02:56.630 --> 00:02:57.240 on their mortgage. 00:02:57.240 --> 00:02:58.840 Some of them are going to pay ahead of time. 00:02:58.840 --> 00:03:01.990 And actually that's what the buyer of the mortgage-backed 00:03:01.990 --> 00:03:04.230 security should try to figure out. 00:03:04.230 --> 00:03:06.220 And all sorts of buyers are going to have all sorts of 00:03:06.220 --> 00:03:07.100 different assumptions. 00:03:07.100 --> 00:03:10.350 And this is what you probably read some articles about, 00:03:10.350 --> 00:03:14.560 these hedge funds with these computer models to value their 00:03:14.560 --> 00:03:15.930 mortgage-backed securities. 00:03:15.930 --> 00:03:17.610 And that's what those computer models do. 00:03:17.610 --> 00:03:21.450 They try to look at historical data and figure out, OK, for a 00:03:21.450 --> 00:03:24.940 given population pool in a given part of the country, 00:03:24.940 --> 00:03:27.530 what percentage of them are able to 00:03:27.530 --> 00:03:28.510 pay off their mortgage? 00:03:28.510 --> 00:03:31.140 What percentage of them default on their mortgage? 00:03:31.140 --> 00:03:34.100 And when they default, what is kind of the recovery? 00:03:34.100 --> 00:03:37.190 Say they default on a $1 million mortgage, and then the 00:03:37.190 --> 00:03:39.930 special purpose entity would get control of that house. 00:03:39.930 --> 00:03:44.070 And then if that house is sold for $500,000 because the 00:03:44.070 --> 00:03:47.610 property value went down, then the recovery would be 50%. 00:03:47.610 --> 00:03:50.430 So that's all of the things that someone needs to factor 00:03:50.430 --> 00:03:52.260 in when they figure out what will be the real return. 00:03:52.260 --> 00:03:54.030 10% is if everyone pays. 00:03:54.030 --> 00:03:58.120 So let's make some very simple assumptions for ourselves. 00:03:58.120 --> 00:03:59.760 Let's say we are thinking about investing in a 00:03:59.760 --> 00:04:02.490 mortgage-backed security and we want to gauge for ourselves 00:04:02.490 --> 00:04:04.490 what we think the return is going to be. 00:04:04.490 --> 00:04:09.420 Well let's say we know that this pool of borrowers that-- 00:04:09.420 --> 00:04:16.450 my pen keeps not working-- that 20% will default. 00:04:16.450 --> 00:04:18.529 We're not going to worry about pre-payment rates and all 00:04:18.529 --> 00:04:19.390 things like that. 00:04:19.390 --> 00:04:20.884 Let's say 20% are going to default. 00:04:24.180 --> 00:04:27.660 Of these 1,000 borrowers, 200 of them are just going to lose 00:04:27.660 --> 00:04:28.480 their job or whatever. 00:04:28.480 --> 00:04:30.730 They can't afford a mortgage anymore. 00:04:30.730 --> 00:04:38.240 And of those 20% that default, we have a 50% recovery. 00:04:38.240 --> 00:04:43.270 So that means borrower X defaulted on his loan. 00:04:43.270 --> 00:04:46.450 And then when we go and get the property-- because the 00:04:46.450 --> 00:04:48.620 loan was secured by the property-- when we auction off 00:04:48.620 --> 00:04:50.980 the property, we only get $500,000 for it. 00:04:50.980 --> 00:04:52.350 So we get a 50% recovery. 00:04:52.350 --> 00:04:55.780 50% of the original value of the loan. 00:04:55.780 --> 00:05:00.820 So if 20% default and then there's a 50% recovery, then 00:05:00.820 --> 00:05:14.570 on average you're going to get 10% of the loan is worthless. 00:05:14.570 --> 00:05:15.910 And I'm going to make some kind of 00:05:15.910 --> 00:05:17.750 handwaving assumptions here. 00:05:17.750 --> 00:05:19.760 But you can assume statistically, and since this 00:05:19.760 --> 00:05:22.230 is a large number of borrowers-- it's 1,000, right? 00:05:22.230 --> 00:05:24.470 If there's only one borrower it would be hard to kind of 00:05:24.470 --> 00:05:26.650 gauge when he defaults, if he defaults at all. 00:05:26.650 --> 00:05:28.590 We would just know that there is a 20% chance. 00:05:28.590 --> 00:05:30.430 But when there's a large number of borrowers, you can 00:05:30.430 --> 00:05:33.460 kind of do the math and say, OK, on average 200 of these 00:05:33.460 --> 00:05:35.750 guys are going to default, and instead of actually getting 00:05:35.750 --> 00:05:40.470 10%, since 10% of the loans are going to be worthless, I'm 00:05:40.470 --> 00:05:43.940 going to get 10% less than this 10%. 00:05:43.940 --> 00:05:45.610 So I'm going to get 9%. 00:05:45.610 --> 00:05:48.210 So this is based on the model that we just 00:05:48.210 --> 00:05:50.500 constructed, right? 00:05:50.500 --> 00:05:51.910 This is the model that we constructed. 00:05:51.910 --> 00:05:55.080 This is a much simpler model than what most people use. 00:05:55.080 --> 00:05:58.950 But based on the model that we just constructed, I think the 00:05:58.950 --> 00:06:00.990 real return we're going to get on this mortgage-backed 00:06:00.990 --> 00:06:02.570 security is 9%. 00:06:02.570 --> 00:06:06.190 If there was another investor who assumed a 50% default 00:06:06.190 --> 00:06:09.040 rate, but with a higher recovery, he or she would have 00:06:09.040 --> 00:06:14.120 a different kind of expected return from this security. 00:06:14.120 --> 00:06:17.000 So why is this even useful? 00:06:17.000 --> 00:06:17.760 Well think about it. 00:06:17.760 --> 00:06:20.540 Before, in the case we did in the first video, when someone 00:06:20.540 --> 00:06:25.620 just borrows from the bank, the bank has very specific 00:06:25.620 --> 00:06:26.670 lending requirements. 00:06:26.670 --> 00:06:29.320 They have their own model. 00:06:29.320 --> 00:06:33.180 So there's a whole class of borrowers that they might have 00:06:33.180 --> 00:06:34.680 not been able to service. 00:06:34.680 --> 00:06:34.970 Right? 00:06:34.970 --> 00:06:37.830 There might be people with really good credit scores, 00:06:37.830 --> 00:06:40.730 really good incomes, who don't have a down payment. 00:06:40.730 --> 00:06:43.210 And if they don't meet what the bank's requirements are, 00:06:43.210 --> 00:06:44.540 they would never get a loan. 00:06:44.540 --> 00:06:46.430 But there are probably some investors out there that would 00:06:46.430 --> 00:06:46.970 say, you know what? 00:06:46.970 --> 00:06:50.310 For the right interest rate and for the right assumptions 00:06:50.310 --> 00:06:53.110 in my model, I'm willing to give anybody a loan, as long 00:06:53.110 --> 00:06:54.840 as I'm compensated for it enough. 00:06:54.840 --> 00:06:57.230 And this is what this mortgage-backed security 00:06:57.230 --> 00:06:58.260 market allows. 00:06:58.260 --> 00:07:02.495 It allows-- let's say this group of borrowers-- let's say 00:07:02.495 --> 00:07:12.520 this pool of borrowers right here actually didn't-- 00:07:12.520 --> 00:07:14.710 This pool of borrowers actually aren't the 00:07:14.710 --> 00:07:22.270 traditional-- they don't have 25% down and they don't have 00:07:22.270 --> 00:07:24.340 kind of the traditional requirements to get a normal 00:07:24.340 --> 00:07:28.290 mortgage-- but if I pool a bunch of people who don't have 00:07:28.290 --> 00:07:29.630 those traditional requirements, but they're good 00:07:29.630 --> 00:07:31.740 in other ways-- they have a high income or high credit 00:07:31.740 --> 00:07:34.940 score-- I can go through this alternate mechanism to find 00:07:34.940 --> 00:07:37.130 investors that are willing to loan them money. 00:07:37.130 --> 00:07:39.840 So essentially, from the borrower's point of view, it 00:07:39.840 --> 00:07:44.260 allows more access to loan funding that they would have 00:07:44.260 --> 00:07:45.790 otherwise not been able to. 00:07:45.790 --> 00:07:48.770 And from an investor point of view, it allows another place 00:07:48.770 --> 00:07:51.880 for me to invest in. 00:07:51.880 --> 00:07:55.170 Maybe I feel that the computer models that I have are really 00:07:55.170 --> 00:07:58.350 good at predicting things like default rates, and recovery 00:07:58.350 --> 00:08:01.320 rates, and what a loan is worth. 00:08:01.320 --> 00:08:05.950 And I feel that I can, in some ways, be a better loan officer 00:08:05.950 --> 00:08:06.480 than the banks. 00:08:06.480 --> 00:08:09.150 And this would be an attractive place for 00:08:09.150 --> 00:08:10.590 me to invest in. 00:08:10.590 --> 00:08:14.890 It also might just have a risk reward characteristic that 00:08:14.890 --> 00:08:17.360 doesn't exist in the market already, and it allows you to 00:08:17.360 --> 00:08:19.600 diversify into one other asset class. 00:08:19.600 --> 00:08:20.740 So that's the value that it has 00:08:20.740 --> 00:08:22.780 across the entire spectrum. 00:08:22.780 --> 00:08:27.050 Now in the next presentation I'm going to show how you can, 00:08:27.050 --> 00:08:30.720 I guess, further complicate this even more, so that you 00:08:30.720 --> 00:08:32.570 can open up the investment to even a 00:08:32.570 --> 00:08:33.820 larger group of investors. 00:08:33.820 --> 00:08:35.320 Because you can think about it right now, there's probably 00:08:35.320 --> 00:08:38.530 some people who say, OK, I already said, some people will 00:08:38.530 --> 00:08:40.760 do these models and try to make their own assumptions and 00:08:40.760 --> 00:08:43.039 say, OK, this is going to give me 9% a year. 00:08:43.039 --> 00:08:44.580 But there's a whole bunch of people who are going to say 00:08:44.580 --> 00:08:46.450 this is just too complicated for me. 00:08:46.450 --> 00:08:47.740 This seems risky. 00:08:47.740 --> 00:08:49.980 I don't have any fancy models. 00:08:49.980 --> 00:08:51.560 I only like to invest in things where I 00:08:51.560 --> 00:08:53.120 know I get my money. 00:08:53.120 --> 00:08:55.580 Very highly rated debt is where I'm going 00:08:55.580 --> 00:08:56.390 to invest my money. 00:08:56.390 --> 00:08:58.850 And there's another group of people who say, OK, 9%, that's 00:08:58.850 --> 00:09:01.670 nice and everything, but I'm a hotshot, I'm a gambler. 00:09:01.670 --> 00:09:04.280 9% isn't the type of returns I want. 00:09:04.280 --> 00:09:07.670 I want to take more risk and more return. 00:09:07.670 --> 00:09:09.690 And so there should be something, maybe, for those 00:09:09.690 --> 00:09:10.500 people as well. 00:09:10.500 --> 00:09:12.650 So that's what we're going to show you in the presentation 00:09:12.650 --> 00:09:15.970 on collateralized debt obligations. 00:09:15.970 --> 00:09:17.460 See you soon.

Mortgage-backed securities II

https://www.youtube.com/watch?v=eYBlfxGIk28

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https://www.youtube.com/api/timedtext?v=eYBlfxGIk28&ei=eWeUZbWzO56np-oPtJSUyAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249834&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8A6B96A66E66BD7CE223DCAAA001C2C3A9F84480.C60860BB59323B7FDFE45E5893215CCCBE444EFF&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.820 --> 00:00:01.700 Welcome back. 00:00:01.700 --> 00:00:04.770 So where we left off is, there was 1,000 people who all 00:00:04.770 --> 00:00:07.800 needed, let's say, $1 million loans each. 00:00:07.800 --> 00:00:11.260 And they were all going to pay 10% on their loans. 00:00:11.260 --> 00:00:12.990 And they borrowed it from this bank. 00:00:12.990 --> 00:00:14.900 It's kind of a standard commercial bank. 00:00:14.900 --> 00:00:18.160 And this bank says, well you know, I just handed out $1 00:00:18.160 --> 00:00:20.040 billion and I'm getting these interest payments. 00:00:20.040 --> 00:00:21.970 And my vaults are empty. 00:00:21.970 --> 00:00:24.450 I want to have money back in my vaults. 00:00:24.450 --> 00:00:27.310 Or I want to have money back on my balance sheet. 00:00:27.310 --> 00:00:28.670 So I'm going to sell these loans. 00:00:28.670 --> 00:00:31.020 I'm going to bundle them all together and sell them to this 00:00:31.020 --> 00:00:32.189 investment bank. 00:00:32.189 --> 00:00:35.140 And now what is this investment bank going to do 00:00:35.140 --> 00:00:35.800 with these loans? 00:00:35.800 --> 00:00:36.700 And why is it doing it? 00:00:36.700 --> 00:00:40.220 So let me delete this. 00:00:40.220 --> 00:00:41.630 So we had the investment bank. 00:00:45.650 --> 00:00:47.430 I randomly picked the color green, but I think it is 00:00:47.430 --> 00:00:49.680 appropriate for the investment bank. 00:00:49.680 --> 00:00:56.250 So now I have all of the, you know, me-- that's a little 00:00:56.250 --> 00:00:57.200 smiley face. 00:00:57.200 --> 00:01:03.270 There's 1,000 of me, and we are now going to pay the 10%. 00:01:03.270 --> 00:01:06.670 We're now going to make all of our mortgage payments, the 10% 00:01:06.670 --> 00:01:10.010 payment, to this bank, this investment bank. 00:01:10.010 --> 00:01:12.280 But the investment bank, they're not in the business of 00:01:12.280 --> 00:01:14.670 servicing loans or keeping loans on their balance sheets. 00:01:14.670 --> 00:01:17.980 So what they do is, they create a corporation. 00:01:17.980 --> 00:01:20.630 They create an entity, maybe a special purpose entity, 00:01:20.630 --> 00:01:21.680 whatever you want to call it. 00:01:21.680 --> 00:01:23.820 So they create a company. 00:01:23.820 --> 00:01:26.170 Let me make that in purple, another color. 00:01:26.170 --> 00:01:27.420 So they created this company. 00:01:32.730 --> 00:01:44.720 And what they do is, they will take their rights to these 00:01:44.720 --> 00:01:45.760 payments that they got. 00:01:45.760 --> 00:01:45.960 Right? 00:01:45.960 --> 00:01:49.160 They paid $1 billion to that first bank in order to get the 00:01:49.160 --> 00:01:51.250 payments from all of these people. 00:01:51.250 --> 00:01:52.030 And they say, you know what? 00:01:52.030 --> 00:01:54.600 The rights on those payments-- that's the asset. 00:01:54.600 --> 00:01:55.460 They bought all these loans. 00:01:55.460 --> 00:01:58.050 The rights on those payments, we are now going to transfer 00:01:58.050 --> 00:02:00.690 to this special purpose entity, to this other 00:02:00.690 --> 00:02:01.560 corporation. 00:02:01.560 --> 00:02:05.150 So now everyone is going to, essentially-- all their 00:02:05.150 --> 00:02:09.500 mortgage payments are going to be funnelled into this entity. 00:02:09.500 --> 00:02:11.190 Right? 00:02:11.190 --> 00:02:12.630 And this is a corporation. 00:02:12.630 --> 00:02:16.020 And so what the bank will do is issue shares in this 00:02:16.020 --> 00:02:16.550 corporation. 00:02:16.550 --> 00:02:20.760 So let's say that it issues, let's sat for simplicity, a 00:02:20.760 --> 00:02:22.010 hundred shares. 00:02:26.050 --> 00:02:27.300 So what's in this corporation? 00:02:29.760 --> 00:02:33.560 The entire corporation gets the mortgage payments on the 00:02:33.560 --> 00:02:35.510 $1 billion in loans, right? 00:02:35.510 --> 00:02:36.975 It has $1 billion of loans outstanding. 00:02:36.975 --> 00:02:38.960 And it is going to get 10% a year. 00:02:38.960 --> 00:02:42.350 So it's going to get $100 million per year right? 00:02:42.350 --> 00:02:45.140 Because it's a thousand loans out there. 00:02:45.140 --> 00:02:47.315 It's going to get $100 million per year for 10 years. 00:02:47.315 --> 00:02:49.220 And at the end of the 10 years it's also going to get $1 00:02:49.220 --> 00:02:50.880 billion, right? 00:02:50.880 --> 00:02:52.020 That's its asset it has. 00:02:52.020 --> 00:02:55.770 Its asset is the rights on those payments streams that 00:02:55.770 --> 00:02:58.210 are going to come into this corporation. 00:02:58.210 --> 00:03:00.530 And it has 100 shares. 00:03:00.530 --> 00:03:02.030 The way I think about is you can split 00:03:02.030 --> 00:03:04.270 this company 100 ways. 00:03:04.270 --> 00:03:06.590 And I'm doing this to further confuse you. 00:03:06.590 --> 00:03:10.280 So what is each of those-- the owner of each of those shares, 00:03:10.280 --> 00:03:12.590 what does it entitle them to? 00:03:12.590 --> 00:03:20.330 Well, it entitles me to 1/100 of what this corporation gets. 00:03:20.330 --> 00:03:23.400 So if I have a share-- let's make it look like a share, a 00:03:23.400 --> 00:03:28.080 stock certificate-- I'm going to get 1/100 of this thing. 00:03:28.080 --> 00:03:29.530 And normally you wouldn't have 100 shares. 00:03:29.530 --> 00:03:31.440 You would have, let's say, a million shares. 00:03:31.440 --> 00:03:32.840 Actually let me make it a million shares, just because I 00:03:32.840 --> 00:03:35.530 think that's, in some strange way, more realistic. 00:03:35.530 --> 00:03:39.600 So let's say there are a million shares. 00:03:39.600 --> 00:03:42.100 So if there are 1 million shares, each share will get 1 00:03:42.100 --> 00:03:44.380 millionth of the cash flow stream that's 00:03:44.380 --> 00:03:46.170 entitled to this entity. 00:03:46.170 --> 00:03:50.180 So instead of getting $100 million every year, it gets 00:03:50.180 --> 00:03:51.110 one millionth of that. 00:03:51.110 --> 00:03:55.390 So it gets $100 per year. 00:03:55.390 --> 00:03:57.940 And then on the last year, instead of getting $1 billion 00:03:57.940 --> 00:04:01.030 it gets $1000. 00:04:01.030 --> 00:04:04.230 So what the bank will do is it'll take these shares and 00:04:04.230 --> 00:04:05.890 then it'll sell it to the general public. 00:04:05.890 --> 00:04:08.380 It'll IPO it, essentially. 00:04:08.380 --> 00:04:09.420 You can think of it that way. 00:04:09.420 --> 00:04:12.360 And tons of people will buy it, especially hedge funds, 00:04:12.360 --> 00:04:15.635 and pension funds, and mutual funds, and bond investors. 00:04:18.370 --> 00:04:22.029 And it's important to think about how the money's flowing. 00:04:22.029 --> 00:04:26.960 So now when they sell these shares in this entity, people 00:04:26.960 --> 00:04:29.620 are going to give them, well, hopefully more than what they 00:04:29.620 --> 00:04:30.840 paid for it, right? 00:04:30.840 --> 00:04:33.830 Maybe there's a lot of demand for this type of asset, where 00:04:33.830 --> 00:04:36.080 I get this type of income stream. 00:04:36.080 --> 00:04:38.950 So maybe once they sell all the shares, they get, I don't 00:04:38.950 --> 00:04:41.760 know, they get $1.1 billion for them, right? 00:04:41.760 --> 00:04:44.702 So this is the investors. 00:04:44.702 --> 00:04:48.360 And the investors collectively buy these 00:04:48.360 --> 00:04:49.700 shares for $1.1 billion. 00:04:49.700 --> 00:04:55.280 Essentially, let's say they paid $1.1 billion for a 00:04:55.280 --> 00:04:59.470 million shares, so they paid $1,100 per share. 00:04:59.470 --> 00:05:00.720 Right? 00:05:02.400 --> 00:05:04.840 Each of the investors paid $1,100 for each of these 00:05:04.840 --> 00:05:08.430 shares, so that $1.1 billion goes into this 00:05:08.430 --> 00:05:09.630 special purpose entity. 00:05:09.630 --> 00:05:11.630 And if you think about it, the bank made out 00:05:11.630 --> 00:05:12.620 like a bandit, right? 00:05:12.620 --> 00:05:17.890 Because the bank paid $1 billion for the rights to 00:05:17.890 --> 00:05:23.520 these mortgage payments, and it's getting $1.1 billion from 00:05:23.520 --> 00:05:24.340 the investors. 00:05:24.340 --> 00:05:27.190 And all the bank has to do is kind of set this whole legal 00:05:27.190 --> 00:05:29.150 structure up and service the loans. 00:05:29.150 --> 00:05:30.350 It actually doesn't even have to service the loans. 00:05:30.350 --> 00:05:31.580 We'll go into that later. 00:05:31.580 --> 00:05:35.520 So let me summarize, I guess. 00:05:35.520 --> 00:05:36.980 Just because I know this can be a little bit 00:05:36.980 --> 00:05:38.230 of a daunting subject. 00:05:41.280 --> 00:05:43.800 Let me summarize. 00:05:43.800 --> 00:05:47.200 And this purple I don't like. 00:05:47.200 --> 00:05:47.830 Anyway. 00:05:47.830 --> 00:05:50.315 So you have tons of investors. 00:05:50.315 --> 00:05:52.730 So, each of these is an investor. 00:05:52.730 --> 00:05:54.690 Actually a mortgage, a borrower. 00:05:54.690 --> 00:05:56.030 All of these people need to buy houses. 00:05:56.030 --> 00:05:59.670 These are all smiley faces. 00:05:59.670 --> 00:06:00.980 They all need to buy houses. 00:06:00.980 --> 00:06:07.370 And then they collectively get $1 billion. 00:06:07.370 --> 00:06:08.830 Right? $1 million each. 00:06:08.830 --> 00:06:13.080 And they each use that $1 million to buy their house. 00:06:13.080 --> 00:06:18.100 And then that $1 billion initially came from just their 00:06:18.100 --> 00:06:19.350 local bank. 00:06:21.790 --> 00:06:25.800 And when the $1 billion came from that local bank, all the 00:06:25.800 --> 00:06:28.640 payments, the interest payments, went to the bank. 00:06:28.640 --> 00:06:31.600 But then an investment bank came along and said, well no, 00:06:31.600 --> 00:06:33.195 I want to buy the rights to those payments. 00:06:36.088 --> 00:06:38.910 And an investment bank came along and says, well, I'm 00:06:38.910 --> 00:06:40.160 going to give you $1 billion. 00:06:44.020 --> 00:06:45.640 And now instead of you getting the 00:06:45.640 --> 00:06:47.720 payments, I get the payments. 00:06:47.720 --> 00:06:52.080 And then the bank sets up a special purpose entity. 00:06:52.080 --> 00:06:54.130 Essentially it sells a bunch of shares. 00:06:54.130 --> 00:06:55.380 Let's say it sells a million shares. 00:07:00.190 --> 00:07:04.560 And let's say it was able to sell each of those shares for 00:07:04.560 --> 00:07:12.950 $1,100 from the investing public. 00:07:12.950 --> 00:07:16.270 So it raises $1.1 billion, right? 00:07:16.270 --> 00:07:20.180 So the value of this company is $1.1 billion that now goes 00:07:20.180 --> 00:07:21.030 to the bank. 00:07:21.030 --> 00:07:24.180 And now the payment stream instead of going to-- let me 00:07:24.180 --> 00:07:28.340 do a different color-- now the payment stream goes to this 00:07:28.340 --> 00:07:32.580 special purpose entity instead of the bank. 00:07:32.580 --> 00:07:35.080 And the bank essentially made out like a bandit because it 00:07:35.080 --> 00:07:40.920 paid $1 billion and it got $1.1 so it made $100 million 00:07:40.920 --> 00:07:42.010 just for doing this transaction. 00:07:42.010 --> 00:07:44.280 I'm not saying that's how much a bank actually would make, 00:07:44.280 --> 00:07:48.150 but this shows you why every person is kind of, what 00:07:48.150 --> 00:07:50.040 they're doing in this value chain. 00:07:50.040 --> 00:07:52.600 And as I said before, this bank also probably did 00:07:52.600 --> 00:07:53.310 something similar. 00:07:53.310 --> 00:07:57.050 They probably took some fees or sold the loans for slightly 00:07:57.050 --> 00:07:59.300 more than they issued the loans for. 00:07:59.300 --> 00:08:07.920 So these shares-- each of these one million shares-- 00:08:07.920 --> 00:08:10.400 this is a mortgage-backed security. 00:08:14.900 --> 00:08:15.400 And it makes sense. 00:08:15.400 --> 00:08:16.040 It's a security. 00:08:16.040 --> 00:08:19.690 A security is an ownership that's tradable in a company. 00:08:19.690 --> 00:08:24.100 And that company has the right to payments that are secured 00:08:24.100 --> 00:08:25.120 by mortgages. 00:08:25.120 --> 00:08:28.420 So if all these people promised they would pay, and 00:08:28.420 --> 00:08:32.590 they're going to pay to this special purpose entity. 00:08:32.590 --> 00:08:36.429 But if, by chance, one of these people lose their jobs 00:08:36.429 --> 00:08:41.000 or they can't pay, or for whatever reason, instead of 00:08:41.000 --> 00:08:44.440 the payments, this entity is going to have the rights to 00:08:44.440 --> 00:08:45.130 their property. 00:08:45.130 --> 00:08:46.890 And that's why we say that it's a 00:08:46.890 --> 00:08:48.240 mortgage-backed security. 00:08:48.240 --> 00:08:50.980 So it's not just a promise to get money. 00:08:50.980 --> 00:08:53.470 The money is actually backed by people's mortgages. 00:08:53.470 --> 00:08:55.660 And of course, then this entity is going to, if this 00:08:55.660 --> 00:08:58.970 guy defaults on his loan-- he's one of a million, so 00:08:58.970 --> 00:09:01.200 statistically you might be able to predict that. 00:09:01.200 --> 00:09:03.900 I don't to put too much stock in these statistical models-- 00:09:03.900 --> 00:09:06.780 then this entity will just have that property auctioned 00:09:06.780 --> 00:09:07.520 off or sold. 00:09:07.520 --> 00:09:10.570 And the cash flow will come back to it. 00:09:10.570 --> 00:09:12.510 So that's what a mortgage-backed security is. 00:09:12.510 --> 00:09:15.980 Hopefully I didn't confuse you too much. 00:09:15.980 --> 00:09:19.260 My next presentation, I'm going to take it to a further 00:09:19.260 --> 00:09:22.430 level of confusion and show you what a collateralized debt 00:09:22.430 --> 00:09:23.430 obligation is. 00:09:23.430 --> 00:09:27.150 And then I'll do a more philosophical video on why 00:09:27.150 --> 00:09:30.190 these things even exist, and why they're useful, and why 00:09:30.190 --> 00:09:35.190 people may benefit or may not benefit from these things.

Mortgage-backed securities I

https://www.youtube.com/watch?v=oosYQHq2hwE

vtt

https://www.youtube.com/api/timedtext?v=oosYQHq2hwE&ei=fGeUZY_oH7K5vdIPzJes0AM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249836&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=68AC20A76393CBF8CE791AD6F7F04CB29F18D12A.AA9DAFDD6ABDC4795D9760F6148070F20BC3E189&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.910 --> 00:00:04.760 Welcome to my presentation on mortgage-backed securities. 00:00:04.760 --> 00:00:05.730 Let's get started. 00:00:05.730 --> 00:00:07.560 And this is going to be part of a whole new series of 00:00:07.560 --> 00:00:10.840 presentations, because I think what's happening right now in 00:00:10.840 --> 00:00:14.670 the credit markets is pretty significant from, I guess, a 00:00:14.670 --> 00:00:16.510 personal finance point of view and just from a 00:00:16.510 --> 00:00:17.730 historic point of view. 00:00:17.730 --> 00:00:20.130 And I want to do a whole set of videos just so people 00:00:20.130 --> 00:00:23.870 understand, I guess, how everything fits together, and 00:00:23.870 --> 00:00:25.470 what the possible repercussions could be. 00:00:25.470 --> 00:00:26.690 But we have to start with the basics. 00:00:26.690 --> 00:00:28.170 So what is a mortgage-backed security? 00:00:28.170 --> 00:00:29.940 You've probably read a lot about these. 00:00:29.940 --> 00:00:33.440 So historically, let's think about what historically 00:00:33.440 --> 00:00:37.100 happens when I went to get a loan for a house, let's say, 00:00:37.100 --> 00:00:37.850 20 years ago. 00:00:37.850 --> 00:00:38.920 And I'm going to simplify some things. 00:00:38.920 --> 00:00:40.250 And later we can do a more nuanced. 00:00:44.000 --> 00:00:46.000 Where'd my pen go? 00:00:46.000 --> 00:00:48.150 Let's say I need $100,000. 00:00:48.150 --> 00:00:50.210 No, let me say $1 million, because that's actually closer 00:00:50.210 --> 00:00:52.710 to how much houses cost now. 00:00:52.710 --> 00:00:57.080 Let's say I need a $1 million loan to buy a house, right? 00:00:57.080 --> 00:00:58.990 This is going to be a mortgage that's going to be 00:00:58.990 --> 00:01:00.160 backed by my house. 00:01:00.160 --> 00:01:03.150 And when I say backed by my house, or secured by my house, 00:01:03.150 --> 00:01:08.230 that means that I'm going to borrow $1 million from a bank, 00:01:08.230 --> 00:01:10.780 and if I can't pay back the loan, then the 00:01:10.780 --> 00:01:11.590 bank gets my house. 00:01:11.590 --> 00:01:13.210 That's all it means. 00:01:13.210 --> 00:01:15.800 And oftentimes it'll only be secured by the house, which 00:01:15.800 --> 00:01:17.310 means that I could just give them back the keys. 00:01:17.310 --> 00:01:19.760 They get the house and I have no other responsibility, but 00:01:19.760 --> 00:01:21.250 of course my credit gets messed up. 00:01:21.250 --> 00:01:22.690 But I need a $1 million loan. 00:01:22.690 --> 00:01:28.280 The traditional way I got a $1 million loan is I would go and 00:01:28.280 --> 00:01:30.550 talk to the bank. 00:01:30.550 --> 00:01:33.260 This is the bank. 00:01:33.260 --> 00:01:34.720 They have the money. 00:01:34.720 --> 00:01:41.830 And then they would give me $1 million and I would pay them 00:01:41.830 --> 00:01:46.340 some type of interest. I'll make up a number. 00:01:46.340 --> 00:01:48.410 The interest rates obviously change, and we'll do future 00:01:48.410 --> 00:01:50.730 presentations on what causes the interest rates to change. 00:01:50.730 --> 00:01:55.890 But let's say I would pay them 10% interest. And for the sake 00:01:55.890 --> 00:01:59.660 of simplicity, I'm going to assume that the loans in this 00:01:59.660 --> 00:02:01.690 presentation are interest-only loans. 00:02:01.690 --> 00:02:04.800 In a traditional mortgage, you actually, your payment has 00:02:04.800 --> 00:02:06.950 some part interest and some part principal. 00:02:06.950 --> 00:02:10.419 Principal is actually when you're paying down the loan. 00:02:10.419 --> 00:02:12.530 The math is a little bit more difficult with that, so what 00:02:12.530 --> 00:02:15.030 we're going to do in this case is assume that I only pay the 00:02:15.030 --> 00:02:17.770 interest portion, and at the end of the loan I pay the 00:02:17.770 --> 00:02:18.800 whole loan amount. 00:02:18.800 --> 00:02:23.470 So let's say that this is a 10-year loan. 00:02:23.470 --> 00:02:27.530 So for each year of the 10 years, I'm going to pay 00:02:27.530 --> 00:02:33.290 $100,000 in interest. $100,000 per year, right? 00:02:33.290 --> 00:02:38.540 And then in year 10, I'm going to pay the $100,000 and I'm 00:02:38.540 --> 00:02:41.490 also going to pay back the $1 million. 00:02:41.490 --> 00:02:44.150 Right? 00:02:44.150 --> 00:02:50.550 Year 1, 2, 3, dot, dot, dot, dot, 9, 10. 00:02:50.550 --> 00:02:52.310 So in year one, I pay $100,000. 00:02:52.310 --> 00:02:53.720 Year two, I pay $100,000. 00:02:53.720 --> 00:02:55.030 Year three, I pay $100,000. 00:02:55.030 --> 00:02:56.030 Dot, dot, dot, dot. 00:02:56.030 --> 00:02:57.650 Year nine, I pay $100,000. 00:02:57.650 --> 00:03:00.340 And then year 10, I pay the $100,000 plus I pay back the 00:03:00.340 --> 00:03:01.030 $1 million. 00:03:01.030 --> 00:03:04.300 So I pay back $1.1 million. 00:03:04.300 --> 00:03:08.180 So that's kind of how the cash is going to be transferred 00:03:08.180 --> 00:03:09.830 between me and the bank. 00:03:09.830 --> 00:03:11.725 And this is how a-- I don't want to say a traditional 00:03:11.725 --> 00:03:13.240 loan, because this isn't a traditional loan, an 00:03:13.240 --> 00:03:17.630 interest-only loan-- but for the sake of this presentation, 00:03:17.630 --> 00:03:19.410 how it's different than a mortgage-backed security, the 00:03:19.410 --> 00:03:21.990 important thing to realize is that the bank would 00:03:21.990 --> 00:03:23.710 have kept the loan. 00:03:23.710 --> 00:03:25.780 These payments I would have been making would have been 00:03:25.780 --> 00:03:27.010 directly to the bank. 00:03:27.010 --> 00:03:28.540 And that's what the business that, 00:03:28.540 --> 00:03:30.800 historically, banks were in. 00:03:30.800 --> 00:03:38.770 Another person, you-- and you have a hat-- let's say you're 00:03:38.770 --> 00:03:40.610 extremely wealthy and you would put $1 00:03:40.610 --> 00:03:42.090 million into the bank. 00:03:42.090 --> 00:03:42.340 Right? 00:03:42.340 --> 00:03:44.320 That's just your life savings or you inherited 00:03:44.320 --> 00:03:45.260 it from your uncle. 00:03:45.260 --> 00:03:47.710 And the bank would pay you, I don't know, 5%. 00:03:51.840 --> 00:03:55.120 And then take that $1 million, give it to me, and get 10% on 00:03:55.120 --> 00:03:56.680 what I just borrowed. 00:03:56.680 --> 00:03:58.330 And then the bank makes the difference, right? 00:03:58.330 --> 00:04:01.360 It's paying you 5% percent and then it's getting 10% from me. 00:04:01.360 --> 00:04:03.790 And we can go later into how they can pull this off, like 00:04:03.790 --> 00:04:05.270 what happens when you have to withdraw the money, 00:04:05.270 --> 00:04:05.960 et cetera, et cetera. 00:04:05.960 --> 00:04:09.220 But the important thing to realize is that these payments 00:04:09.220 --> 00:04:11.210 I make are to the bank. 00:04:11.210 --> 00:04:15.220 That's how loans worked before the mortgage-backed security 00:04:15.220 --> 00:04:17.320 industry really got developed. 00:04:17.320 --> 00:04:20.550 Now let's do the example with a mortgage-backed security. 00:04:23.270 --> 00:04:24.460 Now there's still me. 00:04:24.460 --> 00:04:29.040 I still exist. And I still need $1 million. 00:04:31.950 --> 00:04:34.100 Let's say I still go to the bank. 00:04:34.100 --> 00:04:35.460 Let's say I go to the bank. 00:04:35.460 --> 00:04:36.710 The bank is still there. 00:04:41.740 --> 00:04:45.550 And like before, the bank gives me $1 million. 00:04:48.380 --> 00:04:52.320 And then I give the bank 10% per year. 00:04:52.320 --> 00:04:52.840 Right? 00:04:52.840 --> 00:04:54.940 So it looks very similar to our old model. 00:04:54.940 --> 00:04:57.900 But in the old model, the bank would keep 00:04:57.900 --> 00:04:59.040 these payments itself. 00:04:59.040 --> 00:05:03.650 And that $1 million it had is now used to pay for my house. 00:05:03.650 --> 00:05:05.810 Then there was an innovation. 00:05:05.810 --> 00:05:08.370 Instead of having to get more deposits in order to keep 00:05:08.370 --> 00:05:10.750 giving out loans, the bank said, well, why don't I sell 00:05:10.750 --> 00:05:14.030 these loans to a third party and let them do 00:05:14.030 --> 00:05:15.140 something with it? 00:05:15.140 --> 00:05:16.550 And I know that that might be a little confusing. 00:05:16.550 --> 00:05:17.640 How do you sell a loan? 00:05:17.640 --> 00:05:18.890 Well let's say there's me. 00:05:18.890 --> 00:05:21.510 And let's say there's a thousand of me. 00:05:21.510 --> 00:05:22.020 Right? 00:05:22.020 --> 00:05:24.310 There's a bunch of Sals in the world. 00:05:24.310 --> 00:05:24.820 Right? 00:05:24.820 --> 00:05:28.080 And we each are borrowing money from the bank. 00:05:28.080 --> 00:05:30.740 So there's a thousand of me. 00:05:30.740 --> 00:05:31.050 Right? 00:05:31.050 --> 00:05:32.460 I'm just saying any kind of large number. 00:05:32.460 --> 00:05:34.040 It doesn't have to be a thousand. 00:05:34.040 --> 00:05:36.220 And collectively we have borrowed a 00:05:36.220 --> 00:05:37.430 thousand times a million. 00:05:37.430 --> 00:05:42.190 So we've collectively borrowed $1 billion from the bank. 00:05:42.190 --> 00:05:44.570 And we are collectively paying 10% on that, right? 00:05:44.570 --> 00:05:47.330 Because each of us are going to pay 10% per year, so we're 00:05:47.330 --> 00:05:49.470 each going to pay 10% on that $1 billion. 00:05:49.470 --> 00:05:49.750 Right? 00:05:49.750 --> 00:05:52.930 So 10% on that $1 billion is $100 million in interest. So 00:05:52.930 --> 00:05:58.220 this 10% equals $100 million. 00:05:58.220 --> 00:06:02.810 Now the bank says, OK, all the $1 billion that I had in my 00:06:02.810 --> 00:06:05.830 vaults, or whatever-- I guess now there's no physical money, 00:06:05.830 --> 00:06:09.500 but in my databases-- is now out in people's pockets. 00:06:09.500 --> 00:06:10.800 I want to get more money. 00:06:10.800 --> 00:06:13.470 So what the bank does is it takes all these loans 00:06:13.470 --> 00:06:17.200 together, that $1 billion in loans, and it says, hey, 00:06:17.200 --> 00:06:22.060 investment bank-- so that's another bank-- why don't you 00:06:22.060 --> 00:06:23.750 give me $1 billion? 00:06:26.490 --> 00:06:28.850 So the investment bank gives them $1 billion. 00:06:28.850 --> 00:06:33.110 And then instead of me and the other thousands of me paying 00:06:33.110 --> 00:06:38.430 the money to this bank, we're now paying it to this new 00:06:38.430 --> 00:06:40.990 party, right? 00:06:40.990 --> 00:06:43.020 I'm making my picture very confusing. 00:06:43.020 --> 00:06:44.480 So what just happened? 00:06:44.480 --> 00:06:48.420 When this bank sold the loans-- grouped all of the 00:06:48.420 --> 00:06:50.460 loans together and it folded it into a big, kind of did it 00:06:50.460 --> 00:06:52.750 on a wholesale basis-- it's sold a thousand 00:06:52.750 --> 00:06:54.090 loans to this bank. 00:06:54.090 --> 00:06:58.330 So this bank paid $1 billion for the right to get the 00:06:58.330 --> 00:07:00.870 interest and principal payment on those loans. 00:07:00.870 --> 00:07:06.280 So all that happened is, this guy got the cash and then this 00:07:06.280 --> 00:07:08.850 bank will now get the set of payments. 00:07:08.850 --> 00:07:10.930 So you might wonder, why did this bank do it? 00:07:10.930 --> 00:07:14.210 Well I kind of glazed over the details, but he probably got a 00:07:14.210 --> 00:07:16.550 lot of fees for doing this, or maybe he just likes giving 00:07:16.550 --> 00:07:18.130 loans to his customers, whatever. 00:07:18.130 --> 00:07:20.610 But the actual right answer is that he got 00:07:20.610 --> 00:07:22.300 fees for doing this. 00:07:22.300 --> 00:07:24.110 And he's actually probably going to transfer a little bit 00:07:24.110 --> 00:07:26.130 less value to this guy. 00:07:26.130 --> 00:07:28.630 Now, hopefully you understand the notion of actually 00:07:28.630 --> 00:07:29.610 transferring the loan. 00:07:29.610 --> 00:07:31.980 This guy pays money and now the payments are essentially 00:07:31.980 --> 00:07:33.840 going to be funnelled to him. 00:07:33.840 --> 00:07:35.810 I only have two minutes left in this presentation, so in 00:07:35.810 --> 00:07:39.530 the next presentation I'm going to focus on what this 00:07:39.530 --> 00:07:43.300 guy can now do with the loan to turn it into a 00:07:43.300 --> 00:07:44.630 mortgage-backed security. 00:07:44.630 --> 00:07:46.250 And this guy's an investment bank instead of 00:07:46.250 --> 00:07:48.070 a commercial bank. 00:07:48.070 --> 00:07:49.820 That detail is not that important in understanding 00:07:49.820 --> 00:07:52.020 what a mortgage-backed security is, but that will 00:07:52.020 --> 00:07:54.820 have to wait until the next presentation. 00:07:54.820 --> 00:07:56.340 See you soon.

Return on capital

https://www.youtube.com/watch?v=9T6ZPYYu_Dk

vtt

https://www.youtube.com/api/timedtext?v=9T6ZPYYu_Dk&ei=gmeUZZSoOYyPp-oP5b2ngA8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=46BA869DDA3A2C25650007E44DDD086CCA514C37.7204AC3ECAD138FB62430F05F533CC697C487589&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.160 --> 00:00:04.660 Welcome to my presentation on return on capital. 00:00:04.660 --> 00:00:06.950 Let me write that down. 00:00:06.950 --> 00:00:08.060 I'm using the wrong color. 00:00:08.060 --> 00:00:09.260 Let me use a nicer color. 00:00:09.260 --> 00:00:10.510 Let me go to white. 00:00:20.700 --> 00:00:23.750 I want to do this presentation first, because I think this is 00:00:23.750 --> 00:00:26.420 really going to give you the big picture on how you should 00:00:26.420 --> 00:00:29.100 think about what something is worth. 00:00:29.100 --> 00:00:31.140 Whether you should invest your money into it. 00:00:31.140 --> 00:00:33.890 And how you should weigh the different options you have in 00:00:33.890 --> 00:00:36.450 terms of what you actually have to do with your money, in 00:00:36.450 --> 00:00:37.500 terms of where you want to put it. 00:00:37.500 --> 00:00:38.310 Do you want to put it in the bank? 00:00:38.310 --> 00:00:39.130 Do you want to buy a house? 00:00:39.130 --> 00:00:40.740 Do you want to pay off your credit cards? 00:00:40.740 --> 00:00:41.900 Et cetera, et cetera. 00:00:41.900 --> 00:00:44.860 So let's just define return on capital. 00:00:44.860 --> 00:00:47.570 And just so you know, I'm not necessarily going to be strict 00:00:47.570 --> 00:00:51.270 on the accounting conventions, or the GAP conventions -- 00:00:51.270 --> 00:00:52.840 that's the accounting conventions in this country. 00:00:52.840 --> 00:00:56.290 I'm going to do it more on a hands-on, how Joe Investors 00:00:56.290 --> 00:00:58.470 should think about their money. 00:00:58.470 --> 00:01:03.360 So in this scenario, I define return on capital as just the 00:01:03.360 --> 00:01:17.470 cash you get per year divided by the total cash you put in. 00:01:23.080 --> 00:01:26.730 And, well, I don't want to just say, cash. 00:01:26.730 --> 00:01:28.010 There's other ways to measure return. 00:01:28.010 --> 00:01:30.640 But actually, to keep it simple, let's just say cash. 00:01:30.640 --> 00:01:32.750 So let's think about how this works out. 00:01:32.750 --> 00:01:35.170 Let's say, I have an idea. 00:01:35.170 --> 00:01:36.630 I have a restaurant. 00:01:36.630 --> 00:01:40.530 And that restaurant, it'll cost $1 million. 00:01:40.530 --> 00:01:47.835 It'll cost $1 million investment in this restaurant. 00:01:50.590 --> 00:01:52.210 It's going to be a $1 million investment. 00:01:52.210 --> 00:01:57.500 And let's say that, per year, after paying all the expenses, 00:01:57.500 --> 00:02:01.670 after paying the utility, after paying the employees, 00:02:01.670 --> 00:02:04.690 after repairing, and maintenance, and after paying 00:02:04.690 --> 00:02:07.900 taxes, everything, let's say this restaurant 00:02:07.900 --> 00:02:11.790 makes $100,000 a year. 00:02:11.790 --> 00:02:13.170 And that's after taxes and everything. 00:02:13.170 --> 00:02:14.400 That's what goes into my pocket. 00:02:14.400 --> 00:02:17.470 So in this situation, my return on capital, the way 00:02:17.470 --> 00:02:23.885 I've defined it, is $100,000 divided by $1 million, or we 00:02:23.885 --> 00:02:29.880 could just say a thousand thousand dollars, equals 10%. 00:02:29.880 --> 00:02:30.560 Pretty straightforward. 00:02:30.560 --> 00:02:32.560 You're probably saying, Sal, this is silly. 00:02:32.560 --> 00:02:33.610 Why are you wasting my time? 00:02:33.610 --> 00:02:36.230 Well, maybe it is. 00:02:36.230 --> 00:02:38.890 But I think you'll find that this is going to lay a 00:02:38.890 --> 00:02:42.060 foundation that will eventually blow your mind. 00:02:42.060 --> 00:02:44.590 So let's keep going. 00:02:44.590 --> 00:02:45.840 Let me do another. 00:02:52.510 --> 00:02:55.260 OK, so I said the restaurant -- let's make it a pizza 00:02:55.260 --> 00:03:00.140 restaurant -- let's just say, the restaurant return on 00:03:00.140 --> 00:03:05.110 capital is equal to 10%. 00:03:05.110 --> 00:03:05.570 Right? 00:03:05.570 --> 00:03:09.540 I can put in $1 million and I'll get in $100,000 per year. 00:03:09.540 --> 00:03:10.910 That's where I got 10%. 00:03:10.910 --> 00:03:11.990 Let me write that down. 00:03:11.990 --> 00:03:25.430 I get $100,000 per year off of $1 million investment. 00:03:25.430 --> 00:03:28.720 Now, that's one project. 00:03:28.720 --> 00:03:31.140 I'm not going to factor in things like risk and 00:03:31.140 --> 00:03:32.550 probabilities just yet. 00:03:32.550 --> 00:03:35.380 Let's just say, for sure, I know that if I put my money 00:03:35.380 --> 00:03:38.830 here, I'm going to get 10% on my money. 00:03:38.830 --> 00:03:40.810 And let's say the other option with my 00:03:40.810 --> 00:03:46.365 money is a beauty parlor. 00:03:57.570 --> 00:03:59.330 And let's say that that also costs $1 million. 00:04:08.180 --> 00:04:16.269 And this beauty parlor gets me $50,000 a year. 00:04:16.269 --> 00:04:19.050 I think it's very obvious to you already which investment 00:04:19.050 --> 00:04:20.279 you'd rather invest in. 00:04:20.279 --> 00:04:24.250 Because the return on capital on this beauty parlor is only 00:04:24.250 --> 00:04:27.020 50,000 divided by a million, or 5%. 00:04:27.020 --> 00:04:27.910 So this is obvious. 00:04:27.910 --> 00:04:30.980 You'd rather do the restaurant than a beauty parlor. 00:04:30.980 --> 00:04:35.870 And in general, after adjusting for risk, you always 00:04:35.870 --> 00:04:37.970 want to go with the project that has the 00:04:37.970 --> 00:04:39.910 higher return on capital. 00:04:39.910 --> 00:04:42.670 And, later on, there will be nuances in terms of when you 00:04:42.670 --> 00:04:43.400 get that return. 00:04:43.400 --> 00:04:46.710 Maybe you would rather have a slightly lower return if you 00:04:46.710 --> 00:04:47.900 get the money faster. 00:04:47.900 --> 00:04:52.515 Or a slightly higher return if you're taking on risks, et 00:04:52.515 --> 00:04:53.100 cetera, et cetera. 00:04:53.100 --> 00:04:55.900 Or to compensate for risk. 00:04:55.900 --> 00:04:57.460 So we know we want to do the restaurant. 00:04:57.460 --> 00:05:00.730 But do we definitely want to do the restaurant? 00:05:00.730 --> 00:05:04.990 We'd rather do the restaurant than the beauty parlor, right? 00:05:04.990 --> 00:05:08.490 But my question to you is, do we definitely want to do the 00:05:08.490 --> 00:05:09.280 restaurant? 00:05:09.280 --> 00:05:12.710 And this is where the return on capital becomes 00:05:12.710 --> 00:05:13.180 interesting. 00:05:13.180 --> 00:05:15.340 Because what matters, before we put the money into the 00:05:15.340 --> 00:05:18.070 restaurant, is to think about what the cost of 00:05:18.070 --> 00:05:19.400 that money is to us. 00:05:19.400 --> 00:05:21.930 And this is what I think will be a little bit of a new 00:05:21.930 --> 00:05:22.830 concept to you. 00:05:22.830 --> 00:05:25.730 So I'm going to introduce you, now, to the notion of a cost 00:05:25.730 --> 00:05:27.431 of capital. 00:05:27.431 --> 00:05:29.850 So let me erase this. 00:05:32.350 --> 00:05:33.470 OK. 00:05:33.470 --> 00:05:34.860 So the restaurant costs $1 million. 00:05:40.270 --> 00:05:47.320 And it gives me $100,000 a year. 00:05:47.320 --> 00:05:52.190 And that's a 10% return on capital. 00:05:52.190 --> 00:05:53.680 Now, let's say I have to borrow all the money. 00:05:53.680 --> 00:05:56.360 And there's some bank that's willing to give me all the 00:05:56.360 --> 00:05:57.980 money for this restaurant. 00:05:57.980 --> 00:06:11.280 And the interest rate on this loan is, let's say, 15%. 00:06:11.280 --> 00:06:16.090 Is it still a good idea for me to open up the restaurant? 00:06:16.090 --> 00:06:19.170 Well, if I have a loan and I have to borrow the whole 00:06:19.170 --> 00:06:21.880 amount -- so I'm going to have a loan for $1 million to buy 00:06:21.880 --> 00:06:24.850 that same restaurant. 00:06:24.850 --> 00:06:28.830 And I'm going to be charged 15% in interest every year . 00:06:28.830 --> 00:06:30.690 And I'm not going to take taxes, and the fact that you 00:06:30.690 --> 00:06:32.470 could deduct taxes, et cetera, et cetera , into 00:06:32.470 --> 00:06:33.930 account just yet. 00:06:33.930 --> 00:06:36.580 Let's assume that my total cost is 15% per year in 00:06:36.580 --> 00:06:43.440 interest. So I'm going to have to spend $150,000 per year in 00:06:43.440 --> 00:06:46.640 interest. 00:06:46.640 --> 00:06:50.000 So my question to you is, does it still make sense for me to 00:06:50.000 --> 00:06:51.450 open up this restaurant? 00:06:51.450 --> 00:06:53.920 Every year, I'm going to be making $100,000 from the 00:06:53.920 --> 00:06:55.270 restaurant itself. 00:06:55.270 --> 00:07:00.740 But I'm going to be paying $150,000 a year in interest. 00:07:00.740 --> 00:07:03.870 So you'll probably say, Sal, once again, you have just 00:07:03.870 --> 00:07:04.990 restated the obvious. 00:07:04.990 --> 00:07:06.890 No, you would not want to do this restaurant. 00:07:06.890 --> 00:07:09.890 Because every year, $50,000 will be 00:07:09.890 --> 00:07:11.710 burning out of your pocket. 00:07:11.710 --> 00:07:13.830 Now, you might think that this is obvious, but I'm going to 00:07:13.830 --> 00:07:16.750 show you many, many examples of where people are actively 00:07:16.750 --> 00:07:17.210 doing this. 00:07:17.210 --> 00:07:20.720 People who you would otherwise assume could do 00:07:20.720 --> 00:07:21.660 this type of math. 00:07:21.660 --> 00:07:23.930 And it's especially happening in the housing market. 00:07:23.930 --> 00:07:24.780 But anyway. 00:07:24.780 --> 00:07:27.090 So in this situation, you wouldn't want to invest in it. 00:07:27.090 --> 00:07:30.140 And a very simple way of thinking about this is you'd 00:07:30.140 --> 00:07:35.410 only want to invest, you only want to do a project, if your 00:07:35.410 --> 00:07:41.430 return on capital is greater than your cost of capital. 00:07:46.720 --> 00:07:49.925 This is the only time that you want to invest in a project. 00:07:52.540 --> 00:07:54.770 With that said, I'm not going to go back to 00:07:54.770 --> 00:07:55.730 what we just did. 00:07:55.730 --> 00:07:57.930 I just showed you something that we thought was obvious, 00:07:57.930 --> 00:08:02.250 but I'm going to re-ask you a question. 00:08:02.250 --> 00:08:05.430 So we had the restaurant. 00:08:05.430 --> 00:08:06.725 And we have the beauty parlor. 00:08:06.725 --> 00:08:08.790 Let's call it BP for short. 00:08:08.790 --> 00:08:11.420 They both cost $1 million. 00:08:11.420 --> 00:08:13.630 Let me write ROC. 00:08:13.630 --> 00:08:16.490 The ROC of the restaurant, we said, was 10%. 00:08:16.490 --> 00:08:19.100 And the ROC on the beauty parlor, we said, was 5%. 00:08:19.100 --> 00:08:21.090 So right now, superficially, it looks like the restaurant 00:08:21.090 --> 00:08:24.020 is just a better project. 00:08:24.020 --> 00:08:28.640 But then we said the cost of capital, so the interest rate. 00:08:28.640 --> 00:08:30.680 How much does it cost for us to get that $1 million? 00:08:30.680 --> 00:08:35.530 The interest rate to borrow money for a restaurant is 15%. 00:08:35.530 --> 00:08:38.210 And we said that this is not a good investment. 00:08:38.210 --> 00:08:40.440 Because our cost of capital is higher than 00:08:40.440 --> 00:08:41.080 our return on capital. 00:08:41.080 --> 00:08:42.929 And you could do the math and figure it out. 00:08:42.929 --> 00:08:45.070 But what if there was some kind of government program? 00:08:45.070 --> 00:08:48.390 They just felt that there weren't enough beauty parlors 00:08:48.390 --> 00:08:49.130 in the country. 00:08:49.130 --> 00:08:52.740 And they were willing to give you a really cheap loan to buy 00:08:52.740 --> 00:08:53.820 a beauty parlor. 00:08:53.820 --> 00:08:55.910 And the government program, they said, we're going to give 00:08:55.910 --> 00:09:00.560 you a low-interest loan of 2%. 00:09:00.560 --> 00:09:02.570 So my question to you is, now, which project 00:09:02.570 --> 00:09:03.790 would you rather do? 00:09:03.790 --> 00:09:06.150 Superficially, it looks like the restaurant was better. 00:09:06.150 --> 00:09:09.120 You get a 10% return, as opposed to 5%. 00:09:09.120 --> 00:09:11.660 But your cost of capital, the interest rate you would have 00:09:11.660 --> 00:09:14.000 to pay on a loan for the beauty parlor, all of a sudden 00:09:14.000 --> 00:09:15.600 looks a little bit better. 00:09:15.600 --> 00:09:17.860 In fact, this is actually a good investment. 00:09:17.860 --> 00:09:20.620 Because your cost of capital is less than 00:09:20.620 --> 00:09:21.700 your return on capital. 00:09:21.700 --> 00:09:22.830 We can even do the math. 00:09:22.830 --> 00:09:27.500 Every year the beauty parlor will generate $50,000. 00:09:27.500 --> 00:09:31.260 And you'll be paying $20,000 in interest. So you'll be 00:09:31.260 --> 00:09:34.280 netting $30,000 without having to put any money for yourself. 00:09:34.280 --> 00:09:35.480 You'll be borrowing all the money. 00:09:35.480 --> 00:09:37.510 So clearly this is a good investment. 00:09:37.510 --> 00:09:42.910 So that's it, now, for the intro on return on capital and 00:09:42.910 --> 00:09:43.790 cost of capital. 00:09:43.790 --> 00:09:45.640 And in my next presentations, I'll go into a little bit more 00:09:45.640 --> 00:09:48.810 detail and do a few more nuanced examples.

The unit circle definition of trigonometric function

https://www.youtube.com/watch?v=6Qv_bPlQS8E

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https://www.youtube.com/api/timedtext?v=6Qv_bPlQS8E&ei=fmeUZamGMIO1vdIP1OC5iAQ&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B4F6D2817CC0582EEDFD90FBBA04D25C197D5215.1473D311862A77A1A2BCF34B06EDE958E107083B&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.910 --> 00:00:03.370 We're now going to study the unit circle a little bit more, 00:00:03.370 --> 00:00:08.370 and see how it extends, I guess we could say, the traditional 00:00:08.370 --> 00:00:10.770 SOH-CAH-TOA definitions of functions. 00:00:10.770 --> 00:00:14.590 And how we can actually use it to solve for angles that the 00:00:14.590 --> 00:00:18.280 SOH-CAH-TOA definition of the trig functions actually 00:00:18.280 --> 00:00:19.560 doesn't help us with. 00:00:19.560 --> 00:00:21.830 So let's just, as a review, remember what 00:00:21.830 --> 00:00:22.760 SOH-CAH-TOA told us. 00:00:25.450 --> 00:00:26.670 I'll write it up here in this corner. 00:00:31.460 --> 00:00:32.910 I don't want to get confusing, because I don't want to 00:00:32.910 --> 00:00:33.700 write over too much. 00:00:33.700 --> 00:00:37.260 SOH, CAH, TOA. 00:00:37.260 --> 00:00:37.690 I'm sorry. 00:00:37.690 --> 00:00:39.120 I got it all jumbled up here. 00:00:39.120 --> 00:00:43.730 That told us that if we have a right angle that the sine of an 00:00:43.730 --> 00:00:47.920 angle in the right angle is equal to the opposite side 00:00:47.920 --> 00:00:49.070 over the hypotenuse. 00:00:49.070 --> 00:00:51.210 The cosine of an angle is equal to the adjacent 00:00:51.210 --> 00:00:52.600 side over hypotenuse. 00:00:52.600 --> 00:00:54.810 And the tangent side is equal to the opposite 00:00:54.810 --> 00:00:55.750 over the adjacent side. 00:00:55.750 --> 00:00:56.940 And this worked fine for us. 00:00:56.940 --> 00:00:59.820 But if you think about it, what happens when that angle 00:00:59.820 --> 00:01:00.980 approaches 90 degrees? 00:01:00.980 --> 00:01:03.790 Because you can't have two 90 degree angles in a right angle. 00:01:03.790 --> 00:01:06.890 Or what happens when that angle is greater than 90 degrees? 00:01:06.890 --> 00:01:08.900 Or what if it went negative? 00:01:08.900 --> 00:01:11.780 And that's why, if you remember from the previous video, 00:01:11.780 --> 00:01:14.560 we needed the unit circle definition. 00:01:14.560 --> 00:01:18.100 So let's review the unit circle definition. 00:01:18.100 --> 00:01:21.180 Let me erase this. 00:01:21.180 --> 00:01:21.670 Dum di-dum di-dum. 00:01:25.010 --> 00:01:27.060 I actually got this unit circle-- I think I got 00:01:27.060 --> 00:01:27.850 it from Wikipedia. 00:01:27.850 --> 00:01:30.900 But I want to give due credit to whoever I did 00:01:30.900 --> 00:01:34.610 it get for, this drawing of the unit circle. 00:01:34.610 --> 00:01:37.730 But the unit circle kind of extends that 00:01:37.730 --> 00:01:38.770 SOH-CAH-TOA definition. 00:01:38.770 --> 00:01:41.560 It tells us if we have a unit circle-- and this is a picture 00:01:41.560 --> 00:01:44.290 of a unit circle here-- a unit circle is just a circle 00:01:44.290 --> 00:01:48.100 centered at the origin, centered at the point 0, 0, 00:01:48.100 --> 00:01:49.910 and it has a radius of 1. 00:01:49.910 --> 00:01:54.850 So it intersects the x-axis at 1, 0 and negative 1, 0. 00:01:54.850 --> 00:01:59.440 It intersects the y-axis at 0, 1 and 0, negative 1. 00:01:59.440 --> 00:02:05.000 If we have a unit circle we define-- let's start with the 00:02:05.000 --> 00:02:09.170 cosine of theta-- we define the cosine of theta as let's take 00:02:09.170 --> 00:02:14.960 an angle that's between two radiuses in this unit circle. 00:02:14.960 --> 00:02:19.600 And one radius is going to be the positive x-axis 00:02:19.600 --> 00:02:21.330 between 0 and 1. 00:02:21.330 --> 00:02:24.550 So one radius is going to be this line here. 00:02:24.550 --> 00:02:30.620 And then we have the angle is the angle between you can kind 00:02:30.620 --> 00:02:33.380 of view that as the base radius and some other radius. 00:02:33.380 --> 00:02:37.760 So let's say this case right here. 00:02:37.760 --> 00:02:41.020 And this would be our angle. 00:02:41.020 --> 00:02:45.550 The unit circle definition tells us that the cosine of 00:02:45.550 --> 00:02:51.110 this angle is equal to the x-coordinate where this radius 00:02:51.110 --> 00:02:55.850 intersects the unit circle, and that the sine of this function 00:02:55.850 --> 00:02:59.055 is equal to the y-coordinate where this point intersects 00:02:59.055 --> 00:03:00.370 the unit circle. 00:03:00.370 --> 00:03:04.570 So for example, in this case-- if you can read behind 00:03:04.570 --> 00:03:07.770 my line-- this says 30 degrees equals pi/6. 00:03:07.770 --> 00:03:14.090 So this angle right here is 30 degrees, or pi/6 radians. 00:03:18.000 --> 00:03:21.720 And what this definition tells us is that the sine of 30 00:03:21.720 --> 00:03:27.240 degrees is 1/2, and that the cosine of 30 degrees is 00:03:27.240 --> 00:03:29.990 square root of 3/2. 00:03:29.990 --> 00:03:31.990 And what I want to show you is that this unit circle 00:03:31.990 --> 00:03:35.050 definition actually coincides with our SOH-CAH-TOA 00:03:35.050 --> 00:03:37.490 definition, but then it extends it. 00:03:37.490 --> 00:03:40.320 So let's see how we can get from that SOH-CAH-TOA 00:03:40.320 --> 00:03:42.840 definition to this unit circle definition, and why they're 00:03:42.840 --> 00:03:44.300 actually consistent with each other. 00:03:44.300 --> 00:03:49.410 So let me erase some of what I have written here. 00:03:49.410 --> 00:03:52.340 Let me get the eraser tool. 00:03:52.340 --> 00:03:54.995 I'm going to erase what I had. 00:03:59.270 --> 00:04:08.450 So let me go back to the pen tool, make it small again. 00:04:08.450 --> 00:04:09.310 OK. 00:04:09.310 --> 00:04:11.400 I think I'm all set. 00:04:11.400 --> 00:04:14.150 So let's go back to that theta. 00:04:14.150 --> 00:04:17.410 Let's say that this is the theta. 00:04:17.410 --> 00:04:23.100 And as we said, this angle is 30 degrees or pi/6. 00:04:23.100 --> 00:04:28.710 Let's drop a line from that point to the x-axis. 00:04:28.710 --> 00:04:30.590 And as we see this line is perpendicular, so this 00:04:30.590 --> 00:04:31.740 is a 90 degree angle. 00:04:35.730 --> 00:04:39.510 And if this is a 30 degree angle here-- this is 30. 00:04:39.510 --> 00:04:39.750 Right? 00:04:39.750 --> 00:04:42.450 Theta equals 30. 00:04:42.450 --> 00:04:43.730 This is 30, this is 90. 00:04:43.730 --> 00:04:44.380 What is this angle? 00:04:44.380 --> 00:04:45.955 Well, this is a 60 degree angle, because they 00:04:45.955 --> 00:04:47.100 add up to 180. 00:04:47.100 --> 00:04:49.440 So this is a 30-60-90 triangle. 00:04:49.440 --> 00:04:50.410 Interesting. 00:04:50.410 --> 00:04:53.320 And what do you remember about 30-60-90 triangles? 00:04:53.320 --> 00:04:57.570 Well, the side opposite the 30 degree side is 1/2 the 00:04:57.570 --> 00:04:59.530 length of the hypotenuse. 00:04:59.530 --> 00:05:00.390 I hope you remember that. 00:05:00.390 --> 00:05:02.790 I don't want to confuse you too much. 00:05:02.790 --> 00:05:06.160 So this is the side opposite the 30 degree side. 00:05:06.160 --> 00:05:07.280 Right? 00:05:07.280 --> 00:05:08.390 And what's the hypotenuse? 00:05:08.390 --> 00:05:09.490 This is the hypotenuse. 00:05:09.490 --> 00:05:11.200 And what's the length of this hypotenuse? 00:05:11.200 --> 00:05:13.360 Well it's 1, because this is a unit circle and this is the 00:05:13.360 --> 00:05:14.990 radius of the unit circle. 00:05:14.990 --> 00:05:18.870 So the length of this hypotenuse is 1, and so the 00:05:18.870 --> 00:05:21.300 length of this side, which is opposite the 30 degree 00:05:21.300 --> 00:05:23.490 angle, is going to be 1/2. 00:05:23.490 --> 00:05:23.740 Right? 00:05:23.740 --> 00:05:28.560 And I'm just using the 30-60-90 triangles that we've 00:05:28.560 --> 00:05:30.450 done previous videos on. 00:05:30.450 --> 00:05:34.170 And what's the side opposite the 60 degree side? 00:05:34.170 --> 00:05:36.070 Well once again it's square root of 3/2 00:05:36.070 --> 00:05:38.230 times the hypotenuse. 00:05:38.230 --> 00:05:39.860 And so it's square root of 3/2. 00:05:42.670 --> 00:05:43.180 Right? 00:05:43.180 --> 00:05:46.260 So we can figure out that this side is square root of 3/2, 00:05:46.260 --> 00:05:49.750 and that this side is 1/2. 00:05:49.750 --> 00:05:51.760 So a couple of things we can figure out. 00:05:51.760 --> 00:05:53.650 Just by looking at this we can immediately say, well what's 00:05:53.650 --> 00:05:56.240 the coordinate of this point? 00:05:56.240 --> 00:05:58.440 Well it's x-coordinate is right here. 00:05:58.440 --> 00:05:58.800 Right? 00:05:58.800 --> 00:06:01.510 It's x-coordinate would be square root of 3/2. 00:06:01.510 --> 00:06:02.810 That's this right here. 00:06:02.810 --> 00:06:04.080 This distance. 00:06:04.080 --> 00:06:07.970 And it's y-coordinate would be the length of this side of 00:06:07.970 --> 00:06:09.260 the right triangle, or 1/2. 00:06:09.260 --> 00:06:10.410 And there we have it right here. 00:06:10.410 --> 00:06:11.850 It was already written for us. 00:06:11.850 --> 00:06:13.850 The x-coordinate is the square root of 3/2 and 00:06:13.850 --> 00:06:16.170 the y-coordinate is 1/2. 00:06:16.170 --> 00:06:20.160 And now what I want to show you is why this x-coordinate can be 00:06:20.160 --> 00:06:23.820 taken as the cosine of theta, and why this y-coordinate can 00:06:23.820 --> 00:06:25.340 be taken as the sine of theta. 00:06:25.340 --> 00:06:27.170 Well what does SOH-CAH-TOA tell us? 00:06:27.170 --> 00:06:29.410 Well let's start with the cosine. 00:06:29.410 --> 00:06:30.940 So SOH, CAH, TOA. 00:06:30.940 --> 00:06:31.400 So CAH. 00:06:34.920 --> 00:06:37.795 Cosine is adjacent over hypotenuse, right? 00:06:45.120 --> 00:06:48.790 Well, in this triangle I just drew, what is the adjacent 00:06:48.790 --> 00:06:50.490 side to this angle? 00:06:50.490 --> 00:06:50.613 Right? 00:06:50.613 --> 00:06:52.120 Because we're trying to figure out the cosine of this 00:06:52.120 --> 00:06:54.170 angle, this 30 degrees. 00:06:54.170 --> 00:06:57.220 Well the adjacent side to this angle is, of course, this side. 00:06:57.220 --> 00:06:58.170 Right? 00:06:58.170 --> 00:07:00.260 So adjacent is square root of 3/2. 00:07:00.260 --> 00:07:02.060 We figured that out just now. 00:07:02.060 --> 00:07:03.320 And what's the hypotenuse? 00:07:03.320 --> 00:07:08.050 Well the hypotenuse is this side, which has length 1 00:07:08.050 --> 00:07:10.830 because it was the unit circle and that's the radius of it. 00:07:10.830 --> 00:07:13.540 So the cosine of this angle using the SOH-CAH-TOA 00:07:13.540 --> 00:07:16.630 definition is square root of 3-- the adjacent side-- 00:07:16.630 --> 00:07:18.230 over the hypotenuse 1. 00:07:18.230 --> 00:07:20.935 So square root of 3/2 over 1, which is the square root 00:07:20.935 --> 00:07:25.070 of 3/2, which was the same thing as the x-coordinate. 00:07:25.070 --> 00:07:27.060 Similarly, we can look at SOH. 00:07:27.060 --> 00:07:30.660 Sine equals opposite over hypotenuse. 00:07:35.920 --> 00:07:37.010 Well what's the opposite side? 00:07:37.010 --> 00:07:38.410 It's 1/2. 00:07:38.410 --> 00:07:40.470 And the hypotenuse is 1 here. 00:07:40.470 --> 00:07:43.450 So the sine is just 1/2 over 1. 00:07:43.450 --> 00:07:44.940 And so we have it here. 00:07:44.940 --> 00:07:47.220 So that's why the unit circle definition isn't kind of a 00:07:47.220 --> 00:07:50.180 replacing definition for the SOH-CAH-TOA definition, it's 00:07:50.180 --> 00:07:52.960 really just an extension that allows us-- I mean, for 30 00:07:52.960 --> 00:07:55.940 degrees SOH-CAH-TOA worked, for 45 degrees SOH-CAH-TOA worked, 00:07:55.940 --> 00:07:57.750 for 60 degrees it would work. 00:07:57.750 --> 00:07:59.490 But once you get to 90 it becomes a little bit more 00:07:59.490 --> 00:08:01.980 difficult if you use traditional SOH-CAH-TOA and you 00:08:01.980 --> 00:08:04.310 try to draw a right triangle that has two 90 degree angles 00:08:04.310 --> 00:08:05.410 in it-- because you couldn't. 00:08:05.410 --> 00:08:08.480 And especially once you get to angles that are larger than 90 00:08:08.480 --> 00:08:11.190 or angles that actually could even go negative. 00:08:11.190 --> 00:08:14.650 It's not drawn here in the unit circle, but 330 degrees is the 00:08:14.650 --> 00:08:17.140 same thing as negative 30 degrees, because you could go 00:08:17.140 --> 00:08:18.060 either way in the circle. 00:08:18.060 --> 00:08:19.340 And you could keep going around the circle. 00:08:19.340 --> 00:08:22.480 You could figure out the sine or the cosine of, you know, 1 00:08:22.480 --> 00:08:25.590 million degrees if you just keep going around the circle. 00:08:25.590 --> 00:08:28.460 So hopefully this gives you a sense of the unit circle 00:08:28.460 --> 00:08:31.250 definition of the sine and cosine functions. 00:08:31.250 --> 00:08:33.240 And, of course, the tangent function is always just the 00:08:33.240 --> 00:08:37.310 sine over the cosine, or the y over the x. 00:08:37.310 --> 00:08:39.250 And you could use the unit circle definition 00:08:39.250 --> 00:08:40.440 for that as well. 00:08:40.440 --> 00:08:44.170 And I'll leave it for you as an exercise to try to derive all 00:08:44.170 --> 00:08:48.490 of these other values using this unit circle, and using 00:08:48.490 --> 00:08:52.930 what you already know about 30-60-90 triangles and what you 00:08:52.930 --> 00:08:56.650 already know about 45-45-90 triangles, or what you 00:08:56.650 --> 00:08:57.670 already know about the Pythagorean theorem. 00:08:57.670 --> 00:09:01.320 And you should be able to figure out all of these values 00:09:01.320 --> 00:09:03.570 going around the unit circle. 00:09:03.570 --> 00:09:06.270 Hopefully that was helpful. 00:09:06.270 --> 00:09:07.772 See you soon.

Determining the equation of a trigonometric function

https://www.youtube.com/watch?v=yvW5l9W1hgE

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https://www.youtube.com/api/timedtext?v=yvW5l9W1hgE&ei=fGeUZcWwNM-5mLAPi7OloA8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249836&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1C2053C2A4B77371101943ED0C5707FC06B4897C.C5B982F1EEEC3AAB61A43554A6D79C374004DE8B&key=yt8&lang=en&name=English&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:00.950 --> 00:00:01.800 Hello. 00:00:01.800 --> 00:00:04.820 I'm now going to use the actual Khan Academy website to 00:00:04.820 --> 00:00:05.680 do some more problems. 00:00:05.680 --> 00:00:07.490 And this time we're going to go the other way around. 00:00:07.490 --> 00:00:10.860 We're going to look at the graph of a trigonometric 00:00:10.860 --> 00:00:13.090 function and we're going to try to figure out the equation. 00:00:13.090 --> 00:00:15.950 So let's start with the problem we have in front of us. 00:00:15.950 --> 00:00:18.990 So we have a-- well we don't know if it's a sine curve or a 00:00:18.990 --> 00:00:21.680 cosine curve, but I guess it's fair to say that it's 00:00:21.680 --> 00:00:23.390 one of the two. 00:00:23.390 --> 00:00:25.250 Actually, let's answer that question first. 00:00:25.250 --> 00:00:26.020 What do you think this is? 00:00:26.020 --> 00:00:31.490 Do you think that this is a sine curve or a cosine curve? 00:00:31.490 --> 00:00:32.290 What's the difference? 00:00:32.290 --> 00:00:34.560 Or what's the easiest difference to differentiate 00:00:34.560 --> 00:00:35.840 between the two? 00:00:35.840 --> 00:00:37.870 Well, what was sine of 0? 00:00:37.870 --> 00:00:41.110 Let me get a little chalkboard in here. 00:00:41.110 --> 00:00:42.140 Let me bring it right here. 00:00:45.040 --> 00:00:48.845 Let me make sure my pen tool is correct. 00:00:48.845 --> 00:00:51.640 All right. 00:00:51.640 --> 00:00:55.540 What is sine of 0? 00:00:55.540 --> 00:00:58.420 And it could be 0 degrees or 0 radians. 00:00:58.420 --> 00:00:59.930 Well if you remember from a couple of the other modules, 00:00:59.930 --> 00:01:03.570 or even if you want to use a calculator, sine of 0 is 0. 00:01:03.570 --> 00:01:06.150 And what is cosine of 0? 00:01:06.150 --> 00:01:10.030 Well cosine of 0, if you remember from the last modules 00:01:10.030 --> 00:01:11.840 or you want to use a calculator-- although you 00:01:11.840 --> 00:01:14.010 shouldn't have to use a calculator for cosine of 0-- 00:01:14.010 --> 00:01:16.980 you might remember that it was 1. 00:01:16.980 --> 00:01:23.820 So this graph that we have here, it's definitely when x is 00:01:23.820 --> 00:01:28.530 0, when the x-axis is 0, the function definitely isn't 0. 00:01:28.530 --> 00:01:31.100 In fact, in this case it's 1 1/2. 00:01:31.100 --> 00:01:34.130 So this tells us that this isn't a sine curve-- although 00:01:34.130 --> 00:01:36.710 later with shifts we'll learn that it could be a shifted sine 00:01:36.710 --> 00:01:39.120 curve-- that this isn't a sine curve, that this is 00:01:39.120 --> 00:01:40.120 a cosine curve. 00:01:40.120 --> 00:01:43.080 And then you might ask, well Sal if this is a cosine curve 00:01:43.080 --> 00:01:48.230 why is f of 0 equal to 1 1/2, or 3/2, instead of 1? 00:01:48.230 --> 00:01:52.010 Because I just said here that cosine of 0 is 1. 00:01:52.010 --> 00:01:55.240 Well that's because there must be some type of a coefficient 00:01:55.240 --> 00:01:59.610 here, let's call it A, that is changing the amplitude 00:01:59.610 --> 00:02:01.170 of this cosine curve. 00:02:01.170 --> 00:02:04.160 And if you remember from the last module, what 00:02:04.160 --> 00:02:05.380 do you think this A is? 00:02:05.380 --> 00:02:08.600 Well that A is just literally the amplitude of the curve. 00:02:08.600 --> 00:02:10.710 And what's the amplitude of this curve? 00:02:10.710 --> 00:02:14.950 Well, the amplitude of this curve, if we just see how much 00:02:14.950 --> 00:02:17.710 it moves above and below the x-axis, well it's that 3/2, 00:02:17.710 --> 00:02:19.610 or that 1 1/2, we've been talking about. 00:02:19.610 --> 00:02:23.030 See it moves up 3/2, and it moves down 3/2. 00:02:23.030 --> 00:02:24.530 So let me just write that. 00:02:24.530 --> 00:02:35.410 So we know that this is 3/2 cosine of, well, something x. 00:02:35.410 --> 00:02:36.490 Right? 00:02:36.490 --> 00:02:41.360 We know f of x is equal to 3/2 cosine of something x. 00:02:41.360 --> 00:02:45.140 And we could use the formula that we learned in the previous 00:02:45.140 --> 00:02:49.510 video, that it equals 2pi over the period x. 00:02:49.510 --> 00:02:51.430 So now we just have to look at the graph and try to figure 00:02:51.430 --> 00:02:54.720 out what the period of the graph is. 00:02:54.720 --> 00:02:57.420 Well how many radians does it take for the graph to 00:02:57.420 --> 00:02:59.110 start repeating again? 00:02:59.110 --> 00:03:01.450 Let me click on the hint button, and maybe 00:03:01.450 --> 00:03:03.460 this'll help us. 00:03:03.460 --> 00:03:05.900 When I click hint-- there; drew the period. 00:03:05.900 --> 00:03:07.180 And you could have figured it out on your own. 00:03:07.180 --> 00:03:11.990 If you just go from any point and then follow the curve back 00:03:11.990 --> 00:03:15.540 to the same point again, you'll see how long it's period is. 00:03:15.540 --> 00:03:17.730 And the hint on the Khan icon actually told us 00:03:17.730 --> 00:03:18.430 that the period is 4pi. 00:03:18.430 --> 00:03:20.475 And you could just start from any point to any other point. 00:03:20.475 --> 00:03:23.860 You could have gone from this point and then gone down, 00:03:23.860 --> 00:03:25.350 gone back up, come down. 00:03:25.350 --> 00:03:28.680 And then you would have seen that this distance is also 4pi. 00:03:28.680 --> 00:03:30.200 So we know that the period is 4pi. 00:03:30.200 --> 00:03:32.060 And then I could click hint again and it'll tell us stuff 00:03:32.060 --> 00:03:33.810 that we already figured out. 00:03:33.810 --> 00:03:36.700 The amplitude, we already figured out, was 1 1/2. 00:03:36.700 --> 00:03:38.525 But let's just use a period, because we already knew 00:03:38.525 --> 00:03:39.750 what the amplitude is. 00:03:39.750 --> 00:03:42.370 So the period here we already figured out was 4pi. 00:03:42.370 --> 00:03:43.670 So let's just write that in our equation. 00:03:43.670 --> 00:03:56.400 So f of x is equal to 3/2 cosine of 2pi divided by 00:03:56.400 --> 00:04:01.810 the period-- the period in this case is 4pi-- x. 00:04:01.810 --> 00:04:11.770 That equals 3/2 cosine of 1/2 x. 00:04:11.770 --> 00:04:14.060 Now if you ever forget this formula, which frankly 00:04:14.060 --> 00:04:14.990 I always do forget it. 00:04:14.990 --> 00:04:16.210 I've actually never memorized it. 00:04:16.210 --> 00:04:18.920 I just try to think about what the period would be. 00:04:18.920 --> 00:04:22.680 The way I think about it is the coefficient on the x-term, 00:04:22.680 --> 00:04:28.360 that's a measure of how many cycles does the graph 00:04:28.360 --> 00:04:31.300 do within 2pi radians. 00:04:31.300 --> 00:04:34.150 Let me see if I can explain that within the context 00:04:34.150 --> 00:04:35.240 of this problem. 00:04:35.240 --> 00:04:40.720 So this problem, if we start at 0 and then 2pi is here, how 00:04:40.720 --> 00:04:43.330 many cycles do we complete by the time we get to 2pi? 00:04:43.330 --> 00:04:46.950 We start here, we go back here, and then we're at 2pi. 00:04:46.950 --> 00:04:48.980 Well we only got halfway done through a cycle. 00:04:48.980 --> 00:04:50.420 So that's the coefficient on the x-term. 00:04:50.420 --> 00:04:52.230 And that's how I remember it. 00:04:52.230 --> 00:04:56.980 So I could just say, well that's 3/2 cosine of 00:04:56.980 --> 00:04:59.430 how many cycles do I complete in 2pi radians? 00:04:59.430 --> 00:05:01.810 Well I only complete half a cycle. 00:05:01.810 --> 00:05:04.660 3/2 times the cosine of 1/2 x. 00:05:04.660 --> 00:05:07.580 So that's our f of x. 00:05:07.580 --> 00:05:08.550 Let's do another problem. 00:05:18.490 --> 00:05:19.130 All right. 00:05:19.130 --> 00:05:19.490 OK. 00:05:19.490 --> 00:05:20.630 This one's interesting. 00:05:20.630 --> 00:05:23.520 So the first thing, just by inspection we can figure out 00:05:23.520 --> 00:05:24.590 what this amplitude is. 00:05:24.590 --> 00:05:25.860 This is pretty easy. 00:05:25.860 --> 00:05:26.130 Right? 00:05:26.130 --> 00:05:30.180 How much does it move above and below the x-axis? 00:05:30.180 --> 00:05:33.860 Well it only goes 1, so we know that the coefficient, or the 00:05:33.860 --> 00:05:36.020 multiplier times the sine or the cosine function-- 00:05:36.020 --> 00:05:37.240 whichever this is-- is 1. 00:05:37.240 --> 00:05:38.680 So let's write that down. 00:05:38.680 --> 00:05:48.300 Let's write down that the amplitude is equal to 1. 00:05:48.300 --> 00:05:49.580 Now let's try to figure out if this is a sine 00:05:49.580 --> 00:05:50.800 or a cosine function. 00:05:50.800 --> 00:05:59.290 In the last problem we said sine of 0 is 0 00:05:59.290 --> 00:06:01.880 and cosine of 0 is 1. 00:06:01.880 --> 00:06:04.750 Well f of 0 of this function, whichever it is, is 0. 00:06:04.750 --> 00:06:08.830 So we know this is a non-shifted sine function. 00:06:08.830 --> 00:06:09.870 So there. 00:06:09.870 --> 00:06:11.080 We have another piece of information. 00:06:11.080 --> 00:06:12.070 It's a sine function. 00:06:15.120 --> 00:06:16.680 So the last thing we have to figure out, we can either 00:06:16.680 --> 00:06:20.110 figure out the period or we could use the method that I 00:06:20.110 --> 00:06:23.500 just showed you where we say, well how many times does it 00:06:23.500 --> 00:06:26.800 cycle within 2pi radians? 00:06:26.800 --> 00:06:27.610 So let's do it that way. 00:06:27.610 --> 00:06:28.910 And then we immediately know the coefficient. 00:06:28.910 --> 00:06:30.270 Let's see. 00:06:30.270 --> 00:06:31.770 Well actually, this graph board doesn't even get 00:06:31.770 --> 00:06:32.960 all the way to 2pi. 00:06:32.960 --> 00:06:33.480 But let's see. 00:06:33.480 --> 00:06:38.890 It goes one cycle, two cycles. 00:06:38.890 --> 00:06:41.770 And I did two cycles in only pi radians, right? 00:06:41.770 --> 00:06:43.840 Because I'm only at x equals pi here. 00:06:43.840 --> 00:06:46.830 So if I did two cycles in pi radians, then we must be able 00:06:46.830 --> 00:06:49.820 to do four cycles in 2pi radians. 00:06:49.820 --> 00:06:51.100 Or we could actually start here. 00:06:51.100 --> 00:06:51.750 Actually, this is better. 00:06:51.750 --> 00:06:52.020 Right? 00:06:52.020 --> 00:06:53.230 Going from negative pi to pi. 00:06:53.230 --> 00:06:54.990 That's 2pi radians. 00:06:54.990 --> 00:06:58.560 So we finish one cycle, two cycles, three 00:06:58.560 --> 00:07:00.080 cycles, four cycles. 00:07:00.080 --> 00:07:02.670 So then we know what the coefficient on the x-term is. 00:07:02.670 --> 00:07:04.960 So we know that it is sine of 4x. 00:07:04.960 --> 00:07:10.340 So the answer here is f of x is equal to 1-- because that's the 00:07:10.340 --> 00:07:16.400 amplitude-- the amplitude times sine of 4x. 00:07:16.400 --> 00:07:18.620 I think we have time for one more. 00:07:18.620 --> 00:07:22.410 And I want you to-- don't just mechanically do whatever 00:07:22.410 --> 00:07:22.840 I'm telling you. 00:07:22.840 --> 00:07:25.420 I want you to think about why counting the number of cycles 00:07:25.420 --> 00:07:27.150 within 2pi radians, why that makes sense to you. 00:07:27.150 --> 00:07:29.160 Then think back to the unit circle. 00:07:29.160 --> 00:07:33.790 Or think back why that formula, the 2pi divided by the period, 00:07:33.790 --> 00:07:34.610 is also the coefficient. 00:07:34.610 --> 00:07:36.100 Think about why that makes sense. 00:07:36.100 --> 00:07:38.990 If you realize why it makes sense you'll never have to 00:07:38.990 --> 00:07:41.310 memorize it, and then 20 years later when you're doing it like 00:07:41.310 --> 00:07:44.520 I'm doing it right now you won't be confused. 00:07:44.520 --> 00:07:46.460 You'll be able to re-derive the formulas. 00:07:46.460 --> 00:07:47.320 Let's do one more. 00:07:53.840 --> 00:07:55.080 All right. 00:07:55.080 --> 00:07:56.890 So what's the amplitude here? 00:07:56.890 --> 00:07:57.430 Well, let's see. 00:07:57.430 --> 00:07:59.510 The amplitude is 1/2. 00:07:59.510 --> 00:08:03.462 So let me delete the old stuff that I was writing before. 00:08:03.462 --> 00:08:04.905 For some reason it's not deleting. 00:08:07.900 --> 00:08:09.220 OK. 00:08:09.220 --> 00:08:10.200 Hope I don't confuse you. 00:08:10.200 --> 00:08:13.980 So the amplitude, let's just call it Amplitude, 00:08:13.980 --> 00:08:14.730 is equal to 1/2. 00:08:18.140 --> 00:08:24.040 And how many cycles does it complete within 2pi radians? 00:08:24.040 --> 00:08:26.560 Let's see. 00:08:26.560 --> 00:08:30.180 If we start here it looks like it completes only half a cycle. 00:08:30.180 --> 00:08:30.390 Right? 00:08:30.390 --> 00:08:35.770 Because it takes 4pi radians to complete the entire cycle. 00:08:35.770 --> 00:08:38.820 So it only completes half a cycle. 00:08:38.820 --> 00:08:41.450 So we could think of it either two ways. 00:08:41.450 --> 00:08:45.730 We could say that the period is equal to 4pi, because that's 00:08:45.730 --> 00:08:48.720 how long it takes to complete one cycle, or we could say 00:08:48.720 --> 00:08:54.710 it can only complete half a cycle within 2pi radians. 00:08:54.710 --> 00:08:56.130 The last thing we have to figure out: Is it a sine 00:08:56.130 --> 00:08:57.340 or a cosine function? 00:08:57.340 --> 00:08:59.640 Well, f of 0 is 0. 00:08:59.640 --> 00:09:00.020 Right? 00:09:00.020 --> 00:09:02.250 So it's a non-shifted sine function. 00:09:02.250 --> 00:09:03.500 So then we're done. 00:09:03.500 --> 00:09:09.160 We have f of x is equal to 1/2-- figured out it's a sine 00:09:09.160 --> 00:09:13.950 function-- sine of what? 00:09:13.950 --> 00:09:17.280 How many cycles did it complete in 2pi radians? 00:09:17.280 --> 00:09:18.950 It only completed half a cycle. 00:09:18.950 --> 00:09:21.090 Let me not cover the problem. 00:09:21.090 --> 00:09:23.360 It only completes half a cycle. 00:09:23.360 --> 00:09:25.530 So it's 1/2 sine of 1/2 x. 00:09:25.530 --> 00:09:29.300 Or we could use the formula f of x equals goes 1/2 sine of 00:09:29.300 --> 00:09:31.190 2pi divided by the period x. 00:09:31.190 --> 00:09:31.430 Right? 00:09:31.430 --> 00:09:37.840 Because 2pi divided by the period is equal to 2pi over 00:09:37.840 --> 00:09:42.100 4pi, which also equals 1/2. 00:09:42.100 --> 00:09:44.210 I think that'll give you a sense of how to do 00:09:44.210 --> 00:09:45.850 these problems now. 00:09:45.850 --> 00:09:47.740 And I encourage you to practice them on the 00:09:47.740 --> 00:09:50.140 Khan Academy website. 00:09:50.140 --> 00:09:51.630 Have fun.

More trig graphs

https://www.youtube.com/watch?v=NIG3l8oWKYE

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https://www.youtube.com/api/timedtext?v=NIG3l8oWKYE&ei=f2eUZZ6YFam3mLAP5--46AE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4A4D1AED0E877853D83DAA0F7DB5EC7B9A997022.0E0E3C34A093DB4203408BAB012DB82A5A6C3346&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.280 --> 00:00:03.830 So now we have this graph of this-- what was 00:00:03.830 --> 00:00:04.865 clearly a trig function. 00:00:04.865 --> 00:00:09.300 And our task is to figure out what the function is. 00:00:09.300 --> 00:00:11.040 So let's look at this. 00:00:11.040 --> 00:00:13.210 The first thing I do when I look at something like this, 00:00:13.210 --> 00:00:16.460 I want to figure out its period and its amplitude. 00:00:16.460 --> 00:00:17.380 So what's its amplitude? 00:00:17.380 --> 00:00:18.410 That's always an easy one. 00:00:18.410 --> 00:00:19.230 So the amplitude. 00:00:25.620 --> 00:00:28.240 Well, that's just how much does it move up and down above 00:00:28.240 --> 00:00:29.390 and below the x-axis? 00:00:29.390 --> 00:00:33.760 Well, the amplitude here is how much it moves up the x-axis. 00:00:33.760 --> 00:00:36.750 Well, it moves up 1/2 above and below the x-axis. 00:00:36.750 --> 00:00:38.400 So the amplitude is 1/2. 00:00:38.400 --> 00:00:42.980 And keep in mind, the amplitude is not this whole distance. 00:00:42.980 --> 00:00:44.830 It's not this. 00:00:44.830 --> 00:00:50.370 It's just how much it moves above or below the x-axis. 00:00:50.370 --> 00:00:52.860 So in this case, the amplitude is 1/2. 00:00:52.860 --> 00:00:56.040 And then we have to figure out what the period is. 00:00:56.040 --> 00:01:00.510 The period is, how long does it take-- how many radians does 00:01:00.510 --> 00:01:04.040 it go through for it to go through 1 complete cycle? 00:01:04.040 --> 00:01:08.890 Well, if we start here, and we were to follow the graph, it 00:01:08.890 --> 00:01:11.780 doesn't take until right here until we complete 00:01:11.780 --> 00:01:12.430 1 cycle, right? 00:01:12.430 --> 00:01:13.900 Because here, we're still going down, and now 00:01:13.900 --> 00:01:14.470 we're going below. 00:01:14.470 --> 00:01:15.450 So we're not repeating yet. 00:01:15.450 --> 00:01:17.370 And here's where we start repeating again. 00:01:17.370 --> 00:01:19.240 And then we start repeating again here. 00:01:19.240 --> 00:01:23.130 So every pi radians, we start the cycle over again. 00:01:23.130 --> 00:01:25.010 It happens the same if you go backwards into the 00:01:25.010 --> 00:01:26.540 negative radians. 00:01:26.540 --> 00:01:28.685 So the period is pi, right? 00:01:28.685 --> 00:01:30.420 The period is pi. 00:01:30.420 --> 00:01:31.650 And you could start from any point. 00:01:31.650 --> 00:01:33.760 You could start from this point. 00:01:33.760 --> 00:01:36.350 And if you go, follow the graph, and then come back to 00:01:36.350 --> 00:01:40.070 the same point again, we see once again that the 00:01:40.070 --> 00:01:43.540 period is pi radians. 00:01:43.540 --> 00:01:45.215 Now we have to figure out if this is a sine or 00:01:45.215 --> 00:01:46.335 a cosine function. 00:01:46.335 --> 00:01:50.680 And for now, we'll not think about shifting. 00:01:50.680 --> 00:01:53.930 So let's think about what happens when-- you know, 00:01:53.930 --> 00:01:55.900 we want to know what this function is. 00:01:55.900 --> 00:02:01.010 f of x is equal to question mark. 00:02:01.010 --> 00:02:04.116 Well, we see that f of 0 is 0. 00:02:04.116 --> 00:02:08.800 f of 0 is equal to 0. 00:02:08.800 --> 00:02:09.880 What does that tell us? 00:02:09.880 --> 00:02:12.830 Is this a sine or a cosine function? 00:02:12.830 --> 00:02:15.510 Well, what's cosine of 0? 00:02:15.510 --> 00:02:18.620 Cosine of 0 is 1. 00:02:18.620 --> 00:02:19.630 And what's sine of 0? 00:02:19.630 --> 00:02:21.390 Well, sine of 0 is 0. 00:02:21.390 --> 00:02:23.290 And this function is 0. 00:02:23.290 --> 00:02:26.100 So we know that this is a sine function. 00:02:26.100 --> 00:02:31.050 So we know the formula is going to take the form f of x, it's 00:02:31.050 --> 00:02:38.110 going to equal the amplitude times sine of 2 pi 00:02:38.110 --> 00:02:40.900 over the period x. 00:02:40.900 --> 00:02:43.440 And if we just substitute these numbers we just figured out, we 00:02:43.440 --> 00:02:56.155 know that f of x is equal to 1/2 sine of 2 pi over pi x. 00:02:56.155 --> 00:03:03.650 The pi's cancel out and you get f of x is equal 00:03:03.650 --> 00:03:10.270 to 1/2 sine of 2x. 00:03:15.590 --> 00:03:17.020 Let's define another function. 00:03:17.020 --> 00:03:24.980 Let's define g of x is equal to 1/2 cosine of 2x. 00:03:24.980 --> 00:03:26.370 What would have this looked like? 00:03:26.370 --> 00:03:28.900 Or what would have-- yeah. 00:03:28.900 --> 00:03:30.083 The grammar's a little difficult. 00:03:32.650 --> 00:03:34.660 I picked the wrong color, because f of x is 00:03:34.660 --> 00:03:35.170 actually the pink one. 00:03:35.170 --> 00:03:37.080 This is the one we have now. 00:03:37.080 --> 00:03:38.470 So actually, let me circle that. 00:03:38.470 --> 00:03:40.100 This is f of x. 00:03:40.100 --> 00:03:42.770 f of x is this one right here. 00:03:42.770 --> 00:03:46.500 And now, g of x, I'm going to do--. 00:03:46.500 --> 00:03:52.910 So when x is 0, what is g of 0? 00:03:52.910 --> 00:03:54.800 Let's put 0 in here. 00:03:54.800 --> 00:03:56.230 So this whole term will become 0. 00:03:56.230 --> 00:03:57.360 What's cosine of 0? 00:03:57.360 --> 00:03:59.000 It's 1. 00:03:59.000 --> 00:04:00.000 And 1 times 1/2. 00:04:00.000 --> 00:04:02.300 So g of 0 is 1/2. 00:04:02.300 --> 00:04:08.180 So we would start here, and then we would have-- just 00:04:08.180 --> 00:04:11.220 like the sine function-- we would have a period of pi. 00:04:11.220 --> 00:04:13.500 Because it has the same coefficient here. 00:04:13.500 --> 00:04:15.070 So this'll just look like this. 00:04:20.150 --> 00:04:21.185 I think you get the point. 00:04:24.710 --> 00:04:27.150 It's just like the sine function was just shifted 00:04:27.150 --> 00:04:28.000 to the left of it. 00:04:30.550 --> 00:04:32.920 Well, I'm getting confused on this-- ignore this. 00:04:32.920 --> 00:04:35.120 But if you look at this side, the important thing to realize 00:04:35.120 --> 00:04:41.610 is that it intersects the y-axis at not 1, but 1/2. 00:04:41.610 --> 00:04:43.950 And the reason why it doesn't intersect it at 1, even 00:04:43.950 --> 00:04:47.120 though cosine of 0 is 1, is because we have this 1/2 00:04:47.120 --> 00:04:48.420 coefficient right here. 00:04:48.420 --> 00:04:49.610 I guess you can't call that a coefficient. 00:04:49.610 --> 00:04:55.430 I guess it's a 1/2 times the cosine function. 00:04:55.430 --> 00:04:58.760 Hopefully that gives you a little bit more of a sense of, 00:04:58.760 --> 00:05:00.980 if you just looked at a graph, being able to figure 00:05:00.980 --> 00:05:02.850 out its equation. 00:05:02.850 --> 00:05:05.590 And I'll actually do one more video where we'll actually use 00:05:05.590 --> 00:05:08.900 the Khan Academy trig graphing module to figure 00:05:08.900 --> 00:05:11.360 out a couple more. 00:05:11.360 --> 00:05:12.900 See you soon.

Graphing trig functions

https://www.youtube.com/watch?v=vHYI93UV5Kg

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en

WEBVTT Kind: captions Language: en 00:00:01.110 --> 00:00:04.020 In this presentation we're going to learn how to graph 00:00:04.020 --> 00:00:06.160 trig functions without having to kind of 00:00:06.160 --> 00:00:07.970 graph point by point. 00:00:07.970 --> 00:00:11.410 And hopefully after this presentation you can also look 00:00:11.410 --> 00:00:15.120 at a trig function and be able to figure out the actual 00:00:15.120 --> 00:00:17.740 analytic definition of the function as well. 00:00:17.740 --> 00:00:18.440 So let's start. 00:00:18.440 --> 00:00:19.720 Let's say f of x. 00:00:19.720 --> 00:00:22.550 Let me make sure I'm using all the right tools. 00:00:22.550 --> 00:00:45.200 So let's say that f of x is equal to 2 sine of 1/2 x. 00:00:45.200 --> 00:00:47.890 So when we look at this, a couple interesting things here. 00:00:47.890 --> 00:00:49.980 How is this different than just the regular sine function? 00:00:49.980 --> 00:00:55.170 Well, here we're multiplying the whole function by 2, 00:00:55.170 --> 00:00:57.780 and also the coefficient on the x-term is 1/2. 00:00:57.780 --> 00:01:00.650 And if you've seen some of the videos I've made, you'll know 00:01:00.650 --> 00:01:05.010 that this term affects the amplitude and this term affects 00:01:05.010 --> 00:01:08.070 the period, or the inverse of the period, which 00:01:08.070 --> 00:01:08.730 is the frequency. 00:01:08.730 --> 00:01:09.130 Either way. 00:01:09.130 --> 00:01:11.440 It depends whether you're talking about one or the 00:01:11.440 --> 00:01:12.750 inverse of the other one. 00:01:12.750 --> 00:01:14.710 So let's start with the amplitude. 00:01:14.710 --> 00:01:17.670 This 2 tells us that the amplitude of this function 00:01:17.670 --> 00:01:19.890 is going to be 2. 00:01:19.890 --> 00:01:22.320 Because if it was just a 1 there the amplitude would be 1. 00:01:22.320 --> 00:01:24.000 So it's going to be 2 times that. 00:01:24.000 --> 00:01:34.500 So let's draw a little dotted line up here at y equals 2. 00:01:34.500 --> 00:01:37.230 And then another dotted line at y equals negative 2. 00:01:45.780 --> 00:01:46.740 So we know this is the amplitude. 00:01:46.740 --> 00:01:49.650 We know that the function is going to somehow oscillate 00:01:49.650 --> 00:01:51.710 between these two points, but we have to figure out how fast 00:01:51.710 --> 00:01:53.570 is it going to oscillate between the two points, 00:01:53.570 --> 00:01:55.800 or what's its period. 00:01:55.800 --> 00:01:57.330 And I'll give you a little formula here. 00:02:00.060 --> 00:02:06.390 The function is equal to the amplitude times, let's say, 00:02:06.390 --> 00:02:08.890 sine, but it would also work with cosine. 00:02:08.890 --> 00:02:16.030 The amplitude of the function times sine of 2pi divided 00:02:16.030 --> 00:02:20.800 by the period of the function, times x. 00:02:20.800 --> 00:02:22.850 This right here is a "p." 00:02:22.850 --> 00:02:26.340 So it might not be completely obvious where this comes from. 00:02:26.340 --> 00:02:28.260 But what I want you to do is maybe after this video or 00:02:28.260 --> 00:02:30.950 maybe in future videos we'll experiment when we see what 00:02:30.950 --> 00:02:33.770 happens when we change this coefficient on the x-term. 00:02:33.770 --> 00:02:36.520 And I think it'll start to make sense to you why 00:02:36.520 --> 00:02:38.910 this equation holds. 00:02:38.910 --> 00:02:41.190 But let's just take this as kind of an act of faith right 00:02:41.190 --> 00:02:45.140 now, that 2pi divided by the period is the coefficient on x. 00:02:45.140 --> 00:02:52.670 So if we say that 2pi divided by the period is equal to the 00:02:52.670 --> 00:02:56.100 coefficient, which is 1/2. 00:02:56.100 --> 00:02:57.440 I know this is extremely messy. 00:02:57.440 --> 00:02:59.040 And this is separate from this. 00:02:59.040 --> 00:03:01.760 So 2pi divided by the period is equal to 1/2. 00:03:01.760 --> 00:03:09.150 Or we could say 1/2 the period is equal to 2pi. 00:03:09.150 --> 00:03:13.700 Or, the period is equal to 4pi. 00:03:13.700 --> 00:03:17.110 So we know the amplitude is equal to 2 and the 00:03:17.110 --> 00:03:19.750 period is equal to 4pi. 00:03:19.750 --> 00:03:21.610 And once again, how did we figure out that the 00:03:21.610 --> 00:03:23.050 period is equal to 4pi? 00:03:23.050 --> 00:03:26.670 We used this formula: 2pi divided by the period is the 00:03:26.670 --> 00:03:27.590 coefficient on the x-term. 00:03:27.590 --> 00:03:31.020 So we set 2pi divided by the period equal to 1/2, and then 00:03:31.020 --> 00:03:33.620 we solved that the period is 4pi. 00:03:33.620 --> 00:03:34.650 So where do we start? 00:03:34.650 --> 00:03:40.580 Well, what is f of 0? 00:03:40.580 --> 00:03:43.330 Well, when x is equal to 0 this whole term is 0. 00:03:43.330 --> 00:03:45.210 So what's sine of 0? 00:03:45.210 --> 00:03:48.650 Sine of 0 is 0, if you remember. 00:03:48.650 --> 00:03:49.870 I guess you could use a calculator, but that's 00:03:49.870 --> 00:03:51.320 something you should remember. 00:03:51.320 --> 00:03:53.900 Or you could re-look at the unit circle to remind yourself. 00:03:53.900 --> 00:03:55.990 Sine of 0 is 0. 00:03:55.990 --> 00:03:59.140 And then 0 times 2 is 0. 00:03:59.140 --> 00:04:02.640 So f of 0 is 0. 00:04:02.640 --> 00:04:02.920 Right? 00:04:02.920 --> 00:04:04.510 We'll draw it right there. 00:04:04.510 --> 00:04:07.290 And we know that it has a period of 4pi. 00:04:07.290 --> 00:04:11.300 That means that the function is going to repeat after 4pi. 00:04:11.300 --> 00:04:17.166 So if we go out it should repeat back out here, at 4pi. 00:04:20.020 --> 00:04:21.930 And now we can just kind of draw the function. 00:04:21.930 --> 00:04:24.594 And this will take a little bit of practice, but-- actually I'm 00:04:24.594 --> 00:04:25.780 going to draw it, and then we can explore it a little 00:04:25.780 --> 00:04:26.490 bit more as well. 00:04:26.490 --> 00:04:28.515 So the function's going to look like this. 00:04:33.260 --> 00:04:33.750 Oh, boy. 00:04:33.750 --> 00:04:35.280 This is more difficult than I thought. 00:04:35.280 --> 00:04:37.350 And it'll keep going in this direction as well. 00:04:42.920 --> 00:04:51.980 And notice, the period here you could do it from here to here. 00:04:51.980 --> 00:04:54.170 This distance is 4pi. 00:04:54.170 --> 00:04:56.490 That's how long it takes for the function to repeat, or 00:04:56.490 --> 00:04:57.800 to go through one cycle. 00:04:57.800 --> 00:05:01.840 Or you could also, if you want, you could measure this 00:05:01.840 --> 00:05:04.560 distance to this distance. 00:05:04.560 --> 00:05:06.230 This would also be 4pi. 00:05:06.230 --> 00:05:07.860 And that's the period of the function. 00:05:07.860 --> 00:05:11.270 And then, of course, the amplitude of the function, 00:05:11.270 --> 00:05:15.590 which is this right here, is 2. 00:05:15.590 --> 00:05:17.140 Here's the amplitude. 00:05:17.140 --> 00:05:22.590 And then the period of 4pi we figured out from this equation. 00:05:22.590 --> 00:05:25.100 Another way we could have thought about it, let's say 00:05:25.100 --> 00:05:29.710 that-- let me erase some of the stuff-- let's say I didn't 00:05:29.710 --> 00:05:31.940 have this stuff right here. 00:05:31.940 --> 00:05:37.950 Let's say I didn't know what the function was. 00:05:37.950 --> 00:05:40.090 Let me get rid of all of this stuff. 00:05:40.090 --> 00:05:43.360 And all I saw was this graph, and I asked you 00:05:43.360 --> 00:05:44.990 to go the other way. 00:05:44.990 --> 00:05:48.440 Using this graph, try to figure out what the function is. 00:05:48.440 --> 00:05:51.910 Then we would just see, how long does it take for 00:05:51.910 --> 00:05:52.750 the function to repeat? 00:05:52.750 --> 00:05:56.140 Well, it takes 4pi radians for the function to repeat, so 00:05:56.140 --> 00:05:58.960 you'd be able to just visually realize that the period 00:05:58.960 --> 00:06:00.620 of this function is 4pi. 00:06:00.620 --> 00:06:01.980 And then you would say, well what's the amplitude? 00:06:01.980 --> 00:06:03.180 The amplitude is easy. 00:06:03.180 --> 00:06:06.320 You would just see how high it goes up or down. 00:06:06.320 --> 00:06:09.170 And it goes up 2, right? 00:06:09.170 --> 00:06:11.030 When you're doing the amplitude you don't do the whole swing, 00:06:11.030 --> 00:06:13.120 you just do how much it swings in the positive 00:06:13.120 --> 00:06:14.120 or negative direction. 00:06:14.120 --> 00:06:17.840 So the amplitude is 2. 00:06:17.840 --> 00:06:19.680 I'm using the wrong color. 00:06:19.680 --> 00:06:20.350 The period is 4pi. 00:06:22.930 --> 00:06:24.430 And then your question would be, well this is 00:06:24.430 --> 00:06:26.600 an oscillating, this is a periodic function. 00:06:26.600 --> 00:06:30.080 Is it a sine or is it a cosine function? 00:06:30.080 --> 00:06:34.610 Well, cosine function, assuming we're not doing any shifting-- 00:06:34.610 --> 00:06:38.070 and in a future module I will shift along the x-axis-- but 00:06:38.070 --> 00:06:43.320 assuming we're not doing any shifting, cosine of 0 is 1. 00:06:43.320 --> 00:06:44.500 Right? 00:06:44.500 --> 00:06:47.270 And sine of 0 is 0. 00:06:47.270 --> 00:06:49.900 And what's this function at 0? 00:06:49.900 --> 00:06:50.970 Well, it's 0. 00:06:50.970 --> 00:06:51.370 Right? 00:06:51.370 --> 00:06:53.670 So this is going to be a sine function. 00:06:53.670 --> 00:06:56.290 So we would use this formula here. 00:06:56.290 --> 00:07:00.120 f of x is equal to the amplitude times the sine of 2pi 00:07:00.120 --> 00:07:02.190 divided by the period times x. 00:07:02.190 --> 00:07:07.780 So we would know that the function is f of x is equal to 00:07:07.780 --> 00:07:19.690 the amplitude times sine of 2pi over the period-- 4pi-- x. 00:07:19.690 --> 00:07:22.135 And, of course, these cancel out. 00:07:22.135 --> 00:07:26.900 And then this cancels out and becomes 2 sine of 1/2 x. 00:07:26.900 --> 00:07:30.070 I know this is a little difficult to read. 00:07:30.070 --> 00:07:31.020 My apologies. 00:07:31.020 --> 00:07:31.960 And I'll ask a question. 00:07:31.960 --> 00:07:35.720 What would this function look like? 00:07:35.720 --> 00:07:45.170 f of x equals 2 cosine of 1/2 x. 00:07:45.170 --> 00:07:47.530 Well, it's going to look the same but we're going to 00:07:47.530 --> 00:07:50.930 start at a different point. 00:07:50.930 --> 00:07:52.430 What's cosine of 0? 00:07:52.430 --> 00:07:56.430 When x is equal to 0 this whole term is equal to 0. 00:07:56.430 --> 00:07:59.550 Cosine of 0, we learned before, is 1. 00:07:59.550 --> 00:08:01.500 So f of 0 is equal to 2. 00:08:01.500 --> 00:08:06.470 Let me write that. f of 0 is equal to 2. 00:08:06.470 --> 00:08:07.600 Let me do this in a different color. 00:08:07.600 --> 00:08:09.530 Let me draw the cosine function in a different color. 00:08:09.530 --> 00:08:11.560 We would start here. 00:08:11.560 --> 00:08:14.270 f of 0 is equal to 2, but everything else is the same. 00:08:14.270 --> 00:08:17.670 The amplitude is the same and the period is the same. 00:08:17.670 --> 00:08:20.110 So now it's going to look like this. 00:08:20.110 --> 00:08:21.600 I hope I don't mess this up. 00:08:21.600 --> 00:08:23.190 This is difficult. 00:08:23.190 --> 00:08:26.986 So now the function is going to look like this. 00:08:26.986 --> 00:08:30.090 And you're going to go down here, and you're going 00:08:30.090 --> 00:08:34.030 to rise up again here. 00:08:34.030 --> 00:08:38.580 And on this side you're going to do the same thing. 00:08:38.580 --> 00:08:40.210 And keep going. 00:08:40.210 --> 00:08:42.510 So notice, the cosine and the sine functions 00:08:42.510 --> 00:08:44.120 look awfully similar. 00:08:44.120 --> 00:08:47.780 And the way to differentiate them is what they do-- well, 00:08:47.780 --> 00:08:48.570 what they do in general. 00:08:48.570 --> 00:08:54.020 But the easiest way is, what happens when you input 00:08:54.020 --> 00:08:55.420 a 0 into the function? 00:08:55.420 --> 00:08:59.260 What happens at the y-axis, or when x is equal to 0, or when 00:08:59.260 --> 00:09:02.040 the angle that you input into it is equal 0? 00:09:02.040 --> 00:09:04.270 Unless we're doing shifting-- and don't worry about shifting 00:09:04.270 --> 00:09:07.610 for now, I'll do that in future modules-- sine of 0 is 0 00:09:07.610 --> 00:09:09.910 while cosine of 0 would be 1. 00:09:09.910 --> 00:09:12.920 And since we're multiplying it times this factor right here, 00:09:12.920 --> 00:09:16.120 times this number right here, the 1 becomes a 2. 00:09:16.120 --> 00:09:18.900 And so this is the graph of cosine of x. 00:09:18.900 --> 00:09:20.750 This is this graph of sine of x. 00:09:20.750 --> 00:09:22.600 And this is a little bit of a preview for shifting. 00:09:22.600 --> 00:09:25.760 Notice that the pink graph, or cosine of x, is very 00:09:25.760 --> 00:09:27.210 similar to the green graph. 00:09:27.210 --> 00:09:33.590 And it's just shifted this way by-- well, in this 00:09:33.590 --> 00:09:36.120 case it's shifted by pi. 00:09:36.120 --> 00:09:36.870 Right? 00:09:36.870 --> 00:09:38.790 And this actually has something to do with the period 00:09:38.790 --> 00:09:39.770 of the coefficient. 00:09:39.770 --> 00:09:42.450 In general, cosine of x is actually sine of x shifted 00:09:42.450 --> 00:09:44.810 to the left by pi/2. 00:09:44.810 --> 00:09:47.050 But I don't want to confuse you too much. 00:09:47.050 --> 00:09:49.170 That's all the time I have for this video. 00:09:49.170 --> 00:09:51.070 I will now do another video with a couple of more 00:09:51.070 --> 00:09:53.030 examples like this.

Graphs of trig functions

https://www.youtube.com/watch?v=QmxMPPkZpME

vtt

https://www.youtube.com/api/timedtext?v=QmxMPPkZpME&ei=f2eUZfjzDoachcIPoquZiAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=AEE94334ABD43652202FBEE0F9824EF63C061AE8.9653647B185F7DF6508DF0BD1D5813C6AF3651CB&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.130 --> 00:00:01.950 Welcome. 00:00:01.950 --> 00:00:05.350 Well what I want to do now is actually I'm going to use this 00:00:05.350 --> 00:00:09.920 graphing application to explore the trigonometric functions or 00:00:09.920 --> 00:00:11.450 explore the graphs of them. 00:00:11.450 --> 00:00:14.480 And just to start off or just to let you, this application 00:00:14.480 --> 00:00:17.010 I'm using is from gcalc.net. 00:00:17.010 --> 00:00:18.940 it's G C A L C .net. 00:00:18.940 --> 00:00:21.030 It's not mine, but I want to give them credit because 00:00:21.030 --> 00:00:24.090 that's what I'm using and I hope they don't mind. 00:00:24.090 --> 00:00:26.440 So let's start off just graphing some functions. 00:00:26.440 --> 00:00:28.750 Let's start off with the sine function. 00:00:28.750 --> 00:00:30.820 So let's say sine of x. 00:00:30.820 --> 00:00:32.420 I hope you all can see it. 00:00:32.420 --> 00:00:34.020 I'm typing it in up here. 00:00:34.020 --> 00:00:35.580 So sine x, let me graph that. 00:00:35.580 --> 00:00:35.890 Look at that. 00:00:35.890 --> 00:00:37.260 Like how nice that looks. 00:00:37.260 --> 00:00:38.530 So let's interpret this. 00:00:38.530 --> 00:00:43.250 So it's oscillating between-- well, let's 00:00:43.250 --> 00:00:44.280 just go point by point. 00:00:44.280 --> 00:00:45.780 I guess that's the easiest way to do it. 00:00:45.780 --> 00:00:50.900 So when x is equal to 0, what is the value of this function? 00:00:50.900 --> 00:00:53.720 Well, if we look here, the value of the function is-- 00:00:53.720 --> 00:00:56.510 let me actually trace it. 00:00:56.510 --> 00:00:59.370 When x equals 0 and it has it written down at the 00:00:59.370 --> 00:01:00.320 bottom of this grey area. 00:01:00.320 --> 00:01:05.450 So when x is 0, y is also 0. 00:01:05.450 --> 00:01:08.060 And if you're remember when we looked for the definitions 00:01:08.060 --> 00:01:10.040 in the unit circle that's what we got. 00:01:10.040 --> 00:01:15.080 That the sine of 0 radians is 0. 00:01:15.080 --> 00:01:19.060 And now as we move on, or move along the curve, I 00:01:19.060 --> 00:01:21.620 have the trace function on. 00:01:21.620 --> 00:01:24.820 This is when x is equal-- if we look in the grey area at the 00:01:24.820 --> 00:01:26.650 bottom left it says 1.57. 00:01:26.650 --> 00:01:28.180 But what is that? 00:01:28.180 --> 00:01:33.240 If you're familiar with the-- 1.57 is more 00:01:33.240 --> 00:01:34.580 commonly known as what? 00:01:34.580 --> 00:01:36.510 It's 1/2 of what famous number? 00:01:36.510 --> 00:01:36.840 Right. 00:01:36.840 --> 00:01:38.010 It's half of pi. 00:01:38.010 --> 00:01:39.910 We're at pi over 2. 00:01:39.910 --> 00:01:42.020 And if you want to convert pi over 2 to degrees 00:01:42.020 --> 00:01:43.860 that's 90 degrees. 00:01:43.860 --> 00:01:47.620 So when we're at the angle of pi over 2 radians the sine 00:01:47.620 --> 00:01:49.520 function is equal to 1. 00:01:49.520 --> 00:01:51.410 And if you go back to some previous modules you'll 00:01:51.410 --> 00:01:55.060 remember that that's exactly what the sine function was 00:01:55.060 --> 00:01:57.270 equal to when we looked at the unit circle. 00:01:57.270 --> 00:02:00.045 Because we were essentially at the point 1 comma 0. 00:02:00.045 --> 00:02:02.550 I hope it's not confusing that I keep referring to the unit 00:02:02.550 --> 00:02:03.470 circle that you can't see. 00:02:03.470 --> 00:02:05.460 But we'll keep going. 00:02:05.460 --> 00:02:08.380 But one thing I want to introduce here is the concept 00:02:08.380 --> 00:02:12.840 of the period or the frequency of the sine function. 00:02:12.840 --> 00:02:16.280 It's pretty obvious to you at this point that the function 00:02:16.280 --> 00:02:18.440 keeps repeating itself. 00:02:18.440 --> 00:02:21.910 It goes from 0, moves up to 1, goes back to 0, goes 00:02:21.910 --> 00:02:22.850 down to negative 1. 00:02:22.850 --> 00:02:26.500 Then goes back to 0 and then repeats again. 00:02:26.500 --> 00:02:30.500 So the period of this periodic function because that's what we 00:02:30.500 --> 00:02:33.100 call a function that keeps repeating, the period of this 00:02:33.100 --> 00:02:38.100 periodic function is this distance from here to here. 00:02:38.100 --> 00:02:38.760 And what's that? 00:02:38.760 --> 00:02:41.450 Well, that's 2 pi radians. 00:02:41.450 --> 00:02:42.770 And does that make sense? 00:02:42.770 --> 00:02:47.660 Well sure, because 2 pi radians is one complete revolution 00:02:47.660 --> 00:02:48.990 around the unit circle. 00:02:48.990 --> 00:02:50.650 And then it repeats again. 00:02:50.650 --> 00:02:52.130 And then it goes the other way. 00:02:52.130 --> 00:02:54.450 You go 2 pi radians backwards and things 00:02:54.450 --> 00:02:55.850 start repeating again. 00:02:55.850 --> 00:02:58.580 Pretty interesting, right? 00:02:58.580 --> 00:03:00.200 Oh, and another thing. 00:03:00.200 --> 00:03:02.380 What two numbers does it oscillate between? 00:03:02.380 --> 00:03:05.970 It oscillates between positive 1 and negative 1. 00:03:05.970 --> 00:03:09.450 And that makes sense because in the unit circle you can never 00:03:09.450 --> 00:03:12.750 get to a point on the perimeter of the unit circle that's 00:03:12.750 --> 00:03:16.700 larger than positive 1 or less than negative 1. 00:03:16.700 --> 00:03:19.150 And that's why the sine of x keeps oscillating between 00:03:19.150 --> 00:03:20.990 these two points. 00:03:20.990 --> 00:03:22.760 Let's do the cosine of x. 00:03:22.760 --> 00:03:26.030 Actually, I'm going to leave the sine of x there. 00:03:26.030 --> 00:03:26.520 Interesting. 00:03:26.520 --> 00:03:31.095 It looks almost the same, but it looks shifted. 00:03:31.095 --> 00:03:33.670 It actually looks shifted to the left about 00:03:33.670 --> 00:03:35.360 pi over 2 radians. 00:03:35.360 --> 00:03:37.160 And that's actually the case. 00:03:37.160 --> 00:03:39.550 So let's first think about why. 00:03:39.550 --> 00:03:41.970 We figured out before that sine-- actually, it looks 00:03:41.970 --> 00:03:44.830 like this program is still tracing the sine function. 00:03:44.830 --> 00:03:47.630 That sine of 0 was 0. 00:03:47.630 --> 00:03:51.340 But if you look at the green function, the cosine of 00:03:51.340 --> 00:03:53.580 0 radians is actually 1. 00:03:53.580 --> 00:03:59.470 Let me see if I can-- no. 00:03:59.470 --> 00:04:01.150 I don't know how to trace the cosine function, so 00:04:01.150 --> 00:04:02.130 I'll just do it here. 00:04:02.130 --> 00:04:04.930 The cosine of 0 is 1. 00:04:04.930 --> 00:04:06.110 And why does that make sense? 00:04:06.110 --> 00:04:08.450 Well, the cosine is the x-coordinate on 00:04:08.450 --> 00:04:09.150 the unit circle. 00:04:12.530 --> 00:04:16.900 When you have 0 radians or 0 degrees, you're at the point 1 00:04:16.900 --> 00:04:19.080 comma 0 on the unit circle. 00:04:19.080 --> 00:04:23.270 So 1 is the cosine or is the x-coordinate and 00:04:23.270 --> 00:04:25.380 0 is the sine value. 00:04:25.380 --> 00:04:28.620 And if any of this is confusing, review the video 00:04:28.620 --> 00:04:33.410 where I use the unit circle to solve the various values of the 00:04:33.410 --> 00:04:35.880 trig functions and then this should make sense. 00:04:35.880 --> 00:04:38.840 And notice that this has a period similar to 00:04:38.840 --> 00:04:39.650 the sine function. 00:04:39.650 --> 00:04:43.100 It starts at 1, goes down to negative 1, and then 00:04:43.100 --> 00:04:45.000 comes back to positive 1. 00:04:45.000 --> 00:04:50.560 And it takes 2 pi radians to complete that cycle. 00:04:50.560 --> 00:04:54.210 And just like the sine function it's oscillating between 1 and 00:04:54.210 --> 00:04:59.370 negative 1 because on the unit circle you can't get to a point 00:04:59.370 --> 00:05:01.850 on the perimeter that's higher than that. 00:05:01.850 --> 00:05:04.450 And now to really hit the point home let's do 00:05:04.450 --> 00:05:05.290 the tangent function. 00:05:05.290 --> 00:05:08.280 I think this one might surprise you. 00:05:08.280 --> 00:05:09.470 Well, look at that. 00:05:09.470 --> 00:05:11.860 So the blue line is the tangent function. 00:05:11.860 --> 00:05:14.280 And why does it do this crazy thing? 00:05:14.280 --> 00:05:16.620 Well, if you remember, the tangent function is equal to 00:05:16.620 --> 00:05:19.625 the y over the x on the perimeter of the unit circle. 00:05:23.130 --> 00:05:29.350 Or since the y is the sine and cosine is the x, it also equals 00:05:29.350 --> 00:05:31.260 the sine over the cosine. 00:05:31.260 --> 00:05:34.240 So here, tangent is 0 whatever sine is 0 00:05:34.240 --> 00:05:35.050 because that makes sense. 00:05:35.050 --> 00:05:39.025 Because tangent is equal to sine over cosine. 00:05:39.025 --> 00:05:42.590 So it makes sense that when sine is 0, tangent is 0. 00:05:42.590 --> 00:05:48.770 But then, as the sine function becomes greater and the cosine 00:05:48.770 --> 00:05:52.100 function becomes less, the numerator in the tangent 00:05:52.100 --> 00:05:54.420 function becomes greater because the numerator is sine. 00:05:54.420 --> 00:05:56.690 So we get larger and larger values, all the way to the 00:05:56.690 --> 00:06:02.225 point where the denominator of the tangent function, which is 00:06:02.225 --> 00:06:03.970 the cosine function-- I think this is probably the most 00:06:03.970 --> 00:06:05.870 confusing module I've ever said because I can't really 00:06:05.870 --> 00:06:07.910 write these things down. 00:06:07.910 --> 00:06:10.020 The denominator goes to 0. 00:06:10.020 --> 00:06:11.000 The cosine right here. 00:06:11.000 --> 00:06:14.070 And then tan spikes and it actually approaches infinity. 00:06:14.070 --> 00:06:17.210 And if you look back at the unit circle, that actually 00:06:17.210 --> 00:06:18.750 might make a little bit of sense. 00:06:18.750 --> 00:06:21.980 But like the other functions, actually the tangent 00:06:21.980 --> 00:06:26.210 function has a period of pi instead of pi over 2. 00:06:26.210 --> 00:06:27.130 Instead of 2 pi. 00:06:27.130 --> 00:06:29.050 And I'll leave that as an exercise for you 00:06:29.050 --> 00:06:30.630 to think about. 00:06:30.630 --> 00:06:32.450 But with that drawn out, I'm now going to 00:06:32.450 --> 00:06:33.650 explore something else. 00:06:33.650 --> 00:06:35.380 Let me reset this. 00:06:35.380 --> 00:06:38.070 Yes, I really want to reset. 00:06:38.070 --> 00:06:40.860 I drew the sine function before. 00:06:40.860 --> 00:06:46.650 Let me draw the sine of let's say, 2x. 00:06:46.650 --> 00:06:49.000 Whoops, that's not right. 00:06:49.000 --> 00:06:54.820 sine of 2-- maybe I need to put some parentheses in. 00:06:54.820 --> 00:06:56.090 Oh, there we go. 00:06:56.090 --> 00:06:57.760 Actually, let me reset it. 00:06:57.760 --> 00:06:58.980 Yes, I want to reset. 00:06:58.980 --> 00:07:01.400 So first I'll draw the sine of x and then I'll 00:07:01.400 --> 00:07:04.310 draw the sine of 2x. 00:07:07.250 --> 00:07:09.970 So what's the first thing you notice about the difference 00:07:09.970 --> 00:07:10.570 between these two? 00:07:10.570 --> 00:07:14.760 The brown one is sine of x and the green one is sine of 2x. 00:07:17.680 --> 00:07:20.860 They both oscillate between the same two numbers and just so 00:07:20.860 --> 00:07:25.600 you know, the height of the oscillation is called 00:07:25.600 --> 00:07:27.630 the amplitude of this periodic function. 00:07:27.630 --> 00:07:31.410 So in both cases, the amplitude is 1 because they oscillate 00:07:31.410 --> 00:07:32.950 from 1 to negative 1. 00:07:32.950 --> 00:07:38.550 So the amplitude is 1, but their period is different. 00:07:38.550 --> 00:07:45.860 Sine of x takes 2 pi radians to complete one circle-- one cycle 00:07:45.860 --> 00:07:51.910 while sine of 2x only takes pi radians to complete one cycle. 00:07:51.910 --> 00:07:55.572 So it actually completes it twice as fast. 00:07:55.572 --> 00:07:59.480 And I want you to sit and think about why sine of 2x has 1/2 00:07:59.480 --> 00:08:02.390 the period of sine of x. 00:08:02.390 --> 00:08:03.820 And you can probably guess what happens if 00:08:03.820 --> 00:08:07.420 I type in sine of 3x. 00:08:07.420 --> 00:08:10.030 Actually, let's do sine of 4x. 00:08:10.030 --> 00:08:12.770 It should have 1/2 the period of sine of 2x then. 00:08:12.770 --> 00:08:14.530 And it does, even though this is probably a 00:08:14.530 --> 00:08:18.950 very confusing graph. 00:08:18.950 --> 00:08:19.920 So let's explore. 00:08:19.920 --> 00:08:21.550 So I think you understand what the coefficient 00:08:21.550 --> 00:08:22.650 on the x term does. 00:08:22.650 --> 00:08:25.680 When you have a larger coefficient it kind of 00:08:25.680 --> 00:08:27.570 speeds up the cycles. 00:08:27.570 --> 00:08:30.876 And let's explore a little bit more. 00:08:30.876 --> 00:08:33.670 Let's start off with sine of x again. 00:08:33.670 --> 00:08:36.220 And now, instead of making the coefficient larger, let's 00:08:36.220 --> 00:08:37.380 make the coefficient less. 00:08:37.380 --> 00:08:41.090 Let's make it sine of 0.5x. 00:08:41.090 --> 00:08:41.600 Look at that. 00:08:41.600 --> 00:08:44.960 Now, all of a sudden, it takes 4 pi radians to 00:08:44.960 --> 00:08:46.030 complete one cycle. 00:08:46.030 --> 00:08:47.830 And I want you to think about why that is. 00:08:47.830 --> 00:08:50.480 Because we're now slowing down how fast it cycles 00:08:50.480 --> 00:08:51.275 through the angles. 00:08:54.820 --> 00:08:57.500 Now I want to start playing with the amplitude. 00:08:57.500 --> 00:09:02.310 So we had sine of x, what do you think will happen if I put 00:09:02.310 --> 00:09:09.330 in this 2 times sine of x? 00:09:09.330 --> 00:09:12.840 So here, the period is the same. 00:09:12.840 --> 00:09:16.540 It's still 2 pi, but notice that it oscillates between 2 00:09:16.540 --> 00:09:20.840 and negative 2 instead of between 1 and negative 1. 00:09:20.840 --> 00:09:24.120 So whatever the coefficient, or whatever the number is in front 00:09:24.120 --> 00:09:27.690 of the sine or the cosine function, that actually 00:09:27.690 --> 00:09:28.885 affects its amplitude. 00:09:28.885 --> 00:09:34.030 And similarly, we can look at 0.5 sine-- let's 00:09:34.030 --> 00:09:38.970 say 0.5 sine of 2x. 00:09:38.970 --> 00:09:39.480 Interesting. 00:09:39.480 --> 00:09:44.340 So now it only goes up to 0.5 and down to minus 0.5. 00:09:44.340 --> 00:09:48.780 So it's amplitude is 1/2 or 0.5. 00:09:48.780 --> 00:09:52.290 And it also oscillates twice as fast as the sine function 00:09:52.290 --> 00:09:56.300 because it was 0.5 sine of 2z. 00:09:56.300 --> 00:09:58.840 I think that's all the time I have now. 00:09:58.840 --> 00:10:01.020 I have a feeling this might have confused you more than 00:10:01.020 --> 00:10:04.240 helped, but I'll still put the video up just in case 00:10:04.240 --> 00:10:05.230 it's helpful for someone. 00:10:05.230 --> 00:10:07.950 But in the future I might actually record another video 00:10:07.950 --> 00:10:09.980 where I can actually write things down so it doesn't 00:10:09.980 --> 00:10:10.950 confuse you as much. 00:10:10.950 --> 00:10:13.080 So if it confused you I apologize, but I 00:10:13.080 --> 00:10:14.940 hope it was helpful. 00:10:14.940 --> 00:10:16.470 See you later.

Graph of the sine function

https://www.youtube.com/watch?v=2zoiW4PdVKo

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https://www.youtube.com/api/timedtext?v=2zoiW4PdVKo&ei=fmeUZe6CMeDjxN8PkrCz0Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1C72EA6B11C34C997EFF4ED3B7732F48E02B6C4A.67AA30F6A9747928788806749562F2323D6BEAD4&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:02.150 Hello. 00:00:02.150 --> 00:00:06.100 In the last presentation we kind of re-defined the sine, 00:00:06.100 --> 00:00:09.140 the cosine, and the tangent functions in a broader way 00:00:09.140 --> 00:00:15.580 where we said if we have a unit circle and our theta is, or our 00:00:15.580 --> 00:00:19.795 angle, is -- let me use the right tool -- let's say, and 00:00:19.795 --> 00:00:24.950 our angle is the angle between, say, the x-axis and a radius in 00:00:24.950 --> 00:00:26.570 the unit circle, and this is our radius. 00:00:30.525 --> 00:00:33.870 the coordinate of the point where this radius intersects 00:00:33.870 --> 00:00:39.070 the unit circle is x comma y. 00:00:39.070 --> 00:00:45.490 Our new definition of the trig functions was that sine of 00:00:45.490 --> 00:00:49.000 theta is equal to the y-coordinate, right, this is 00:00:49.000 --> 00:00:50.920 y-coordinate where it intersects the unit circle. 00:00:50.920 --> 00:00:52.400 And remember, this is the unit circle. 00:00:52.400 --> 00:00:56.260 It's not just any circle, which means it has a radius of 1. 00:00:56.260 --> 00:01:03.990 Cosine of theta is equal to the x-coordinate of this point. 00:01:03.990 --> 00:01:06.360 This is the x-coordinate. 00:01:06.360 --> 00:01:12.550 And tangent of theta equaled opposite over 00:01:12.550 --> 00:01:14.050 adjacent or y over x. 00:01:17.190 --> 00:01:19.665 That's interesting because that's also equal to sine of 00:01:19.665 --> 00:01:22.060 theta over cosine of theta. 00:01:22.060 --> 00:01:23.000 I'll just do that. 00:01:23.000 --> 00:01:26.000 I wasn't even planning on covering that, but just it 00:01:26.000 --> 00:01:29.370 leaves you something to think about. 00:01:29.370 --> 00:01:32.850 So with that said, let's take a look or let's try to see how 00:01:32.850 --> 00:01:34.940 this defines these functions. 00:01:34.940 --> 00:01:37.120 And I guess a good place to start is just with the sine 00:01:37.120 --> 00:01:39.450 function and we can try to graph it. 00:01:39.450 --> 00:01:42.080 So let's write, let's do a little table like we always do 00:01:42.080 --> 00:01:45.470 when we define a function. 00:01:45.470 --> 00:01:50.190 Let's put in values of theta, and let's figure 00:01:50.190 --> 00:01:53.690 out what sine of theta is. 00:01:53.690 --> 00:01:57.370 So when theta is equal to 0 radians, what is sine of theta? 00:01:57.370 --> 00:02:05.430 So when theta's 0, right, then the radius between it -- this 00:02:05.430 --> 00:02:07.600 is the radius and this is the point where the radius 00:02:07.600 --> 00:02:09.210 intersects the unit circle. 00:02:09.210 --> 00:02:12.720 And this point has a coordinate 1 comma 0, right? 00:02:15.340 --> 00:02:19.120 And so if where it intersects the unit circle is at 1 comma 00:02:19.120 --> 00:02:22.460 0, then sine of theta is just the y-coordinate. 00:02:22.460 --> 00:02:25.620 So sine of theta is 0. 00:02:25.620 --> 00:02:31.570 If we said what is sine of theta when theta is 00:02:31.570 --> 00:02:32.870 equal to pi over 2. 00:02:36.840 --> 00:02:39.370 So now our radius is this radius and we intersect the 00:02:39.370 --> 00:02:45.180 unit circle right here at the point 0 comma 1. 00:02:45.180 --> 00:02:47.010 And what's the y-coordinate at 0 comma 1? 00:02:47.010 --> 00:02:49.900 Well it's 1. 00:02:49.900 --> 00:02:54.910 What happens when we have theta is equal to pi radians? 00:02:54.910 --> 00:02:59.350 So at pi radians we intersect the unit circle right here. 00:02:59.350 --> 00:03:00.700 We're at pi radian. 00:03:00.700 --> 00:03:03.600 This is the angle, pi. 00:03:03.600 --> 00:03:08.520 We intersect with unit circle at negative 1 comma 0. 00:03:08.520 --> 00:03:10.950 Because once again, this is the unit circle. 00:03:10.950 --> 00:03:13.790 So at negative 1 comma 0, what's the y-coordinate? 00:03:13.790 --> 00:03:16.410 Well, it's 0. 00:03:16.410 --> 00:03:18.710 So sine of pi is equal to 0. 00:03:18.710 --> 00:03:20.750 Let's just keep going around the circle. 00:03:20.750 --> 00:03:32.185 When we have the angle, when theta is equal to 3 pi over 4 00:03:32.185 --> 00:03:37.370 -- no, sorry, 3 pi over 2. 00:03:37.370 --> 00:03:40.220 Because this is pi and this is another half pi. 00:03:40.220 --> 00:03:42.310 So this is 3 pi over 2, sorry. 00:03:42.310 --> 00:03:46.720 So when theta is equal to 3 pi over 2, what is sine of theta? 00:03:46.720 --> 00:03:49.430 Well, now we intersect the unit circle down here at the 00:03:49.430 --> 00:03:51.920 point 0 comma negative 1. 00:03:51.920 --> 00:03:56.040 So now sine of theta is equal to negative 1. 00:03:56.040 --> 00:03:59.320 Then if we go all the way around the circle to 2 00:03:59.320 --> 00:04:03.680 pi radians, we're back at this point again. 00:04:03.680 --> 00:04:09.300 So sine of theta, so we're at 2 pi, sine of theta 00:04:09.300 --> 00:04:11.960 is now 0 once again. 00:04:11.960 --> 00:04:15.210 So let's graph these points out and then we'll try to figure 00:04:15.210 --> 00:04:16.970 out what the points in between look like, and I'll show 00:04:16.970 --> 00:04:19.050 you the graph of a sine function is. 00:04:24.200 --> 00:04:25.800 So let's draw the x-axis. 00:04:28.310 --> 00:04:29.450 This is my x-axis. 00:04:32.350 --> 00:04:36.595 And let's draw the y-axis. 00:04:41.180 --> 00:04:43.700 Not as clean as I wanted to draw it. 00:04:43.700 --> 00:04:44.735 This is y. 00:04:48.320 --> 00:04:49.410 And that's x. 00:04:49.410 --> 00:04:51.360 But in this case instead of saying that's the x-axis, let's 00:04:51.360 --> 00:04:54.910 call that the theta axis, because we defined theta as the 00:04:54.910 --> 00:04:59.370 input or our domains in terms of theta. 00:04:59.370 --> 00:05:02.170 So this is the theta axis. 00:05:02.170 --> 00:05:04.290 Now we're going to graph sine of theta. 00:05:04.290 --> 00:05:07.565 So when we said when theta equaled 0, sine 00:05:07.565 --> 00:05:09.030 of theta is equal to 0. 00:05:09.030 --> 00:05:12.260 So that's this point right here, 0 comma 0. 00:05:12.260 --> 00:05:24.760 When theta is equal to pi over 2, sine of theta is equal to 1. 00:05:24.760 --> 00:05:30.680 So this is the point pi over 2 comma 1, right? 00:05:30.680 --> 00:05:32.270 That's just this 1. 00:05:32.270 --> 00:05:40.610 When theta is equal to pi, sine of theta is 0 again. 00:05:40.610 --> 00:05:44.660 So this is the point pi comma 0. 00:05:44.660 --> 00:05:56.540 And when theta equaled 3 pi over 2, what was sine of theta? 00:05:56.540 --> 00:05:58.420 I equaled negative 1. 00:05:58.420 --> 00:06:00.730 Interesting. 00:06:00.730 --> 00:06:04.330 Then when we got to 2 pi -- when we got to theta equal 00:06:04.330 --> 00:06:07.700 to 2 pi, sine of theta, again, equaled 0. 00:06:07.700 --> 00:06:11.570 So we know that these points are on the graph 00:06:11.570 --> 00:06:12.680 of sine of theta. 00:06:12.680 --> 00:06:15.720 And if you actually tried the points in between, and as 00:06:15.720 --> 00:06:18.480 an exercise it might be interesting for you to do so. 00:06:18.480 --> 00:06:20.140 You could actually figure out a lot of the points using 00:06:20.140 --> 00:06:23.650 30-60-90 triangles or using the Pythagorean Theorem. 00:06:23.650 --> 00:06:25.930 But you actually get a curve that looks something -- let me 00:06:25.930 --> 00:06:30.630 use a nicer color than this kind of drab grey -- you 00:06:30.630 --> 00:06:35.360 get a graph that looks something like this. 00:06:39.870 --> 00:06:43.430 And you've probably seen that before. 00:06:43.430 --> 00:06:45.990 The term for this function is actually a sine wave. 00:06:45.990 --> 00:06:47.870 It looks like something that's oscillating or 00:06:47.870 --> 00:06:49.140 that's moving up and down. 00:06:49.140 --> 00:06:53.390 And actually if you were to put in thetas that were less than 00:06:53.390 --> 00:06:58.470 0, the sine wave will keep going into the 00:06:58.470 --> 00:06:59.520 negative theta axis. 00:06:59.520 --> 00:07:01.220 It keeps going forever in both directions. 00:07:01.220 --> 00:07:05.940 It keeps oscillating between 1 and negative 1 and 00:07:05.940 --> 00:07:07.960 the points in between. 00:07:07.960 --> 00:07:11.180 So that's the graph of the sign function. 00:07:11.180 --> 00:07:13.390 In the next module I'll actually do the graph of the 00:07:13.390 --> 00:07:15.350 cosine function, or actually I might just show you the graph 00:07:15.350 --> 00:07:16.840 of the cosine function. 00:07:16.840 --> 00:07:20.580 Then I'll show you how they relate and how these can 00:07:20.580 --> 00:07:27.480 describe any kind of, or many types of oscillatory things in 00:07:27.480 --> 00:07:31.150 the world and how it relates to frequency and amplitude. 00:07:31.150 --> 00:07:32.740 So I'll see you in the next module. 00:07:32.740 --> 00:07:36.270 And just for fun you might want to sit down with a piece of 00:07:36.270 --> 00:07:39.920 paper and try to graph the cosine function or the 00:07:39.920 --> 00:07:41.680 tangent function as well. 00:07:41.680 --> 00:07:43.160 Have fun.

Unit Circle Definition of Trig Functions

https://www.youtube.com/watch?v=ZffZvSH285c

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https://www.youtube.com/api/timedtext?v=ZffZvSH285c&ei=fWeUZdnLAc20vdIPmrm1gAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249837&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=748EAD9AC1E5D656160C4D49CF0FE8329F50B480.9A82681B78462D7F94B1A52C25EDC7B066D04E4D&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.280 --> 00:00:02.210 Hello. 00:00:02.210 --> 00:00:06.150 Well, welcome to the next presentation in the 00:00:06.150 --> 00:00:08.260 trigonometry modules. 00:00:08.260 --> 00:00:10.340 Just to start off a little bit, let's review what 00:00:10.340 --> 00:00:13.070 we've done so far. 00:00:13.070 --> 00:00:15.320 In the last couple modules, we learned the definitions-- or at 00:00:15.320 --> 00:00:17.720 least, I guess, we could call it a partial definition-- of 00:00:17.720 --> 00:00:19.880 the sine, the cosine, and the tangent functions. 00:00:19.880 --> 00:00:24.250 And the mnemonic we used to memorize that was sohcahtoa. 00:00:24.250 --> 00:00:26.190 Let me write that down. 00:00:26.190 --> 00:00:26.620 Sohcahtoa. 00:00:32.490 --> 00:00:35.320 And what that told us is, let's say we had a right triangle. 00:00:35.320 --> 00:00:36.705 Let me draw a right triangle. 00:00:39.720 --> 00:00:40.850 This is a right angle here. 00:00:40.850 --> 00:00:42.355 This is the hypotenuse. 00:00:42.355 --> 00:00:44.680 Let me label the hypotenuse, h. 00:00:44.680 --> 00:00:49.570 Let me label this-- and so we want to figure out, we want to 00:00:49.570 --> 00:00:50.730 use this angle right here. 00:00:50.730 --> 00:00:52.110 Theta, we'll call this theta. 00:00:52.110 --> 00:00:52.780 Whatever. 00:00:52.780 --> 00:00:57.650 Then this is the adjacent side, and this is the opposite side. 00:00:57.650 --> 00:00:58.770 And that's an o. 00:00:58.770 --> 00:01:02.310 So soh tells us that sine is equal to opposite 00:01:02.310 --> 00:01:03.880 over hypotenuse. 00:01:03.880 --> 00:01:07.600 Cosine is equal to adjacent over hypotenuse. 00:01:07.600 --> 00:01:11.250 And tangent is equal to opposite over adjacent. 00:01:11.250 --> 00:01:14.310 And I think, by this point-- and especially if you did some 00:01:14.310 --> 00:01:17.590 of the exercises on the Khan Academy-- that should be second 00:01:17.590 --> 00:01:19.470 nature and should make a lot of sense to you. 00:01:19.470 --> 00:01:23.460 But this definition, using a right triangle like this, 00:01:23.460 --> 00:01:25.140 actually breaks down at certain points. 00:01:25.140 --> 00:01:26.390 Actually, at a lot of points. 00:01:26.390 --> 00:01:29.640 For example, what happens as this angle right here 00:01:29.640 --> 00:01:31.160 approaches 90 degrees? 00:01:31.160 --> 00:01:34.280 You can't have two 90 degree angles in a right 00:01:34.280 --> 00:01:35.330 triangle, can you? 00:01:35.330 --> 00:01:37.620 Then it would be like a rectangle or something. 00:01:37.620 --> 00:01:39.340 But you could actually probably figure out what happens as 00:01:39.340 --> 00:01:40.510 it approaches 90 degrees. 00:01:40.510 --> 00:01:42.980 But the definition, this definition, breaks 00:01:42.980 --> 00:01:44.340 down for that. 00:01:44.340 --> 00:01:46.280 Also, what happens if this angle is negative? 00:01:46.280 --> 00:01:49.655 Or what happens if this angle is more than 90 degrees? 00:01:49.655 --> 00:01:52.600 Or what happens if it's 800 degrees? 00:01:52.600 --> 00:01:55.600 Or you know, 8 pi radians? 00:01:55.600 --> 00:01:57.830 Not that 800 and 8 pi radians are the same thing. 00:01:57.830 --> 00:02:00.180 But obviously, this definition starts to break down. 00:02:00.180 --> 00:02:02.100 Because we couldn't even draw a right triangle that 00:02:02.100 --> 00:02:03.890 has those properties. 00:02:03.890 --> 00:02:07.630 So now I'm going to introduce you to an extension 00:02:07.630 --> 00:02:08.500 of this definition. 00:02:08.500 --> 00:02:12.920 It's really the same thing, but it allows the sine, the cosine, 00:02:12.920 --> 00:02:16.970 and the tangent functions to be defined for angles greater than 00:02:16.970 --> 00:02:21.530 or equal to pi over 2, or 90 degrees, or less than 0. 00:02:21.530 --> 00:02:22.900 So let's draw a unit circle. 00:02:22.900 --> 00:02:25.290 So this is just the coordinate axis, and here is a 00:02:25.290 --> 00:02:27.700 circle of radius 1. 00:02:27.700 --> 00:02:30.540 And let's make-- let me see. 00:02:30.540 --> 00:02:32.920 Let me make sure I'm using the correct pen tool. 00:02:32.920 --> 00:02:33.670 OK. 00:02:33.670 --> 00:02:39.930 So let's call this right here-- so this is theta. 00:02:39.930 --> 00:02:42.220 This is an angle, right? 00:02:42.220 --> 00:02:45.640 Between the x-axis and this line I just drew here. 00:02:45.640 --> 00:02:47.010 And this is a radius, right? 00:02:47.010 --> 00:02:49.590 And we said that this has a radius 1. 00:02:49.590 --> 00:02:51.600 So the length of this line is 1, right? 00:02:51.600 --> 00:02:54.020 Because it just goes from the origin to the 00:02:54.020 --> 00:02:54.820 outside of the circle. 00:02:54.820 --> 00:02:56.940 So it has a radius of 1. 00:02:56.940 --> 00:02:59.890 And now I'm going to draw a right triangle again. 00:02:59.890 --> 00:03:02.500 Let me just drop a line from here. 00:03:02.500 --> 00:03:05.920 So there I have a right triangle. 00:03:05.920 --> 00:03:07.950 So if we use the old definition we learned before. 00:03:07.950 --> 00:03:10.010 Let's just focus on sine for now. 00:03:10.010 --> 00:03:17.920 So sine is equal to opposite over hypotenuse. 00:03:17.920 --> 00:03:20.280 Let's apply that to this right triangle right here. 00:03:20.280 --> 00:03:22.030 This is the right angle. 00:03:22.030 --> 00:03:24.170 So what's the opposite angle of this? 00:03:24.170 --> 00:03:25.840 What's the opposite side from this angle? 00:03:29.200 --> 00:03:32.090 I'm going to change to yellow. 00:03:32.090 --> 00:03:32.960 It's this side, right? 00:03:32.960 --> 00:03:34.630 This is the opposite side. 00:03:34.630 --> 00:03:37.040 And what's the hypotenuse? 00:03:37.040 --> 00:03:42.500 The hypotenuse is just this radius, right? 00:03:42.500 --> 00:03:45.440 And let's just say that this point, where it intersects 00:03:45.440 --> 00:03:54.550 the circle-- let's call this point right here x comma y. 00:03:54.550 --> 00:04:00.450 So what's the height of this opposite side? 00:04:00.450 --> 00:04:01.300 Well, it's y, right? 00:04:01.300 --> 00:04:05.130 Because it's just the height of that point. 00:04:05.130 --> 00:04:06.570 This is of height y. 00:04:06.570 --> 00:04:13.170 So sine of this angle right here, sine of theta, is 00:04:13.170 --> 00:04:15.400 going to equal the opposite side-- which is this yellow 00:04:15.400 --> 00:04:18.340 side, which is just the y-coordinate-- is going to 00:04:18.340 --> 00:04:21.490 equal y over the hypotenuse. 00:04:21.490 --> 00:04:23.930 The hypotenuse is this pink side here. 00:04:23.930 --> 00:04:25.350 And what's the length of the hypotenuse? 00:04:25.350 --> 00:04:27.750 Well, it's the radius of this unit circle. 00:04:27.750 --> 00:04:29.600 So it's 1. 00:04:29.600 --> 00:04:31.100 And y divided by 1? 00:04:31.100 --> 00:04:31.860 Well, that's just y. 00:04:31.860 --> 00:04:40.040 So we see that sine of theta is equal to y. 00:04:40.040 --> 00:04:42.520 Let's do the same thing for cosine of theta. 00:04:42.520 --> 00:04:49.700 Well, we know that cosine is equal to adjacent 00:04:49.700 --> 00:04:51.430 over hypotenuse. 00:04:51.430 --> 00:04:55.410 Well, what's the adjacent side here? 00:04:55.410 --> 00:04:56.470 I'm running out of colors. 00:04:56.470 --> 00:05:00.200 The adjacent side is this bottom side, right here. 00:05:00.200 --> 00:05:02.390 So that would equal-- so if I said-- I'm running 00:05:02.390 --> 00:05:03.760 out of space, too. 00:05:03.760 --> 00:05:09.460 Cosine of theta would equal this gray side-- which is 00:05:09.460 --> 00:05:11.250 the adjacent side-- and what is that? 00:05:11.250 --> 00:05:14.350 What is this length? 00:05:14.350 --> 00:05:16.610 What is the length of this side? 00:05:16.610 --> 00:05:18.430 Well, it's just x, right? 00:05:18.430 --> 00:05:23.070 If this is the point x, y then this distance here is x and we 00:05:23.070 --> 00:05:25.250 already learned this distance-- or we already observed-- 00:05:25.250 --> 00:05:26.740 that this distance is y. 00:05:26.740 --> 00:05:29.100 So this distance being just x, we know that the 00:05:29.100 --> 00:05:31.280 adjacent length is x. 00:05:31.280 --> 00:05:34.980 So we say cosine of theta is equal to x over the hypotenuse. 00:05:34.980 --> 00:05:37.360 And once again, the hypotenuse is 1. 00:05:37.360 --> 00:05:43.330 So cosine of theta is equal to x. 00:05:43.330 --> 00:05:44.120 I know what you're thinking. 00:05:44.120 --> 00:05:45.820 Sal, that's very nice and cute. 00:05:45.820 --> 00:05:48.280 Cosine of theta equals x, sine of theta equals y. 00:05:48.280 --> 00:05:49.760 But how is this really different from what we 00:05:49.760 --> 00:05:51.670 were doing before? 00:05:51.670 --> 00:05:56.160 Well, if I define it this way, now all of a sudden when the 00:05:56.160 --> 00:06:02.940 angle becomes 90 degrees, now I can actually define 00:06:02.940 --> 00:06:04.980 what sine of theta is. 00:06:04.980 --> 00:06:07.540 Sine of theta now is just y. 00:06:07.540 --> 00:06:10.190 Is just the y-coordinate, which is 1. 00:06:10.190 --> 00:06:13.400 If theta is equal to-- I'm going to make sure it's very 00:06:13.400 --> 00:06:15.610 messy right here-- if theta is equal to 90 degrees, 00:06:15.610 --> 00:06:18.000 or pi over 2 radians. 00:06:18.000 --> 00:06:20.680 This is pi over 2. 00:06:20.680 --> 00:06:22.870 This angle right here. 00:06:22.870 --> 00:06:26.590 And similar, cosine of pi over 2 is 0. 00:06:26.590 --> 00:06:31.650 Because the x-coordinate right here is 0. 00:06:31.650 --> 00:06:33.060 Let me do it with a couple more examples. 00:06:33.060 --> 00:06:35.050 Oh, I'm forgetting the tangent function. 00:06:35.050 --> 00:06:36.740 And you could probably figure out now, what is the 00:06:36.740 --> 00:06:39.650 definition now we can use for the tangent function? 00:06:39.650 --> 00:06:41.600 Well, going back-- let's use this green theta here. 00:06:41.600 --> 00:06:43.490 Because it's kind of a normal angle. 00:06:43.490 --> 00:06:45.830 So in this green angle here, tangent is 00:06:45.830 --> 00:06:47.680 opposite over adjacent. 00:06:47.680 --> 00:06:54.040 So tangent now, we can define as y/x. 00:06:54.040 --> 00:06:57.760 And remember, these y's and x's that we're using are the point 00:06:57.760 --> 00:07:01.780 on the unit circle where the angle that's defined by this-- 00:07:01.780 --> 00:07:06.440 by whatever-- where the radius that is subtended by this 00:07:06.440 --> 00:07:09.150 angle, or I guess the arc, intersects-- actually, 00:07:09.150 --> 00:07:10.940 I'm getting confused with terminology. 00:07:10.940 --> 00:07:13.810 It's where this line intersects the circumference. 00:07:13.810 --> 00:07:17.800 The coordinate of that-- the sine of theta is equal to y. 00:07:17.800 --> 00:07:19.750 The cosine of theta is equal to x. 00:07:19.750 --> 00:07:22.740 And the tangent of theta is equal to y/x. 00:07:22.740 --> 00:07:27.020 Let's do a couple of examples and hopefully this'll make a 00:07:27.020 --> 00:07:28.110 little bit more sense to you. 00:07:31.550 --> 00:07:35.246 Let me try to really fast draw a new unit circle. 00:07:37.780 --> 00:07:39.030 So that's my unit circle. 00:07:42.810 --> 00:07:44.960 And here's the coordinate axis. 00:07:44.960 --> 00:07:46.530 It's one of them. 00:07:46.530 --> 00:07:48.000 And here is the other one. 00:07:58.180 --> 00:08:04.010 So if we use the angle-- let's use the angle pi over 2, right? 00:08:04.010 --> 00:08:06.450 Theta equals pi over 2. 00:08:06.450 --> 00:08:09.670 Well, pi over 2 is right here. 00:08:09.670 --> 00:08:12.860 It's a 90 degree angle, if you wanted to use degrees. 00:08:12.860 --> 00:08:15.300 And now, we just figure out where it intersects 00:08:15.300 --> 00:08:15.840 the unit circle. 00:08:15.840 --> 00:08:17.715 And once again, this is a unit circle, so it 00:08:17.715 --> 00:08:20.300 has a radius of 1. 00:08:20.300 --> 00:08:29.210 So we can see that sine of pi over 2 equals the y-coordinate 00:08:29.210 --> 00:08:32.080 where it intersects the unit circle. 00:08:32.080 --> 00:08:34.630 So that's just 1. 00:08:34.630 --> 00:08:36.090 What's cosine of pi over 2? 00:08:39.340 --> 00:08:41.090 Well, it's just the x-coordinate, where you 00:08:41.090 --> 00:08:42.220 intersect the unit circle. 00:08:42.220 --> 00:08:46.060 And the x-coordinate here is 0. 00:08:46.060 --> 00:08:47.900 And what's the tangent of pi over 2? 00:08:47.900 --> 00:08:49.700 This is interesting. 00:08:49.700 --> 00:08:52.770 The tangent of pi over 2. 00:08:52.770 --> 00:08:55.450 Well, the tangent we defined now as y/x. 00:08:55.450 --> 00:08:58.910 So the y-coordinate, this is the point 0, 1, right? 00:08:58.910 --> 00:09:01.610 The y-coordinate is 1. 00:09:01.610 --> 00:09:02.460 So it equals 1/0. 00:09:05.400 --> 00:09:06.750 So this is undefined. 00:09:06.750 --> 00:09:09.800 So still, we don't have a tangent function that can 00:09:09.800 --> 00:09:11.640 define itself at certain points. 00:09:11.640 --> 00:09:14.660 But in the next module, we're actually going to graph this. 00:09:14.660 --> 00:09:17.470 And you'll see that it approaches infinity. 00:09:17.470 --> 00:09:21.580 And similarly, we could try to find the functions for when 00:09:21.580 --> 00:09:24.100 theta equals pi, right? 00:09:24.100 --> 00:09:27.060 That's like 180 degrees. 00:09:27.060 --> 00:09:28.570 That's this point right here. 00:09:28.570 --> 00:09:31.450 So sine of pi. 00:09:31.450 --> 00:09:33.820 What's the y-coordinate at this point? 00:09:33.820 --> 00:09:37.720 Well, this point is negative 1 comma 0. 00:09:37.720 --> 00:09:39.880 So the y-coordinate is 0. 00:09:39.880 --> 00:09:40.940 What's the x-coordinate? 00:09:40.940 --> 00:09:43.520 Cosine of pi. 00:09:43.520 --> 00:09:45.260 That's negative 1. 00:09:45.260 --> 00:09:49.570 And of course, what's the tangent of pi radians? 00:09:49.570 --> 00:09:51.220 It's y/x. 00:09:51.220 --> 00:09:54.400 So it's 0 over negative 1, which equals 0. 00:09:54.400 --> 00:09:55.760 Hopefully this makes sense. 00:09:55.760 --> 00:09:58.730 Now in the next module I'll actually graph these points. 00:09:58.730 --> 00:10:04.450 And you'll see how it all comes together and why it is useful 00:10:04.450 --> 00:10:09.010 to define the sine, the cosine, the tangent functions this way. 00:10:09.010 --> 00:10:10.160 See you soon. 00:10:10.160 --> 00:10:11.460 Bye.

Using Trig Functions Part II

https://www.youtube.com/watch?v=RoXmKYjpLGk

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https://www.youtube.com/api/timedtext?v=RoXmKYjpLGk&ei=fmeUZbzLL4Wrp-oP5Yu0qAs&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=1DE4BE9B80A732C78355101BFABAEEE04E86BB84.611E3FD431F7E8D4DBF4721A8AD503CE195A96A9&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.980 --> 00:00:01.580 Welcome back. 00:00:01.580 --> 00:00:05.340 I just want to do a couple more of these problems, just so you, 00:00:05.340 --> 00:00:09.440 I guess, see a few couple more problems and get some practice. 00:00:09.440 --> 00:00:12.040 So let's start with another problem. 00:00:12.040 --> 00:00:16.440 As always, let me draw my triangle. 00:00:16.440 --> 00:00:18.270 I always like to draw it a little different 00:00:18.270 --> 00:00:21.240 to confuse you. 00:00:21.240 --> 00:00:34.410 Let's say that this angle up here is 0.36 radians. 00:00:34.410 --> 00:00:38.990 An angle is 0.36 radians, and then this side right here, this 00:00:38.990 --> 00:00:44.390 top side-- let me do it in a different color-- this top side 00:00:44.390 --> 00:00:53.190 in pink is square root of 73 units long or whatever. 00:00:53.190 --> 00:00:54.800 It could be inches or feet or whatever. 00:00:54.800 --> 00:00:57.690 But it's square root of 73 units long. 00:00:57.690 --> 00:01:05.060 So my question is, what is this side here, this green side? 00:01:05.060 --> 00:01:08.350 Well, how did we do this type of problem last time? 00:01:08.350 --> 00:01:11.600 We should figure out what sides we're dealing with. 00:01:11.600 --> 00:01:14.040 Are we dealing with the adjacent and the opposite, the 00:01:14.040 --> 00:01:16.460 adjacent and the hypotenuse, the opposite and the 00:01:16.460 --> 00:01:20.050 hypotenuse, and then we'll know which trig functions we 00:01:20.050 --> 00:01:21.370 should be dealing with. 00:01:21.370 --> 00:01:25.960 So we know this side, this pink side-- and this pink side, it 00:01:25.960 --> 00:01:28.090 should be almost second nature by now, is the 00:01:28.090 --> 00:01:28.450 hypotenuse, right? 00:01:28.450 --> 00:01:31.050 It's the longest side, and it's opposite the right angle. 00:01:31.050 --> 00:01:34.600 So we know the hypotenuse, and what are we solving for? 00:01:34.600 --> 00:01:36.480 Well, this is the angle we know, so we're solving 00:01:36.480 --> 00:01:38.000 for the opposite side. 00:01:38.000 --> 00:01:40.520 So we're solving for the opposite side, and we know the 00:01:40.520 --> 00:01:45.000 hypotenuse, so what trig function will we probably use? 00:01:45.000 --> 00:01:47.360 Let's write out our mnemonic. 00:01:47.360 --> 00:01:52.540 SOH CAH TOA. 00:01:52.540 --> 00:01:54.120 So what did I say before? 00:01:54.120 --> 00:01:56.600 We're solving for the opposite sign, and we know the 00:01:56.600 --> 00:02:01.245 hypotenuse, so we are going to use the opposite and 00:02:01.245 --> 00:02:02.470 the hypotenuse. 00:02:02.470 --> 00:02:04.600 So which of these is that? 00:02:04.600 --> 00:02:06.200 The opposite and the hypotenuse, the o 00:02:06.200 --> 00:02:07.420 and the h, right? 00:02:07.420 --> 00:02:10.460 It's SOH, right? 00:02:10.460 --> 00:02:13.250 And SOH says that sine is equal to the opposite 00:02:13.250 --> 00:02:15.180 over the hypotenuse. 00:02:15.180 --> 00:02:20.050 Sine of an angle-- in this case, it's this angle-- sine of 00:02:20.050 --> 00:02:24.900 0.36 radians-- remember, this is radians we're dealing with, 00:02:24.900 --> 00:02:28.910 not degrees-- is equal to the opposite side. 00:02:28.910 --> 00:02:31.730 And the opposite side is this green side right here, so I'll 00:02:31.730 --> 00:02:34.920 just write opposite instead of writing o because o would look 00:02:34.920 --> 00:02:38.720 like a 0-- is equal to the opposite side over the 00:02:38.720 --> 00:02:40.470 hypotenuse, right? 00:02:40.470 --> 00:02:41.920 This is just sine is equal to opposite over hypotenuse. 00:02:41.920 --> 00:02:43.130 Well, what's the hypotenuse length? 00:02:43.130 --> 00:02:44.240 Well, it's the square root of 73. 00:02:47.600 --> 00:02:48.970 These are p's, by the way. 00:02:48.970 --> 00:02:52.850 I know they don't look like p's, but opposite. 00:02:52.850 --> 00:02:58.350 So the opposite side-- we're just multiplying both sides by 00:02:58.350 --> 00:03:03.680 the square root of 73-- is equal to the square root of 73 00:03:03.680 --> 00:03:11.200 times the sine of 0.36 radians. 00:03:11.200 --> 00:03:14.450 Now, once again, I don't know what the sign of 0.36 radians 00:03:14.450 --> 00:03:17.440 is in my head, but I'll tell you the answer. 00:03:17.440 --> 00:03:23.000 The sine of 0.36 radians is equal to-- I'm just rewriting 00:03:23.000 --> 00:03:26.630 this-- the sine of 0.36 radians, if you looked it up on 00:03:26.630 --> 00:03:32.550 a table or if you used your calculator in radian mode, is 3 00:03:32.550 --> 00:03:36.870 square roots of 73 over 73. 00:03:36.870 --> 00:03:38.560 And, of course, your calculator is going to give you 00:03:38.560 --> 00:03:41.420 something-- it'll give you some decimal number. 00:03:41.420 --> 00:03:43.390 I won't write it this way. 00:03:43.390 --> 00:03:46.470 So just remember, this is this. 00:03:46.470 --> 00:03:47.490 And I just looked that up. 00:03:47.490 --> 00:03:49.270 There's no magic there. 00:03:49.270 --> 00:03:52.740 And the square root-- or you could use a calculator. 00:03:52.740 --> 00:03:55.020 On the Khan Academy, when you do problems, it'll actually 00:03:55.020 --> 00:03:58.270 tell you what it is, so you don't have to use a calculator. 00:03:58.270 --> 00:03:59.350 So now we just simplify. 00:03:59.350 --> 00:04:03.660 Square root of 73 times square root of 73 is 73 over 73 is 00:04:03.660 --> 00:04:05.760 equal to 1, so these all cancel out. 00:04:05.760 --> 00:04:09.340 And we get the answer of 3. 00:04:09.340 --> 00:04:13.510 So this side right here is 3. 00:04:13.510 --> 00:04:15.900 And just out of curiosity, if you wanted to solve for this 00:04:15.900 --> 00:04:18.460 side, there's two ways we could do it, right? 00:04:18.460 --> 00:04:21.680 We could use the Pythagorean theorem, because, you know, a 00:04:21.680 --> 00:04:24.520 squared plus b squared is equal to c squared, or we 00:04:24.520 --> 00:04:26.510 could use trigonometry. 00:04:26.510 --> 00:04:28.400 I'll let you guess what trig function-- 00:04:28.400 --> 00:04:29.490 actually, let's do that. 00:04:29.490 --> 00:04:33.330 Let's figure out that side using trigonometry, and then 00:04:33.330 --> 00:04:37.200 let's figure out that side using the Pythagorean theorem, 00:04:37.200 --> 00:04:41.600 just to show that everything fits together in math. 00:04:41.600 --> 00:04:44.660 So I wrote that 3 there, so I can erase all of this stuff. 00:04:51.610 --> 00:04:53.050 Let me erase it. 00:04:58.110 --> 00:04:59.840 I shouldn't have erased the SOHCAHTOA. 00:04:59.840 --> 00:05:02.570 Actually, we should have that memorized by now. 00:05:02.570 --> 00:05:02.920 SOHCAHTOA. 00:05:07.240 --> 00:05:10.560 All right, so let's figure out what this orange side is here. 00:05:10.560 --> 00:05:13.150 And if you think about it, we could do it a bunch of ways. 00:05:13.150 --> 00:05:15.110 We could say, well, this is the adjacent side, right? 00:05:15.110 --> 00:05:16.990 Because we know the opposite and we know the hypotenuse. 00:05:16.990 --> 00:05:22.020 So we could either use-- we know the opposite, so we could 00:05:22.020 --> 00:05:26.260 say what trig function uses the opposite and the adjacent? 00:05:26.260 --> 00:05:28.540 Well, that's tangent function, right? 00:05:28.540 --> 00:05:35.970 So we could say tangent of 0.36-- let's call 00:05:35.970 --> 00:05:37.110 this side A, right? 00:05:37.110 --> 00:05:38.720 A for adjacent. 00:05:38.720 --> 00:05:44.210 Tangent of 0.36 is equal to the opposite, 3, over 00:05:44.210 --> 00:05:46.770 the adjacent, over A. 00:05:46.770 --> 00:05:49.720 Is there another trigonometry way we could think about this? 00:05:49.720 --> 00:05:51.580 Well, we also know the hypotenuse. 00:05:51.580 --> 00:05:55.730 What trig function uses the hypotenuse and the adjacent? 00:05:55.730 --> 00:05:59.940 Well, if you remember, SOH CAH TOA. 00:05:59.940 --> 00:06:03.730 CAH, cosine is adjacent over hypotenuse. 00:06:03.730 --> 00:06:11.260 So we could say cosine of 0.36 is equal to adjacent 00:06:11.260 --> 00:06:16.040 over square root of 73. 00:06:16.040 --> 00:06:18.630 And I'll just write SOHCAHTOA here, just so you can 00:06:18.630 --> 00:06:21.900 confirm what I'm doing. 00:06:21.900 --> 00:06:27.660 The TOA says the tangent is equal to the opposite, 3, over 00:06:27.660 --> 00:06:32.840 the adjacent, and CAH tells us the cosine is equal to the 00:06:32.840 --> 00:06:36.810 adjacent over the hypotenuse. 00:06:36.810 --> 00:06:38.300 So we could solve either one of these. 00:06:38.300 --> 00:06:41.820 If we use the second formula, we would get the adjacent side 00:06:41.820 --> 00:06:48.980 is equal to the square root of 73 times the cosine of 0.36, 00:06:48.980 --> 00:06:51.610 and then if you use your calculator in radian mode, or 00:06:51.610 --> 00:07:01.650 I'll just tell you, that cosine of 0.36 radians is equal to 8 00:07:01.650 --> 00:07:06.810 square roots of 73 over 73. 00:07:06.810 --> 00:07:09.300 And you can confirm that by getting a decimal number and 00:07:09.300 --> 00:07:11.780 then-- making sure once again your calculator is not in 00:07:11.780 --> 00:07:12.880 degree mode, but in radian mode. 00:07:12.880 --> 00:07:14.370 I think that's actually the default mode in a lot of 00:07:14.370 --> 00:07:15.960 calculators-- and solving for this. 00:07:15.960 --> 00:07:19.720 But once again, the 73, this square root of 73 times this 00:07:19.720 --> 00:07:23.100 square root of 73, is equal to 73, and then divided by 73. 00:07:23.100 --> 00:07:24.720 These all cancel out. 00:07:24.720 --> 00:07:26.320 How convenient, huh? 00:07:26.320 --> 00:07:28.460 And you get 8. 00:07:28.460 --> 00:07:31.220 So the adjacent side is equal to 8. 00:07:31.220 --> 00:07:33.500 And so if we'd solved for A here using the tangent 00:07:33.500 --> 00:07:36.300 function, we should've also gotten the adjacent 00:07:36.300 --> 00:07:37.570 side is equal to 8. 00:07:37.570 --> 00:07:42.510 And just to show you that everything works out from other 00:07:42.510 --> 00:07:49.250 concepts, let me show you this using the Pythagorean theorem. 00:07:53.990 --> 00:08:03.100 So 8 squared plus 3 squared should equal 00:08:03.100 --> 00:08:03.970 the hypotenuse squared. 00:08:03.970 --> 00:08:08.100 The square root of 73 squared. 00:08:08.100 --> 00:08:13.530 Well, 8 squared is 64 plus 9 should equal-- what's the 00:08:13.530 --> 00:08:14.820 square root of 73 squared? 00:08:14.820 --> 00:08:16.690 Right, it's 73. 00:08:16.690 --> 00:08:19.250 And, of course, 64 plus 9 is 73. 00:08:19.250 --> 00:08:22.600 And sure enough, that equals 73, so it works. 00:08:22.600 --> 00:08:26.470 Isn't that interesting how math just kind of fits together? 00:08:26.470 --> 00:08:29.180 I think at this point you're ready to try the modules, the 00:08:29.180 --> 00:08:32.790 Trigonometry II modules, and I guess let me know if you 00:08:32.790 --> 00:08:35.510 have any problems, or if you want to see more videos. 00:08:35.510 --> 00:08:37.010 Have fun!

Using Trig Functions

https://www.youtube.com/watch?v=znR9tW4AiZI

vtt

https://www.youtube.com/api/timedtext?v=znR9tW4AiZI&ei=fmeUZcGTMby-mLAPqJKWuAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8FD5F92DE6EB07122128DE823459F8DF014B85D9.6EF1CD560DF2224C9550102F2598D970A746FDBB&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.180 --> 00:00:03.150 We're now going to do a few examples to actually show 00:00:03.150 --> 00:00:06.320 you why the trig functions are actually useful. 00:00:06.320 --> 00:00:08.760 So let's get started with a problem. 00:00:08.760 --> 00:00:11.065 Let's say I have this right triangle. 00:00:15.860 --> 00:00:17.040 That's my right triangle. 00:00:20.990 --> 00:00:22.470 There's the right angle. 00:00:22.470 --> 00:00:25.950 And let's say I know that the measure of this angle 00:00:25.950 --> 00:00:31.710 is pi over 4 radians. 00:00:31.710 --> 00:00:33.980 And I'll just write rad for short. 00:00:33.980 --> 00:00:37.230 If the measure of this angle is pi over 4 radians, and I also 00:00:37.230 --> 00:00:40.900 know that this side of the triangle-- this side right 00:00:40.900 --> 00:00:48.080 here-- is 10 square roots of 2. 00:00:48.080 --> 00:00:51.970 So I know this side of the triangle. 00:00:51.970 --> 00:00:54.690 I know this angle, which is pi over 4 radians. 00:00:54.690 --> 00:01:00.420 And now, the question is, what is this side of the triangle? 00:01:00.420 --> 00:01:01.410 I'm going to highlight that. 00:01:01.410 --> 00:01:03.680 And let me make it in orange. 00:01:06.190 --> 00:01:07.900 So let's figure out what we know and what we 00:01:07.900 --> 00:01:09.390 need to figure out. 00:01:09.390 --> 00:01:11.500 We know the angle, pi over 4 radians. 00:01:11.500 --> 00:01:13.970 And actually, turns out if you were to convert that 00:01:13.970 --> 00:01:16.290 to degrees, it would be 45 degrees. 00:01:16.290 --> 00:01:18.120 And we know-- what side is this? 00:01:18.120 --> 00:01:21.440 This is the hypotenuse of the triangle, right? 00:01:21.440 --> 00:01:23.130 And what are we trying to figure out? 00:01:23.130 --> 00:01:26.720 Are we trying to figure out the hypotenuse, the adjacent 00:01:26.720 --> 00:01:30.390 side to the angle, or the opposite side to the angle? 00:01:30.390 --> 00:01:32.540 Well, this is the hypotenuse, we already know that. 00:01:32.540 --> 00:01:33.430 This is the opposite side. 00:01:39.510 --> 00:01:41.090 This is the opposite side. 00:01:41.090 --> 00:01:43.080 And this yellow side is the adjacent side, right? 00:01:43.080 --> 00:01:45.260 It's just adjacent to this angle. 00:01:45.260 --> 00:01:50.150 So we know the angle, we know the hypotenuse, and we want to 00:01:50.150 --> 00:01:52.060 figure out the adjacent side. 00:01:52.060 --> 00:01:54.240 So let me ask you a question. 00:01:54.240 --> 00:01:58.910 What trig function deals with the adjacent side 00:01:58.910 --> 00:02:00.360 and the hypotenuse? 00:02:00.360 --> 00:02:01.620 Because we have the adjacent side is what we want 00:02:01.620 --> 00:02:04.005 to figure out, and we know the hypotenuse. 00:02:04.005 --> 00:02:05.770 Well, let's write down our mnemonic, just in 00:02:05.770 --> 00:02:08.390 case you forgot it. 00:02:08.390 --> 00:02:08.836 SOHCAHTOA. 00:02:16.820 --> 00:02:19.650 So which one uses adjacent and hypotenuse? 00:02:19.650 --> 00:02:19.990 Right? 00:02:19.990 --> 00:02:21.750 It's CAH. 00:02:21.750 --> 00:02:24.770 And CAH, the c is for what? 00:02:24.770 --> 00:02:27.150 The c is for cosine. 00:02:27.150 --> 00:02:30.980 Cosine of an angle-- let's just call it any angle-- is equal to 00:02:30.980 --> 00:02:35.870 the adjacent over the hypotenuse. 00:02:35.870 --> 00:02:40.460 So let's use this information to try to solve for this orange 00:02:40.460 --> 00:02:42.790 side, or this yellow side. 00:02:42.790 --> 00:02:48.350 So we know that cosine of pi over 4 radians-- so let's say 00:02:48.350 --> 00:02:55.580 cosine of pi over 4-- must equal this adjacent 00:02:55.580 --> 00:02:56.820 side right here. 00:02:56.820 --> 00:03:00.190 Let's just call that a. a for adjacent. 00:03:00.190 --> 00:03:04.030 The adjacent side divided by the hypotenuse. 00:03:04.030 --> 00:03:05.540 The hypotenuse is this side. 00:03:05.540 --> 00:03:07.590 And in the problem, we were given that it's 00:03:07.590 --> 00:03:08.730 10 square roots of 2. 00:03:15.250 --> 00:03:17.790 So we can solve for a by multiplying both sides of 00:03:17.790 --> 00:03:20.130 this equation by 10 square roots of 2. 00:03:20.130 --> 00:03:22.180 And we will get-- because, right? 00:03:22.180 --> 00:03:25.650 If we just multiply times 10 square root of 2, 00:03:25.650 --> 00:03:26.580 these cancel out. 00:03:26.580 --> 00:03:29.650 And then you get a 10 square root of 2 here. 00:03:29.650 --> 00:03:37.840 So you get a is equal to 10 square roots of 2 times 00:03:37.840 --> 00:03:42.710 the cosine of pi over 4. 00:03:42.710 --> 00:03:45.860 Now you're probably saying, Sal, this does not look too 00:03:45.860 --> 00:03:50.170 simple, and I don't know how big the cosine of pi over 4 is. 00:03:50.170 --> 00:03:51.450 What do I do? 00:03:51.450 --> 00:03:54.650 Well, no one has the trig functions, or the values of 00:03:54.650 --> 00:03:56.450 the trig functions memorized. 00:03:56.450 --> 00:03:57.730 There's a couple of ways to do it. 00:03:57.730 --> 00:04:00.710 Either I could give you what the cosine of pi over 4 is. 00:04:00.710 --> 00:04:02.270 That's sometimes given in a problem. 00:04:02.270 --> 00:04:05.400 Or you can make sure that your calculator is set to radians 00:04:05.400 --> 00:04:09.400 and you can just type in pi divided by 4-- which is roughly 00:04:09.400 --> 00:04:12.530 0.79-- and then press the cosine button. 00:04:12.530 --> 00:04:13.960 You finally know what it's good for. 00:04:13.960 --> 00:04:15.250 And you'll get a value. 00:04:15.250 --> 00:04:17.370 Or-- and this is kind of the old school way of doing it-- 00:04:17.370 --> 00:04:20.554 there are trig tables where you could look up what cosine 00:04:20.554 --> 00:04:22.540 of pi over 4 is in a table. 00:04:22.540 --> 00:04:25.430 Since I don't have any of that at my disposal right now, 00:04:25.430 --> 00:04:29.040 I'll just tell you what the cosine of pi over 4 is. 00:04:29.040 --> 00:04:35.760 The cosine of pi over 4 is square root of 2 over 2. 00:04:35.760 --> 00:04:39.970 So a, which is the adjacent side-- a for adjacent-- is 00:04:39.970 --> 00:04:46.470 equal to 10 square roots of 2 times square root of 2 over 2. 00:04:46.470 --> 00:04:48.750 Remember, to get the square root of 2 over 2, you might 00:04:48.750 --> 00:04:49.330 be a little confused. 00:04:49.330 --> 00:04:51.210 You're like, how did Sal get that? 00:04:51.210 --> 00:04:54.420 All I said is, the cosine of pi over 4 is square 00:04:54.420 --> 00:04:55.090 root of 2 over 2. 00:04:55.090 --> 00:04:57.010 And that's not something that-- well, actually, this one you 00:04:57.010 --> 00:04:59.420 might know offhand, because of the 45 degree angle. 00:04:59.420 --> 00:05:01.490 But this isn't something that people memorize. 00:05:01.490 --> 00:05:03.275 This is something you would look up, or it's given in 00:05:03.275 --> 00:05:05.660 the problem, or you'd use a calculator for. 00:05:05.660 --> 00:05:07.610 And a calculator, of course, wouldn't give you square 00:05:07.610 --> 00:05:08.020 root of 2 over 2. 00:05:08.020 --> 00:05:11.910 It'd give you a decimal number that's not obviously 00:05:11.910 --> 00:05:13.180 square root of 2 over 2. 00:05:13.180 --> 00:05:15.800 But anyway, I told you that the cosine of pi over 4 is the 00:05:15.800 --> 00:05:17.310 square root of 2 over 2. 00:05:17.310 --> 00:05:20.050 And so if we multiply, what's the square root of 2 over 2? 00:05:20.050 --> 00:05:22.560 What's the square root of 2 times the square root of 2? 00:05:22.560 --> 00:05:23.140 It's 2. 00:05:23.140 --> 00:05:26.690 So that's 2, and then that cancels with that 2. 00:05:26.690 --> 00:05:29.570 And so everything cancels except for the 10. 00:05:29.570 --> 00:05:32.340 So the adjacent side is equal to 10. 00:05:35.230 --> 00:05:36.140 Let's do another one. 00:05:43.200 --> 00:05:45.230 Let me delete this. 00:05:50.260 --> 00:05:52.180 Give me 1 second. 00:05:52.180 --> 00:05:54.550 I'm actually-- this is one of the few modules that I'm not 00:05:54.550 --> 00:05:56.770 generating the problems on the fly, because I need to make 00:05:56.770 --> 00:05:59.200 sure that I actually have the trig function values 00:05:59.200 --> 00:06:00.920 before I do the problem. 00:06:00.920 --> 00:06:05.820 So let's say I have another right triangle. 00:06:05.820 --> 00:06:07.470 I probably shouldn't have deleted that last one. 00:06:07.470 --> 00:06:11.320 So let's see, this is my right triangle. 00:06:11.320 --> 00:06:13.350 How much time do I have-- about 4 minutes left. 00:06:13.350 --> 00:06:14.700 Should be enough. 00:06:14.700 --> 00:06:16.880 So this is my right triangle. 00:06:16.880 --> 00:06:23.470 And I know the angle-- let's call this--. 00:06:23.470 --> 00:06:29.970 I know this angle right here is 0.54 radians. 00:06:29.970 --> 00:06:38.240 And I also know that this side right here is 3 units long. 00:06:38.240 --> 00:06:42.680 And I want to figure out this side. 00:06:42.680 --> 00:06:45.390 So what do I know? 00:06:45.390 --> 00:06:49.300 Well, this side is what side relative to the angle? 00:06:49.300 --> 00:06:50.930 It's the opposite side, right? 00:06:50.930 --> 00:06:52.760 Because the angle is here, and we go opposite the angle. 00:06:52.760 --> 00:06:55.110 So this is the opposite side. 00:06:55.110 --> 00:06:56.080 And what's this side? 00:06:56.080 --> 00:06:59.220 Is this the adjacent side, or is it the hypotenuse? 00:06:59.220 --> 00:07:00.370 Well, this is the hypotenuse, right? 00:07:00.370 --> 00:07:02.600 The long side, and it's opposite the right angle. 00:07:02.600 --> 00:07:05.070 So this is the adjacent side. 00:07:05.070 --> 00:07:08.510 So what trig function uses opposite and adjacent? 00:07:08.510 --> 00:07:10.400 Let's write down SOHCAHTOA again. 00:07:10.400 --> 00:07:10.760 SOHCAHTOA. 00:07:15.220 --> 00:07:17.160 TOA uses opposite and adjacent. 00:07:17.160 --> 00:07:17.420 OA. 00:07:20.620 --> 00:07:22.540 So T for tangent, right? 00:07:22.540 --> 00:07:23.840 TOA. 00:07:23.840 --> 00:07:30.280 So tangent is equal to opposite over adjacent. 00:07:30.280 --> 00:07:31.520 So let's use that. 00:07:31.520 --> 00:07:35.040 So let's take the tangent of 0.54 radians. 00:07:35.040 --> 00:07:44.320 So the tangent of 0.54 will equal the side opposite to it. 00:07:44.320 --> 00:07:46.600 So that's 3, right? 00:07:46.600 --> 00:07:48.070 The opposite side is 3. 00:07:48.070 --> 00:07:49.750 Over the adjacent side. 00:07:49.750 --> 00:07:51.550 Well, once again, the adjacent side is what we don't know. 00:07:51.550 --> 00:07:55.520 So we have to solve for a. 00:07:55.520 --> 00:08:03.020 So if we multiply both sides by a, we get a tan of 0.54-- 00:08:03.020 --> 00:08:08.860 we could do that because we know it's not 0-- equals 3. 00:08:08.860 --> 00:08:18.240 Or a is equal to 3 divided by the tangent of 0.54. 00:08:18.240 --> 00:08:22.500 So once again, I don't have memorized what the tangent of 00:08:22.500 --> 00:08:29.770 0.54 is, but I will tell you what it is because you also 00:08:29.770 --> 00:08:30.480 don't have it memorized. 00:08:30.480 --> 00:08:31.990 Or you could use a calculator to figure it out if you 00:08:31.990 --> 00:08:34.060 had a radian function. 00:08:34.060 --> 00:08:39.730 The tangent of 0.54 is equal to-- let me make sure 00:08:39.730 --> 00:08:41.080 I have this right. 00:08:41.080 --> 00:08:41.960 Oh, right. 00:08:41.960 --> 00:08:45.270 The tangent of 0.54 is 3/5. 00:08:45.270 --> 00:08:51.830 So then a is equal to 3 over 3/5. 00:08:54.500 --> 00:08:56.850 Right, the adjacent side-- now, once again, how 00:08:56.850 --> 00:08:57.810 did I get this 3/5? 00:08:57.810 --> 00:08:58.820 Well, I just told you. 00:08:58.820 --> 00:09:00.200 Or you can use a calculator to know that the 00:09:00.200 --> 00:09:03.250 tangent of 0.54 is 3/5. 00:09:03.250 --> 00:09:05.130 And of course, I'm using numbers that work out 00:09:05.130 --> 00:09:07.400 well, just so that the fractions all cancel. 00:09:07.400 --> 00:09:09.580 So we know that the adjacent side is equal to-- when you 00:09:09.580 --> 00:09:12.550 divide by fractions, it's like multiplying by the numerator. 00:09:12.550 --> 00:09:14.420 Multiplying by the inverse. 00:09:14.420 --> 00:09:17.430 So times 5/3. 00:09:17.430 --> 00:09:20.370 So the adjacent side is equal to 5. 00:09:20.370 --> 00:09:21.120 There. 00:09:21.120 --> 00:09:21.930 There you go. 00:09:21.930 --> 00:09:23.870 So let's just think about what I always do. 00:09:23.870 --> 00:09:27.190 I think about what I have, what sides I have, and what 00:09:27.190 --> 00:09:28.210 side I want to solve for. 00:09:28.210 --> 00:09:30.130 And in this case, it was the opposite side I had, and I 00:09:30.130 --> 00:09:31.740 wanted to solve for the adjacent side. 00:09:31.740 --> 00:09:35.170 And I said, what trig function involves those 2 sides? 00:09:35.170 --> 00:09:36.940 The opposite and the adjacent. 00:09:36.940 --> 00:09:37.880 I wrote down SOHCAHTOA. 00:09:37.880 --> 00:09:39.410 I said, oh, TOA. 00:09:39.410 --> 00:09:40.370 Opposite and adjacent. 00:09:40.370 --> 00:09:41.320 That's tan. 00:09:41.320 --> 00:09:43.500 So I took the tan of the angle. 00:09:43.500 --> 00:09:45.960 And then I said, the tan of the angle is equal to the opposite 00:09:45.960 --> 00:09:47.820 side divided by the adjacent side. 00:09:47.820 --> 00:09:48.780 That's right here. 00:09:48.780 --> 00:09:50.430 And then I just solved for the adjacent side. 00:09:50.430 --> 00:09:52.740 And of course, I used a calculator, or I told you 00:09:52.740 --> 00:09:55.880 what the tangent of 0.54 is. 00:09:55.880 --> 00:09:58.110 I think I'll do a couple more of these problems in the next 00:09:58.110 --> 00:09:59.960 module, but I'm out of time for now. 00:09:59.960 --> 00:10:01.480 Have fun.

Radian and degree

https://www.youtube.com/watch?v=9zspW8u6kQM

vtt

https://www.youtube.com/api/timedtext?v=9zspW8u6kQM&ei=fWeUZa2zBO--mLAP8uqGgAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249837&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=8CDD206EAC67345A9F54BD7AD56AF722EA222C22.7604A6F1A6D84820097BEF959D91AEDF2B488075&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.910 --> 00:00:03.770 Welcome to the presentation on radians and degrees. 00:00:03.770 --> 00:00:07.010 So you all are probably already reasonably familiar with 00:00:07.010 --> 00:00:07.950 the concept of degrees. 00:00:07.950 --> 00:00:10.310 I think in our angles models we actually drill you 00:00:10.310 --> 00:00:12.820 through a bunch of problems. 00:00:12.820 --> 00:00:23.460 You're probably familiar that a right angle is 90 degrees. 00:00:23.460 --> 00:00:28.650 Or half a right angle -- 45 degrees. 00:00:28.650 --> 00:00:32.630 And you're also probably familiar with the concept that 00:00:32.630 --> 00:00:36.610 in a circle -- and that's my best adept at a circle -- in a 00:00:36.610 --> 00:00:41.010 circle, there are 360 degrees. 00:00:41.010 --> 00:00:44.750 So today I'm going to introduce you to another measure or unit 00:00:44.750 --> 00:00:47.270 for angles and this is called a radian. 00:00:52.160 --> 00:00:53.450 So what is a radian? 00:00:53.450 --> 00:00:55.650 So I'll start with the definition and I think this 00:00:55.650 --> 00:00:57.105 might give you a little intuition for why it's 00:00:57.105 --> 00:00:59.910 even called radian. 00:00:59.910 --> 00:01:02.850 Let me use this circle tool and actually draw a nice circle. 00:01:10.060 --> 00:01:14.270 I'm still using the radian tool, the circle tool. 00:01:14.270 --> 00:01:14.530 OK. 00:01:19.430 --> 00:01:21.630 This is a radius of length r. 00:01:21.630 --> 00:01:25.500 A radian is the angle that subtends an arc. 00:01:25.500 --> 00:01:30.210 And all subtend means is if this is angle, and this is 00:01:30.210 --> 00:01:34.520 the arc, this angle subtends this arc and this arc 00:01:34.520 --> 00:01:36.020 subtends this angle. 00:01:36.020 --> 00:01:41.050 So a radian -- one radian -- is the angle that subtends an arc 00:01:41.050 --> 00:01:44.130 that's the length of the radius. 00:01:44.130 --> 00:01:46.780 So the length of this is also r. 00:01:46.780 --> 00:01:50.440 And this angle is one radian. 00:01:50.440 --> 00:01:51.140 i think that's messy. 00:01:51.140 --> 00:01:52.490 Let me draw a bigger circle. 00:01:55.010 --> 00:01:56.640 Here you go. 00:01:56.640 --> 00:01:57.860 And I'm going to do this because I was wondering 00:01:57.860 --> 00:01:58.780 why they do radians. 00:01:58.780 --> 00:02:00.300 We all know degrees. 00:02:00.300 --> 00:02:02.090 But actually when you think about it it actually makes a 00:02:02.090 --> 00:02:03.100 reasonable amount of sense. 00:02:03.100 --> 00:02:05.873 So let me use the line tool now. 00:02:12.980 --> 00:02:18.990 And let's say that this radius is a length r and that this arc 00:02:18.990 --> 00:02:21.460 right here is also length r. 00:02:21.460 --> 00:02:28.210 Then this angle, what's called theta, is equal to one radian. 00:02:28.210 --> 00:02:30.220 And now it makes sense that they call it a radian. 00:02:30.220 --> 00:02:32.440 It's kind of like a radius. 00:02:32.440 --> 00:02:35.100 So let me ask a question: how many radians are 00:02:35.100 --> 00:02:37.420 there in a circle? 00:02:37.420 --> 00:02:41.300 Well, if this is r, what is the whole circumference 00:02:41.300 --> 00:02:42.050 of a circle? 00:02:44.630 --> 00:02:46.540 It's 2 pi r, right? 00:02:46.540 --> 00:02:50.050 You know that from the basic geometry module. 00:02:50.050 --> 00:02:55.850 So if the radian is the angle that sub tends an arc of r, 00:02:55.850 --> 00:03:03.650 then the angle that subtends an arc of 2 pi r is 2 pi radians. 00:03:03.650 --> 00:03:06.970 So this angle is 2 pi radians. 00:03:12.510 --> 00:03:14.820 If you're still confused, think of it this way. 00:03:14.820 --> 00:03:20.390 An angle of 2 pi radians going all the way around subtends 00:03:20.390 --> 00:03:22.650 an arc of 2 pi radiuses. 00:03:22.650 --> 00:03:23.500 Or radii. 00:03:23.500 --> 00:03:26.460 I don't know how to say the plural of radius. 00:03:26.460 --> 00:03:27.110 Maybe it's radians. 00:03:27.110 --> 00:03:30.130 And I don't know. 00:03:30.130 --> 00:03:32.630 So why am I going through all of this mess and confusing you? 00:03:32.630 --> 00:03:35.580 I just want to one, give you an intuition for why it's called 00:03:35.580 --> 00:03:38.130 a radian and kind of how it relates to a circle. 00:03:38.130 --> 00:03:41.890 And then given that there 2 pi radians in a circle, we can now 00:03:41.890 --> 00:03:46.980 figure out a relationship between radians and degrees. 00:03:46.980 --> 00:03:49.920 Let me delete this. 00:03:49.920 --> 00:03:54.190 So we said in a circle, there are 2 pi radians. 00:03:57.340 --> 00:03:58.970 And how many degrees are there in a circle? 00:03:58.970 --> 00:04:00.800 If we went around a whole circle how many degrees? 00:04:00.800 --> 00:04:04.360 Well that's equal to 360 degrees. 00:04:07.080 --> 00:04:07.520 So there. 00:04:07.520 --> 00:04:09.620 We have an equation that sets up a conversion between 00:04:09.620 --> 00:04:10.950 radians and degrees. 00:04:10.950 --> 00:04:19.390 So one radian is equal to 360 over 2 pi degrees. 00:04:19.390 --> 00:04:22.570 I just divided both sides by 2 pi. 00:04:22.570 --> 00:04:27.040 Which equals 180 over pi degrees. 00:04:29.710 --> 00:04:31.080 Similarly, we could have done the other way. 00:04:31.080 --> 00:04:34.000 We could have divided both sides by 360 and we could have 00:04:34.000 --> 00:04:38.530 said 1 degree -- I'm just going to divide both sides but 00:04:38.530 --> 00:04:39.970 360 and I'm flipping it. 00:04:39.970 --> 00:04:45.410 1 degree is equal to 2 pi over 360 radians. 00:04:48.570 --> 00:04:53.260 Which equals pi over 180 radians. 00:04:53.260 --> 00:05:00.440 So then we have a conversion: 1 radian equals 180 over pi 00:05:00.440 --> 00:05:05.220 degrees and 1 degree equals pi over 180 radians. 00:05:05.220 --> 00:05:06.980 Amd if you ever forget these, it doesn't hurt 00:05:06.980 --> 00:05:08.740 to to memorize this. 00:05:08.740 --> 00:05:12.520 But if you ever forget it, I always go back to this. 00:05:12.520 --> 00:05:15.810 That 2 pi radians is equal to 360 degrees. 00:05:15.810 --> 00:05:21.450 Or another way that actually makes the algebra a little 00:05:21.450 --> 00:05:26.545 simpler is if you just think of a half circle. 00:05:26.545 --> 00:05:31.550 A half circle -- this angle -- is a 180 degrees, right? 00:05:35.210 --> 00:05:36.120 That's a degree sign. 00:05:36.120 --> 00:05:37.810 I could also write degrees out. 00:05:37.810 --> 00:05:39.680 And that's also equal to pi radians. 00:05:42.680 --> 00:05:46.250 So pi radians equal 180 degrees and we can get to see the math. 00:05:46.250 --> 00:05:57.250 1 radian equals 180 over pi degrees or 1 degree is equal 00:05:57.250 --> 00:06:00.940 to pi over 180 radians. 00:06:00.940 --> 00:06:02.495 So let's do a couple of problems were you'll get 00:06:02.495 --> 00:06:03.540 the intuition for this. 00:06:03.540 --> 00:06:09.010 If I asked you 45 degrees -- to convert that into radians. 00:06:12.440 --> 00:06:18.410 Well, we know that 1 degree os pi over 180 radians. 00:06:18.410 --> 00:06:32.910 So 45 degrees is equal to 45 times pi over 180 radians. 00:06:32.910 --> 00:06:36.850 And let's see, 45 divided by 180. 00:06:36.850 --> 00:06:42.360 45 goes into 180 four times so this equals pi over 4 radians. 00:06:45.650 --> 00:06:49.600 45 degrees is equal to pi over 4 radians. 00:06:49.600 --> 00:06:52.610 And just keep in mind, these are just two different units 00:06:52.610 --> 00:06:55.070 or two different ways of measuring angles. 00:06:55.070 --> 00:06:56.590 And the reason why I do this is this is actually the 00:06:56.590 --> 00:06:59.700 mathematical standard for measuring angles, although most 00:06:59.700 --> 00:07:01.690 of us are more familiar with degrees just from 00:07:01.690 --> 00:07:03.030 everyday life. 00:07:03.030 --> 00:07:04.920 Let's do a couple of other examples. 00:07:04.920 --> 00:07:06.690 Just always remember: this 1 radian equals 00:07:06.690 --> 00:07:08.400 180 over pi degrees. 00:07:08.400 --> 00:07:10.200 1 degree equals pi over 180 radians. 00:07:10.200 --> 00:07:12.630 If you ever get confused, just write this out. 00:07:12.630 --> 00:07:15.400 this is what I do because I always forget whether it's 00:07:15.400 --> 00:07:17.570 pi over 180 or 180 over pi. 00:07:17.570 --> 00:07:21.550 I just remember pi radians is equal to 180 degrees. 00:07:21.550 --> 00:07:23.840 Let's do another one. 00:07:23.840 --> 00:07:33.060 So if I were to say pi over 2 radians equals 00:07:33.060 --> 00:07:33.765 how many degrees? 00:07:37.480 --> 00:07:40.660 Well I already forgot what I had just written so I just 00:07:40.660 --> 00:07:45.565 remind myself that pi radians is equal to 180 degrees. 00:07:55.720 --> 00:07:57.930 Oh, my wife just got home, so I'm just going to have to leave 00:07:57.930 --> 00:08:02.670 the presentation like that and I will continue it later. 00:08:02.670 --> 00:08:05.120 Actually, let me just finish this problem and then I'll 00:08:05.120 --> 00:08:07.270 go attend to my wife. 00:08:07.270 --> 00:08:12.140 But we know that pi radians is equal to 180 degrees, right? 00:08:12.140 --> 00:08:18.840 So one radian is equal to 180 over -- that's one radian -- is 00:08:18.840 --> 00:08:21.660 equal to 180 over pi degrees. 00:08:21.660 --> 00:08:23.470 I just figure out the formula again because 00:08:23.470 --> 00:08:24.490 I always forget it. 00:08:24.490 --> 00:08:25.500 So let's go back here. 00:08:25.500 --> 00:08:33.160 So pi over 2 radians is equal to pi over 2 times 00:08:33.160 --> 00:08:38.510 180 over pi degrees. 00:08:38.510 --> 00:08:41.585 And that equals 90 degrees. 00:08:47.240 --> 00:08:48.830 I'll do one more example. 00:08:54.480 --> 00:08:55.915 Let's say 30 degrees. 00:09:00.950 --> 00:09:03.200 Once again, I forgot the formula so I just remember 00:09:03.200 --> 00:09:10.960 that pi radians is equal to 180 degrees. 00:09:10.960 --> 00:09:19.150 So 1 degree is equal to pi over 180 radians. 00:09:19.150 --> 00:09:27.220 So 30 degrees is equal to 30 times pi over 180 radians 00:09:27.220 --> 00:09:31.320 which equals -- 30 goes into 180 six times. 00:09:31.320 --> 00:09:36.160 That equals pi over 6 radians. 00:09:36.160 --> 00:09:39.630 Hopefully you have a sense of how to convert between degrees 00:09:39.630 --> 00:09:42.070 and radians now and even why it's called a radian because 00:09:42.070 --> 00:09:45.880 it's very closely related to a radius and you'll feel 00:09:45.880 --> 00:09:50.210 comfortable when someone asks you to, I don't know, deal with 00:09:50.210 --> 00:09:52.410 radians as opposed to degrees. 00:09:52.410 --> 00:09:54.671 I'll see you in the next presentation.

Basic Trigonometry II

https://www.youtube.com/watch?v=QS4r_mqs-rY

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en

WEBVTT Kind: captions Language: en 00:00:01.060 --> 00:00:03.280 Welcome to the second part of the presentation on 00:00:03.280 --> 00:00:03.770 basic trigonometry. 00:00:03.770 --> 00:00:06.580 In the last 10 minutes, I had trouble getting in a lot of 00:00:06.580 --> 00:00:09.280 examples, so I want to do a couple more with you guys. 00:00:09.280 --> 00:00:09.780 OK. 00:00:09.780 --> 00:00:13.000 So, let me start over just because this got messy. 00:00:15.850 --> 00:00:17.740 And we're just going to do what we did in the last time around. 00:00:17.740 --> 00:00:20.310 So let me just draw another right triangle. 00:00:20.310 --> 00:00:21.845 And make sure it's not going to be too big. 00:00:25.410 --> 00:00:27.860 Here's my right triangle. 00:00:27.860 --> 00:00:30.270 And let me just throw out some random sides. 00:00:30.270 --> 00:00:31.660 Let me say that this is 6. 00:00:35.930 --> 00:00:39.960 Let's make this side 5. 00:00:39.960 --> 00:00:42.730 And then, if this is a right triangle, Pythagorean Theorem 00:00:42.730 --> 00:00:45.890 tells us that this would be the square root of what? 00:00:45.890 --> 00:00:55.420 36 plus 25 is equal to the square root of 61. 00:00:55.420 --> 00:00:56.600 I think that's right. 00:00:56.600 --> 00:00:59.270 I've gotten feedback on some of my other videos that I tend to 00:00:59.270 --> 00:01:01.840 get this type of addition wrong. 00:01:01.840 --> 00:01:03.390 I malfunction sometimes. 00:01:03.390 --> 00:01:03.950 But anyway. 00:01:03.950 --> 00:01:06.230 So this side is the square root of 61, and that's 00:01:06.230 --> 00:01:07.830 the hypotenuse. 00:01:07.830 --> 00:01:10.450 So let's get started with some problems. 00:01:10.450 --> 00:01:12.720 If I were to give you-- if I were to ask you. 00:01:12.720 --> 00:01:13.300 Well, let's see. 00:01:13.300 --> 00:01:18.560 Let's call this angle theta. 00:01:18.560 --> 00:01:25.070 And I want to know what is the tangent of theta? 00:01:25.070 --> 00:01:29.390 And we'll shorten that as tangent of-- tan of theta. 00:01:29.390 --> 00:01:33.080 What is the tangent of this angle right here? 00:01:33.080 --> 00:01:35.580 Well, you probably already forgot what the definition 00:01:35.580 --> 00:01:36.160 of tangent is. 00:01:36.160 --> 00:01:37.560 So I will repeat it. 00:01:37.560 --> 00:01:39.580 In fact, I will write up in this corner. 00:01:39.580 --> 00:01:40.700 Soh cah toa. 00:01:44.980 --> 00:01:47.610 So I think now your brain might be refreshed and you'll 00:01:47.610 --> 00:01:51.760 remember that toa is the mnemonic for tangent. 00:01:51.760 --> 00:01:54.000 And it says that tangent is equal to the opposite 00:01:54.000 --> 00:01:55.600 over the adjacent. 00:01:55.600 --> 00:01:58.660 So the tangent of theta is equal to the opposite side-- 00:01:58.660 --> 00:02:01.900 well, that's this side, that's the side of length 5-- 00:02:01.900 --> 00:02:04.160 over the adjacent side. 00:02:04.160 --> 00:02:05.350 That's this side. 00:02:05.350 --> 00:02:08.010 The side of length 6. 00:02:08.010 --> 00:02:09.110 That's pretty easy, huh? 00:02:09.110 --> 00:02:12.910 The tangent of theta is 5/6. 00:02:12.910 --> 00:02:14.410 And we'll just do a couple more. 00:02:14.410 --> 00:02:14.860 All right? 00:02:14.860 --> 00:02:16.550 We'll just go through all of the trig functions, or at least 00:02:16.550 --> 00:02:18.300 the basic trig functions. 00:02:18.300 --> 00:02:23.250 What is the sine of theta? 00:02:23.250 --> 00:02:24.780 Well, let's go back to our mnemonic. 00:02:24.780 --> 00:02:25.480 Soh cah toa. 00:02:25.480 --> 00:02:27.180 This is one of the few things in mathematics that you 00:02:27.180 --> 00:02:28.630 should probably memorize. 00:02:28.630 --> 00:02:30.680 It's kind of a funny word anyway. 00:02:30.680 --> 00:02:33.710 And soh-- to find sine. 00:02:33.710 --> 00:02:36.090 It tells us that sine is opposite over hypotenuse. 00:02:40.670 --> 00:02:44.840 Well, the opposite side, once again, is 5. 00:02:44.840 --> 00:02:46.420 And what's the hypotenuse? 00:02:46.420 --> 00:02:47.970 Well, the hypotenuse, we just figured out, was 00:02:47.970 --> 00:02:49.145 the square root of 61. 00:02:53.900 --> 00:02:57.170 And a lot of people don't like irrational denominators. 00:02:57.170 --> 00:02:59.150 So we can rationalize the denominator. 00:02:59.150 --> 00:03:01.390 And we do that by multiplying the numerator and the 00:03:01.390 --> 00:03:03.280 denominator by the square root of 61. 00:03:03.280 --> 00:03:09.570 So if we say that this is equal to 5 over the square root of 61 00:03:09.570 --> 00:03:13.080 times the square root of 61, over the square 00:03:13.080 --> 00:03:13.720 root of 61, right? 00:03:13.720 --> 00:03:14.600 We're just multiplying it by 1. 00:03:14.600 --> 00:03:17.940 Because this is the same thing top and bottom. 00:03:17.940 --> 00:03:24.440 This equals 5 square roots of 61 over-- what's the square 00:03:24.440 --> 00:03:26.350 root of 61 times the square root of 61? 00:03:26.350 --> 00:03:29.140 Oh yeah, it's 61. 00:03:29.140 --> 00:03:35.110 So the sine of theta is 5 square roots of 61 over 61. 00:03:35.110 --> 00:03:38.610 And then finally, let me make some space here. 00:03:38.610 --> 00:03:39.915 Let me erase some stuff. 00:03:43.550 --> 00:03:46.420 Let me erase this one right here. 00:03:46.420 --> 00:03:48.930 And you're probably still wondering, OK, I kind 00:03:48.930 --> 00:03:52.090 of get this whole sine, tangent, cosine thing. 00:03:52.090 --> 00:03:53.430 What is it useful for? 00:03:53.430 --> 00:03:57.140 And all I can tell you right now is, get to know how to use 00:03:57.140 --> 00:03:59.500 these, soh cah toa, and in the next presentation and onwards, 00:03:59.500 --> 00:04:01.410 we're going to show you that trigonometry is actually 00:04:01.410 --> 00:04:03.220 probably one of the most obviously useful 00:04:03.220 --> 00:04:04.660 things in math. 00:04:04.660 --> 00:04:05.900 You can figure out all sorts of things. 00:04:05.900 --> 00:04:08.930 How far planets are, how tall buildings are. 00:04:08.930 --> 00:04:10.130 I mean, there's tons of things you could figure 00:04:10.130 --> 00:04:12.010 out with trigonometry. 00:04:12.010 --> 00:04:14.260 And then later, we'll study sine waves and cosine 00:04:14.260 --> 00:04:15.460 waves, and all that. 00:04:15.460 --> 00:04:19.650 You'll learn that it actually describes almost everything. 00:04:19.650 --> 00:04:20.990 But anyway. 00:04:20.990 --> 00:04:22.440 Going back to the problem. 00:04:22.440 --> 00:04:25.850 All we have left now is cosine. 00:04:25.850 --> 00:04:28.200 Oh, look how big that is. 00:04:28.200 --> 00:04:33.770 Cosine of theta equals-- we'll go back to our mnemonic. 00:04:33.770 --> 00:04:34.840 Soh cah toa. 00:04:34.840 --> 00:04:37.820 Well, cosine is adjacent over hypotenuse. 00:04:37.820 --> 00:04:40.300 So once again, what's the adjacent side? 00:04:40.300 --> 00:04:42.530 Well, this is the angle we're finding the cosine of, so the 00:04:42.530 --> 00:04:44.480 adjacent side is right here. 00:04:44.480 --> 00:04:46.100 So length 6. 00:04:46.100 --> 00:04:49.200 So it equals the adjacent side, which is 6, right? 00:04:49.200 --> 00:04:51.300 And we figured out what the hypotenuse was. 00:04:51.300 --> 00:04:52.200 That's this side. 00:04:52.200 --> 00:04:54.940 And its length, square root of 61. 00:04:54.940 --> 00:04:58.130 And if we rationalize this denominator, we get 6 square 00:04:58.130 --> 00:05:01.320 roots of 61 over 61. 00:05:01.320 --> 00:05:03.110 It's kind of messy numbers. 00:05:03.110 --> 00:05:07.730 But I think now you get the hang of figuring out-- if you 00:05:07.730 --> 00:05:12.390 know the sides of a triangle-- figuring out what the sine, the 00:05:12.390 --> 00:05:16.380 cosine, or the tangent of any given angle in that 00:05:16.380 --> 00:05:17.290 right triangle is. 00:05:17.290 --> 00:05:21.890 And obviously, you can't figure it out for this angle, because 00:05:21.890 --> 00:05:25.010 for this angle the opposite and the hypotenuse are 00:05:25.010 --> 00:05:26.400 actually the same number. 00:05:26.400 --> 00:05:27.420 So actually-- never mind. 00:05:27.420 --> 00:05:28.950 You actually can figure it out. 00:05:28.950 --> 00:05:30.350 But it actually gives something-- an 00:05:30.350 --> 00:05:32.180 interesting number. 00:05:32.180 --> 00:05:36.830 So with that said, I will finish this presentation. 00:05:36.830 --> 00:05:40.560 And in the next presentation, I will show you how-- if we know 00:05:40.560 --> 00:05:44.620 what the sine, or the cosine, or the tangent of an angle is, 00:05:44.620 --> 00:05:48.860 and we know one of the sides-- how we can figure out 00:05:48.860 --> 00:05:51.500 the other sides. 00:05:51.500 --> 00:05:53.020 See you in the next presentation. 00:05:53.020 --> 00:05:54.320 Bye.

Basic Trigonometry

https://www.youtube.com/watch?v=F21S9Wpi0y8

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WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:04.630 Welcome to the presentation on basic trigonometry. 00:00:04.630 --> 00:00:07.250 Sorry it's taken so long to get a new video out, but I had 00:00:07.250 --> 00:00:08.440 a lot of family in town. 00:00:08.440 --> 00:00:11.120 So let's get started with trigonometry. 00:00:11.120 --> 00:00:13.590 Let me get the pen tools all set up. 00:00:13.590 --> 00:00:14.830 I'm a little rusty. 00:00:14.830 --> 00:00:17.200 I'll use green. 00:00:17.200 --> 00:00:19.820 Trigonometry. 00:00:19.820 --> 00:00:27.790 I think it means-- I think it's from Ancient Greek, and it 00:00:27.790 --> 00:00:30.230 means triangle measure. 00:00:30.230 --> 00:00:32.290 I think that I read it on Wikipedia a couple days ago, 00:00:32.290 --> 00:00:33.610 so I believe that's the case. 00:00:33.610 --> 00:00:38.040 But all trigonometry is is really the study of right 00:00:38.040 --> 00:00:40.570 triangles and the relationship between the sides and the 00:00:40.570 --> 00:00:42.520 angles of a right triangle. 00:00:42.520 --> 00:00:43.890 That might sound a little confusing, but 00:00:43.890 --> 00:00:45.300 I'll get started. 00:00:45.300 --> 00:00:47.850 And actually, you've probably seen a lot of these things that 00:00:47.850 --> 00:00:49.940 we're going to go over now, and you'll finally know what 00:00:49.940 --> 00:00:52.840 those buttons on the calculator actually do. 00:00:52.840 --> 00:00:56.730 So let's start with a right triangle. 00:00:56.730 --> 00:00:59.016 Let's see, so it's a triangle. 00:01:01.620 --> 00:01:05.420 And it's a right triangle. 00:01:05.420 --> 00:01:08.470 Just for simplicity, let's say that this side is 3, this side 00:01:08.470 --> 00:01:12.730 is 4, and the hypotenuse is 5. 00:01:12.730 --> 00:01:17.540 So the trig functions tell you that for any angle, it tells 00:01:17.540 --> 00:01:20.170 you what the ratios of the sides of the triangle are 00:01:20.170 --> 00:01:21.130 relative to that angle. 00:01:21.130 --> 00:01:22.550 Let me give you an example. 00:01:22.550 --> 00:01:25.490 Let's call this angle theta. 00:01:25.490 --> 00:01:29.510 Theta is the Greek alphabet people tend to use for the 00:01:29.510 --> 00:01:33.000 angle that you're going to find the trig function of. 00:01:33.000 --> 00:01:41.530 Let's say you wanted to find the sine and s-i-n 00:01:41.530 --> 00:01:44.020 is short for sine. 00:01:44.020 --> 00:01:48.310 Let's say you wanted to find the sine of theta. 00:01:48.310 --> 00:01:50.500 So before we solve the sine of theta, I'm just going to 00:01:50.500 --> 00:01:53.310 throw out a mnemonic that I remembered when I was learning 00:01:53.310 --> 00:01:55.700 this in school, and I carried it every time, and every time I 00:01:55.700 --> 00:01:59.320 do a trig problem, I actually write it down on the page, or I 00:01:59.320 --> 00:02:00.940 at least repeat it to myself. 00:02:00.940 --> 00:02:09.500 And this is SOH CAH TOA. 00:02:09.500 --> 00:02:11.960 I have vague memories of my math teacher in high school 00:02:11.960 --> 00:02:14.990 telling a story about some Indian princess, who was named 00:02:14.990 --> 00:02:16.290 Sohcahtoa, but I forget. 00:02:16.290 --> 00:02:19.560 But all you have to remember is SOHCAHTOA. 00:02:19.560 --> 00:02:21.680 Now you might say, well, what it's SOHCAHTOA? 00:02:21.680 --> 00:02:26.150 Well, SOHCAHTOA says that sine is opposite over hypotenuse, 00:02:26.150 --> 00:02:31.160 cosine is adjacent over hypotenuse, and tangent is 00:02:31.160 --> 00:02:33.110 opposite over adjacent. 00:02:33.110 --> 00:02:35.000 Now, that's going to be confusing right now, but 00:02:35.000 --> 00:02:36.420 we're going to do a lot of examples, and I think 00:02:36.420 --> 00:02:37.510 it's going to make sense. 00:02:37.510 --> 00:02:38.760 So let's go back to this problem. 00:02:38.760 --> 00:02:40.280 We want to know what's the sine of theta. 00:02:40.280 --> 00:02:43.380 Theta is this angle in the triangle. 00:02:43.380 --> 00:02:45.300 So let's go to our mnemonic SOHCAHTOA. 00:02:45.300 --> 00:02:46.700 So which one is sine? 00:02:46.700 --> 00:02:49.930 Well, S for sine, so we use SOH. 00:02:49.930 --> 00:02:56.040 And we know that sine from this mnemonic, sine of, let's 00:02:56.040 --> 00:03:01.150 say, theta, is equal to opposite over hypotenuse. 00:03:01.150 --> 00:03:04.980 Opposite over hypotenuse. 00:03:04.980 --> 00:03:06.860 OK, so let's just figure out what the opposite 00:03:06.860 --> 00:03:08.590 and the hypotenuse are. 00:03:08.590 --> 00:03:11.790 Well, what is the opposite side of this angle? 00:03:14.450 --> 00:03:17.500 Well, if we just go opposite the angle, let's go here, 00:03:17.500 --> 00:03:20.890 the opposite side is 4, is this length of 4. 00:03:20.890 --> 00:03:23.100 I'll make that in a color. 00:03:23.100 --> 00:03:25.110 Oh, I thought I was changing colors. 00:03:25.110 --> 00:03:30.150 Yeah, so this side is the opposite, and I'll circle it. 00:03:30.150 --> 00:03:31.380 Now, which side is the hypotenuse? 00:03:31.380 --> 00:03:32.620 And you know this one. 00:03:32.620 --> 00:03:35.610 We've been doing this in the Pythagorean theorem modules. 00:03:35.610 --> 00:03:38.320 The long side, or the side opposite the right angle, 00:03:38.320 --> 00:03:40.640 is the hypotenuse. 00:03:40.640 --> 00:03:43.160 So that is the hypotenuse. 00:03:43.160 --> 00:03:45.370 So now I think we're ready to figure out what 00:03:45.370 --> 00:03:47.750 the sine of theta is. 00:03:47.750 --> 00:03:50.390 The sine-- whoops, I stayed in pink. 00:03:50.390 --> 00:03:55.920 Sine of theta is equal to the opposite side, 4, over the 00:03:55.920 --> 00:03:59.670 hypotenuse, which is 5. 00:03:59.670 --> 00:04:01.160 We're done. 00:04:01.160 --> 00:04:04.960 Let's figure out what-- let me erase part of this, and we'll 00:04:04.960 --> 00:04:08.652 figure out some more things about this triangle. 00:04:08.652 --> 00:04:09.840 Let me erase this. 00:04:14.800 --> 00:04:17.210 I think if you practice this, you'll realize that this is 00:04:17.210 --> 00:04:18.660 probably one of the easier things you learn in 00:04:18.660 --> 00:04:22.090 mathematics, and it's actually shocking that they take-- that 00:04:22.090 --> 00:04:27.790 they wait until Precalculus to teach this, because a smart 00:04:27.790 --> 00:04:30.210 middle-schooler could, I think, easily handle this. 00:04:30.210 --> 00:04:32.860 Not to make you feel bad if you're not getting it, just to 00:04:32.860 --> 00:04:35.130 give you confidence that you will get it, and you'll realize 00:04:35.130 --> 00:04:38.450 that it is very simple. 00:04:38.450 --> 00:04:43.520 OK, so let's figure out what the cosine-- and 00:04:43.520 --> 00:04:46.010 cos is short for cosine. 00:04:46.010 --> 00:04:50.530 I'll write it out, but I'm sure you've seen it before. 00:04:50.530 --> 00:04:54.500 So what is the cosine of theta? 00:04:54.500 --> 00:04:58.390 Well, we go back to our mnemonic: SOHCAHTOA. 00:04:58.390 --> 00:05:01.770 Well, cosine is the CAH, right? 00:05:01.770 --> 00:05:05.570 And that tells us that cosine of theta is equal to 00:05:05.570 --> 00:05:06.680 adjacent over hypotenuse. 00:05:11.730 --> 00:05:14.780 Well, once again, let's figure out what the adjacent side is. 00:05:14.780 --> 00:05:20.000 Well, the adjacent side-- this side was the opposite side, 00:05:20.000 --> 00:05:22.390 right, because it's opposite the angle. 00:05:22.390 --> 00:05:24.480 This side is the hypotenuse, because it's the longest side, 00:05:24.480 --> 00:05:27.250 and then, you could, just by deductive reasoning, but also 00:05:27.250 --> 00:05:29.740 just by looking at it, you see that this side right here, the 00:05:29.740 --> 00:05:32.080 side of length 3, is adjacent to the angle, right? 00:05:32.080 --> 00:05:34.600 Adjacent means right beside it. 00:05:34.600 --> 00:05:36.950 So that's the adjacent side. 00:05:36.950 --> 00:05:40.110 We already figured out that the hypotenuse is that 00:05:40.110 --> 00:05:42.400 side that I wrote in pink. 00:05:42.400 --> 00:05:46.770 So we're ready to figure out what cosine of theta equals. 00:05:46.770 --> 00:05:54.610 Cosine of theta is equal to the adjacent side, that's 3, over 00:05:54.610 --> 00:05:59.060 the hypotenuse, which is this pink side, 5. 00:05:59.060 --> 00:06:00.910 Pretty straightforward, isn't it? 00:06:00.910 --> 00:06:03.320 Let's do another one. 00:06:03.320 --> 00:06:05.050 OK, I don't want to erase the whole thing. 00:06:05.050 --> 00:06:07.670 I just want to erase part of the page because I want to 00:06:07.670 --> 00:06:08.773 keep using this triangle. 00:06:18.310 --> 00:06:19.920 OK, one left. 00:06:19.920 --> 00:06:21.860 The TOA. 00:06:21.860 --> 00:06:26.150 So if you remember what I said a little while ago-- well, 00:06:26.150 --> 00:06:27.000 we'll figure it out. 00:06:27.000 --> 00:06:31.870 But what is the-- oh, look how big that is. 00:06:31.870 --> 00:06:37.140 What is the tan of theta, or the tangent of theta? 00:06:37.140 --> 00:06:39.740 Well, let's go back to our mnemonic. 00:06:39.740 --> 00:06:41.320 TOA, right? 00:06:41.320 --> 00:06:43.510 TOA is for tangent, or t for tangent. 00:06:43.510 --> 00:06:47.630 So it tells us that tangent is the opposite over the adjacent. 00:06:47.630 --> 00:06:54.220 So tan of theta is equal to opposite over adjacent. 00:06:54.220 --> 00:06:56.960 Well, that equals-- what was the opposite side? 00:06:56.960 --> 00:07:00.360 Right, the opposite side was 4. 00:07:00.360 --> 00:07:01.130 And what was the adjacent side? 00:07:01.130 --> 00:07:01.780 Well, we just saw that. 00:07:01.780 --> 00:07:03.520 It was 3. 00:07:03.520 --> 00:07:08.890 So the tangent of this angle is 4/3. 00:07:08.890 --> 00:07:10.910 Now let's do another angle on this. 00:07:10.910 --> 00:07:17.180 Let's call this angle here-- I don't know. 00:07:17.180 --> 00:07:18.990 Let's call it x. 00:07:18.990 --> 00:07:21.700 I don't know any other Greek letters. 00:07:21.700 --> 00:07:23.840 Let's call that angle x. 00:07:23.840 --> 00:07:27.570 So if we want to figure out the tan of x, let's see if it's 00:07:27.570 --> 00:07:29.770 the same as the tan of theta. 00:07:29.770 --> 00:07:31.620 The tan of x. 00:07:31.620 --> 00:07:33.780 Well, now what's the opposite side? 00:07:33.780 --> 00:07:36.340 Well, now the opposite side is the white side, right? 00:07:36.340 --> 00:07:39.980 Because opposite this angle is the 3 side. 00:07:39.980 --> 00:07:43.890 So we see here tan is opposite over adjacent, so opposite is 00:07:43.890 --> 00:07:48.130 3, and then adjacent is 4. 00:07:48.130 --> 00:07:49.910 This is interesting. 00:07:49.910 --> 00:07:54.230 The tangent of this angle is the inverse of the 00:07:54.230 --> 00:07:54.830 tangent to that angle. 00:07:54.830 --> 00:07:56.520 I don't want to confuse you too much, but I just want to show 00:07:56.520 --> 00:07:59.930 you that when you take the trig functions, it matters which 00:07:59.930 --> 00:08:03.050 angle of the right angle you're taking the functions of. 00:08:03.050 --> 00:08:05.310 And you might be saying, well, this is all good and well, 00:08:05.310 --> 00:08:06.810 Sal, but what use is this? 00:08:06.810 --> 00:08:10.990 Well, we'll later show you that if you have some of the 00:08:10.990 --> 00:08:13.390 information, so you know an angle, and you know a side, or 00:08:13.390 --> 00:08:15.940 you know a couple of sides, you can figure out-- and if you 00:08:15.940 --> 00:08:20.920 have either a slide ruler or a trig table or a good 00:08:20.920 --> 00:08:24.320 calculator, you can figure out-- given the sides of a 00:08:24.320 --> 00:08:26.510 triangle, you can figure out the angles, or given an angle 00:08:26.510 --> 00:08:28.220 and a side, you could figure out other sides. 00:08:28.220 --> 00:08:31.010 And we're actually going to do that in the next module. 00:08:31.010 --> 00:08:33.020 So, hopefully, this gives you a little bit of an introduction. 00:08:33.020 --> 00:08:36.900 I'm almost out of time on the YouTube 10-minute limit, so I'm 00:08:36.900 --> 00:08:39.130 going to wait to do a couple more examples in 00:08:39.130 --> 00:08:40.840 the next video. 00:08:40.840 --> 00:08:42.900 See you in the next presentation. 00:08:42.900 --> 00:08:44.200 Bye!

Quadratic inequalities (visual explanation)

https://www.youtube.com/watch?v=xdiBjypYFRQ

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WEBVTT Kind: captions Language: en 00:00:01.850 --> 00:00:05.900 Welcome to the presentation on quadratic inequalities. 00:00:05.900 --> 00:00:09.660 Before we get to quadratic inequalities, let's just start 00:00:09.660 --> 00:00:11.945 graphing some functions and interpret them and then we'll 00:00:11.945 --> 00:00:13.880 slowly move to the inequalities. 00:00:13.880 --> 00:00:26.360 Let's say I had f of x is equal to x squared plus x minus 6. 00:00:26.360 --> 00:00:29.210 Well, if we wanted to figure out where this function 00:00:29.210 --> 00:00:33.070 intersects the x-axis or the roots of it, we learned in our 00:00:33.070 --> 00:00:35.210 factoring quadratics that we could just set f of x 00:00:35.210 --> 00:00:36.735 is equal to 0, right? 00:00:36.735 --> 00:00:39.460 Because f of x equals 0 when you're intersecting the x-axis. 00:00:39.460 --> 00:00:46.990 So you would say x squared plus x minus 6 is equal to 0. 00:00:46.990 --> 00:00:48.800 And you just factor this quadratic. 00:00:48.800 --> 00:00:57.470 x plus 3 times x minus 2 equals 0. 00:00:57.470 --> 00:01:03.270 And you would learn that the roots of this quadratic 00:01:03.270 --> 00:01:11.600 function are x is equal to minus 3, and x is equal to 2. 00:01:11.600 --> 00:01:13.970 How would we visualize this? 00:01:13.970 --> 00:01:16.970 Well let's draw this quadratic function. 00:01:21.380 --> 00:01:24.510 Those are my very uneven lines. 00:01:24.510 --> 00:01:27.200 So the roots are x is equal to negative 3. 00:01:27.200 --> 00:01:33.180 So this is, right here, x is at minus 3y0 -- by definition one 00:01:33.180 --> 00:01:36.790 of the roots is where f of x is equal to 0. 00:01:36.790 --> 00:01:41.370 So the y, or the f of x axis here is 0. 00:01:41.370 --> 00:01:42.600 The coordinate is 0. 00:01:42.600 --> 00:01:47.130 And this point here is 2 comma 0. 00:01:47.130 --> 00:01:53.360 Once again, this is the x-axis, and this is the f of x-axis. 00:01:53.360 --> 00:01:56.270 We also know that the y intercept is minus 6. 00:01:56.270 --> 00:01:57.800 This isn't the vertex, this is the y intercept. 00:01:57.800 --> 00:02:03.710 And that the graph is going to look something like this -- not 00:02:03.710 --> 00:02:05.640 as bumpy as what I'm drawing, which I think you get the 00:02:05.640 --> 00:02:10.180 general idea if you've ever seen a clean parabola. 00:02:10.180 --> 00:02:16.200 It looks like that with x minus 3 here, and x is 2 here. 00:02:16.200 --> 00:02:17.080 Pretty straightforward. 00:02:17.080 --> 00:02:19.480 We figured out the roots, we figured out what it looks like. 00:02:19.480 --> 00:02:22.170 Now what if we, instead of wanting to know where f of x is 00:02:22.170 --> 00:02:24.960 equal to 0, which is these two points, what if we wanted 00:02:24.960 --> 00:02:29.290 to know where f of x is greater than 0? 00:02:29.290 --> 00:02:33.200 What x values make f of x greater than 0? 00:02:33.200 --> 00:02:36.050 Or another way of saying it, what values make 00:02:36.050 --> 00:02:37.410 the statement true? 00:02:37.410 --> 00:02:42.740 x squared plus x minus 6 is greater than 0, Right, 00:02:42.740 --> 00:02:44.730 this is just f of x. 00:02:44.730 --> 00:02:49.470 Well if we look at the graph, when is f of x greater than 0? 00:02:49.470 --> 00:02:52.010 Well this is the f of x axis, and when are we 00:02:52.010 --> 00:02:52.910 in positive territory? 00:02:52.910 --> 00:02:55.360 Well f of x is greater than 0 here -- let me draw that 00:02:55.360 --> 00:03:00.890 another color -- is greater than 0 here, right? 00:03:00.890 --> 00:03:04.980 Because it's above the x-axis. 00:03:04.980 --> 00:03:06.830 And f of x is greater than 0 here. 00:03:11.560 --> 00:03:17.070 So just visually looking at it, what x values make this true? 00:03:17.070 --> 00:03:23.580 Well, this is true whenever x is less than minus 3, right, or 00:03:23.580 --> 00:03:26.440 whenever x is greater than 2. 00:03:26.440 --> 00:03:31.560 Because when x is greater than 2, f of x is greater than 0, 00:03:31.560 --> 00:03:35.910 and when x is less than negative 3, f of x 00:03:35.910 --> 00:03:37.430 is greater than 0. 00:03:37.430 --> 00:03:41.460 So we would say the solution to this quadratic inequality, and 00:03:41.460 --> 00:03:46.910 we pretty much solved this visually, is x is less than 00:03:46.910 --> 00:03:52.970 minus 3, or x is greater than 2. 00:03:52.970 --> 00:03:53.890 And you could test it out. 00:03:53.890 --> 00:03:56.990 You could try out the number minus 4, and you should get f 00:03:56.990 --> 00:03:58.800 of x being greater than 0. 00:03:58.800 --> 00:04:00.890 You could try it out here. 00:04:00.890 --> 00:04:04.110 Or you could try the number 3 and make sure that this works. 00:04:04.110 --> 00:04:06.820 And you can just make sure that, you could, for example, 00:04:06.820 --> 00:04:10.410 try out the number 0 and make sure that 0 doesn't work, 00:04:10.410 --> 00:04:12.790 right, because 0 is between the two roots. 00:04:12.790 --> 00:04:15.030 It actually turns out that when x is equal to 0, f 00:04:15.030 --> 00:04:19.040 of x is minus 6, which is definitely less than 0. 00:04:19.040 --> 00:04:22.400 So I think this will give you a visual intuition of what this 00:04:22.400 --> 00:04:24.000 quadratic inequality means. 00:04:24.000 --> 00:04:26.510 Now with that visual intuition in the back of your mind, let's 00:04:26.510 --> 00:04:29.090 do some more problems and maybe we won't have to go through the 00:04:29.090 --> 00:04:32.600 exercise of drawing it, but maybe I will draw it just to 00:04:32.600 --> 00:04:35.190 make sure that the point hits home. 00:04:35.190 --> 00:04:37.140 Let me give you a slightly trickier problem. 00:04:37.140 --> 00:04:49.120 Let's say I had minus x squared minus 3x plus 28, let me 00:04:49.120 --> 00:04:52.190 say, is greater than 0. 00:04:52.190 --> 00:04:53.600 Well I want to get rid of this negative sign in 00:04:53.600 --> 00:04:54.350 front of the x squared. 00:04:54.350 --> 00:04:56.420 I just don't like it there because it makes it look 00:04:56.420 --> 00:04:58.080 more confusing to factor. 00:04:58.080 --> 00:05:00.140 I'm going to multiply everything by negative 1. 00:05:00.140 --> 00:05:00.780 Both sides. 00:05:00.780 --> 00:05:08.210 I get x squared plus 3x minus 28, and when you multiply or 00:05:08.210 --> 00:05:10.160 divide by a negative, with any inequality you have 00:05:10.160 --> 00:05:11.440 to swap the sign. 00:05:11.440 --> 00:05:16.860 So this is now going to be less than 0. 00:05:16.860 --> 00:05:25.130 And if we were to factor this, we get x plus 7 times x 00:05:25.130 --> 00:05:29.880 minus 4 is less than 0. 00:05:29.880 --> 00:05:32.460 So if this was equal to 0, we would know that the two roots 00:05:32.460 --> 00:05:37.400 of this function -- let's define the function f of x -- 00:05:37.400 --> 00:05:40.540 let's define the function as f of x is equal to -- well we can 00:05:40.540 --> 00:05:42.670 define it as this or this because they're the same thing. 00:05:42.670 --> 00:05:47.080 But for simplicity let's define it as x plus 7 times x minus 4. 00:05:47.080 --> 00:05:49.390 That's f of x, right? 00:05:49.390 --> 00:05:53.260 Well, after factoring it, we know that the roots of this, 00:05:53.260 --> 00:06:05.600 the roots are x is equal to minus 7, and x is equal to 4. 00:06:05.600 --> 00:06:07.900 Now what we want to know is what x values make 00:06:07.900 --> 00:06:10.000 this inequality true? 00:06:10.000 --> 00:06:12.060 If this was any equality we'd be done. 00:06:12.060 --> 00:06:14.650 But we want to know what makes this inequality true. 00:06:14.650 --> 00:06:17.600 I'll give you a little bit of a trick, it's always going to be 00:06:17.600 --> 00:06:21.160 the numbers in between the two roots or outside 00:06:21.160 --> 00:06:23.120 of the two roots. 00:06:23.120 --> 00:06:25.810 So what I do whenever I'm doing this on a test or something, I 00:06:25.810 --> 00:06:28.510 just test numbers that are either between the roots or 00:06:28.510 --> 00:06:30.610 outside of the two roots. 00:06:30.610 --> 00:06:34.660 So let's pick a number that's between x equals minus 00:06:34.660 --> 00:06:36.360 7 and x equals 4. 00:06:36.360 --> 00:06:41.560 Well let's try x equals 0. 00:06:41.560 --> 00:06:46.660 Well, f of 0 is equal to -- we could do it right here -- f of 00:06:46.660 --> 00:06:57.360 0 is 0 plus 7 times 0 minus 4 is just 7 times minus 00:06:57.360 --> 00:07:00.280 4, which is minus 28. 00:07:00.280 --> 00:07:04.100 So f of 0 is minus 28. 00:07:04.100 --> 00:07:08.800 Now is this -- this is the function we're working with 00:07:08.800 --> 00:07:11.790 -- is this less than 0? 00:07:11.790 --> 00:07:13.140 Well yeah, it is. 00:07:13.140 --> 00:07:16.090 So it actually turns that a number, an x value between 00:07:16.090 --> 00:07:17.470 the two roots works. 00:07:17.470 --> 00:07:19.940 So actually I immediately know that the answer here 00:07:19.940 --> 00:07:23.340 is all of the x's that are between the two roots. 00:07:23.340 --> 00:07:29.170 So we could say that the solution to this is 00:07:29.170 --> 00:07:34.720 minus 7 is less than x which is less than 4. 00:07:34.720 --> 00:07:35.460 Because now the other way. 00:07:35.460 --> 00:07:38.240 You could have tried a number that's outside of the roots, 00:07:38.240 --> 00:07:41.460 either less than minus 7 or greater than 4 and 00:07:41.460 --> 00:07:42.650 have tried it out. 00:07:42.650 --> 00:07:45.820 Let's say if you had tried out 5. 00:07:45.820 --> 00:07:48.140 Try x equals 5. 00:07:48.140 --> 00:07:55.780 Well then f of 5 would be 12 times 1, right, 00:07:55.780 --> 00:07:58.690 which is equal to 12. 00:07:58.690 --> 00:07:59.570 f of 5 is 12. 00:07:59.570 --> 00:08:02.300 Is that less than 0? 00:08:02.300 --> 00:08:03.110 No. 00:08:03.110 --> 00:08:04.060 So that wouldn't have worked. 00:08:04.060 --> 00:08:06.000 So once again, that gives us a confidence that we 00:08:06.000 --> 00:08:07.260 got the right interval. 00:08:07.260 --> 00:08:11.750 And if we wanted to think about this visually, because we got 00:08:11.750 --> 00:08:14.920 this answer, when you do it visually it actually makes, I 00:08:14.920 --> 00:08:18.640 think, a lot of sense, but maybe I'm biased. 00:08:26.180 --> 00:08:28.770 If you look at it visually it looks like this. 00:08:35.130 --> 00:08:40.890 If you drive visually and this is the parabola, this is f of 00:08:40.890 --> 00:08:52.580 x, the roots here are minus 7, 0 and 4, 0, we're saying that 00:08:52.580 --> 00:08:56.230 for all x values between these two numbers, f of 00:08:56.230 --> 00:08:57.390 x is less than 0. 00:08:57.390 --> 00:08:59.770 And that makes sense, because when is f of x less than 0? 00:08:59.770 --> 00:09:02.420 Well this is the graph of f of x. 00:09:06.110 --> 00:09:07.570 And when is f of x less than 0? 00:09:07.570 --> 00:09:08.480 Right here. 00:09:08.480 --> 00:09:10.880 So what x values give us that? 00:09:10.880 --> 00:09:14.140 Well the x values that give us that are right here. 00:09:14.140 --> 00:09:15.410 I hope I'm not confusing you too much with 00:09:15.410 --> 00:09:16.960 these visual graphs. 00:09:16.960 --> 00:09:19.140 And you're probably saying, well how do I know 00:09:19.140 --> 00:09:20.180 I don't include 0? 00:09:20.180 --> 00:09:23.116 Well you could try it out, but if you -- oh, well how come 00:09:23.116 --> 00:09:24.690 I don't include the roots? 00:09:24.690 --> 00:09:28.030 Well at the roots, f of x is equal to 0. 00:09:28.030 --> 00:09:31.640 So if this was this, if this was less than or equal to 0, 00:09:31.640 --> 00:09:36.290 then the answer would be negative 7 is less than 00:09:36.290 --> 00:09:39.230 or equal to x is less than or equal to 4. 00:09:39.230 --> 00:09:40.620 I hope that gives you a sense. 00:09:40.620 --> 00:09:42.460 You pretty much just have to try number in between the 00:09:42.460 --> 00:09:45.250 roots, and try number outside of the roots, and that tells 00:09:45.250 --> 00:09:49.300 you what interval will make the inequality true. 00:09:49.300 --> 00:09:51.640 I'll see you in the next presentation.

Introduction to Logarithms

https://www.youtube.com/watch?v=mQTWzLpCcW0

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WEBVTT Kind: captions Language: en 00:00:01.040 --> 00:00:03.750 Welcome to the logarithm presentation. 00:00:03.750 --> 00:00:06.000 Let me write down the word logarithm just because it is 00:00:06.000 --> 00:00:08.930 another strange and unusual word like hypotenuse and it's 00:00:08.930 --> 00:00:10.700 good to at least, see it once. 00:00:10.700 --> 00:00:14.520 Let me get the pen tool working. 00:00:14.520 --> 00:00:14.980 Logarithm. 00:00:19.680 --> 00:00:24.620 This is one of my most misspelled words. 00:00:24.620 --> 00:00:27.300 I went to MIT and actually one of the a cappella groups there, 00:00:27.300 --> 00:00:30.050 they were called the Logarhythms. 00:00:30.050 --> 00:00:31.710 Like rhythm, like music. 00:00:31.710 --> 00:00:33.990 But anyway, I'm digressing. 00:00:33.990 --> 00:00:35.730 So what is a logarithm? 00:00:35.730 --> 00:00:37.790 Well, the easiest way to explain what a logarithm is is 00:00:37.790 --> 00:00:41.370 to have first-- I guess it's just to say it's the inverse of 00:00:41.370 --> 00:00:43.140 taking the exponent of something. 00:00:43.140 --> 00:00:44.250 Let me explain. 00:00:44.250 --> 00:00:49.900 If I said that 2 to the third power-- well, we know that 00:00:49.900 --> 00:00:52.120 from the exponent modules. 00:00:52.120 --> 00:00:54.770 2 the third power, well that's equal to 8. 00:00:54.770 --> 00:00:56.830 And once again, this is a 2, it's not a z. 00:00:56.830 --> 00:00:59.530 2 to the third power is 8, so it actually turns out that 00:00:59.530 --> 00:01:04.580 log-- and log is short for the word logarithm. 00:01:04.580 --> 00:01:13.200 Log base 2 of eight is equal to 3. 00:01:13.200 --> 00:01:15.130 I think when you look at that you're trying to say oh, 00:01:15.130 --> 00:01:17.210 that's trying to make a little bit of sense. 00:01:17.210 --> 00:01:22.730 What this says, if I were to ask you what log base 2 of 00:01:22.730 --> 00:01:27.770 8 is, this says 2 to the what power is equal to 8? 00:01:27.770 --> 00:01:31.060 So the answer to a logarithm-- you can say the answer to this 00:01:31.060 --> 00:01:33.880 logarithm expression, or if you evaluate this logarithm 00:01:33.880 --> 00:01:36.440 expression, you should get a number that is really the 00:01:36.440 --> 00:01:42.320 exponent that you would have to raised 2 to to get 8. 00:01:42.320 --> 00:01:43.970 And once again, that's 3. 00:01:43.970 --> 00:01:47.560 Let's do a couple more examples and I think you might get it. 00:01:47.560 --> 00:01:54.690 If I were to say log-- what happened to my pen? 00:01:54.690 --> 00:02:03.710 log base 4 of 64 is equal to x. 00:02:03.710 --> 00:02:09.820 Another way of rewriting this exact equation is to say 4 to 00:02:09.820 --> 00:02:14.230 the x power is equal to 64. 00:02:14.230 --> 00:02:16.790 Or another way to think about it, 4 to what 00:02:16.790 --> 00:02:18.290 power is equal to 64? 00:02:18.290 --> 00:02:20.960 Well, we know that 4 to the third power is 64. 00:02:20.960 --> 00:02:25.950 So we know that in this case, this equals 3. 00:02:25.950 --> 00:02:36.120 So log base 4 of 64 is equal to 3. 00:02:36.120 --> 00:02:39.320 Let me do a bunch of more examples and I think the more 00:02:39.320 --> 00:02:42.260 examples you see, it'll start to make some sense. 00:02:42.260 --> 00:02:45.720 Logarithms are a simple idea, but I think they can get 00:02:45.720 --> 00:02:48.980 confusing because they're the inverse of exponentiation, 00:02:48.980 --> 00:02:52.390 which is sometimes itself, a confusing concept. 00:02:52.390 --> 00:03:05.780 So what is log base 10 of let's say, 1,000,000. 00:03:05.780 --> 00:03:08.540 Put some commas here to make sure. 00:03:08.540 --> 00:03:12.490 So this equals question mark. 00:03:12.490 --> 00:03:15.960 Well, all we have to ask ourselves is 10 to what power 00:03:15.960 --> 00:03:17.770 is equal to 1,000,000. 00:03:17.770 --> 00:03:22.060 And 10 to any power is actually equal to 1 followed by the 00:03:22.060 --> 00:03:24.900 power of-- if you say 10 of the fifth power, that's equal 00:03:24.900 --> 00:03:26.930 to 1 followed by five 0's. 00:03:26.930 --> 00:03:29.550 So if we have 1 followed by six 0's this is the same thing 00:03:29.550 --> 00:03:31.350 as 10 to the sixth power. 00:03:31.350 --> 00:03:34.590 So 10 to the sixth power is equal to 1,000,000. 00:03:34.590 --> 00:03:47.170 So since 10 to the sixth power is equal to 1,000,000 log base 00:03:47.170 --> 00:03:54.060 10 of 1,000,000 is equal to 6. 00:03:54.060 --> 00:03:57.740 Just remember, this 6 is an exponent that we raise 10 00:03:57.740 --> 00:03:59.640 to to get the 1,000,000. 00:03:59.640 --> 00:04:01.460 I know I'm saying this in a hundred different ways and 00:04:01.460 --> 00:04:04.200 hopefully, one or two of these million different ways that I'm 00:04:04.200 --> 00:04:06.310 explaining it actually will make sense. 00:04:06.310 --> 00:04:08.830 Let's do some more. 00:04:08.830 --> 00:04:12.570 Actually, I'll do even a slightly confusing one. 00:04:12.570 --> 00:04:19.790 log base 1/2 of 1/8. 00:04:23.252 --> 00:04:25.820 Let's say that that equals x. 00:04:25.820 --> 00:04:27.760 So let's just remind ourselves, that's just 00:04:27.760 --> 00:04:32.050 like saying 1/2-- whoops. 00:04:32.050 --> 00:04:32.670 1/2. 00:04:32.670 --> 00:04:34.280 That's supposed to be parentheses. 00:04:34.280 --> 00:04:37.020 To the x power is equal to 1/8. 00:04:40.500 --> 00:04:44.490 Well, we know that 1/2 to the third power is equal to 1/8. 00:04:44.490 --> 00:04:54.766 So log base 1/2 of 1/8 is equal to 3. 00:04:54.766 --> 00:04:56.275 Let me do a bunch of more problems. 00:05:00.850 --> 00:05:02.290 Actually, let me mix it up a little bit. 00:05:02.290 --> 00:05:13.680 Let's say that log base x of 27 is equal to 3. 00:05:13.680 --> 00:05:16.480 What's x? 00:05:16.480 --> 00:05:20.520 Well, just like what we did before, this says that x to the 00:05:20.520 --> 00:05:22.790 third power is equal to 27. 00:05:25.350 --> 00:05:34.060 Or x is equal to the cubed root of 27. 00:05:34.060 --> 00:05:36.170 And all that means is that there's some number times 00:05:36.170 --> 00:05:38.160 itself three times that equals 27. 00:05:38.160 --> 00:05:39.740 And I think at this point you know that that 00:05:39.740 --> 00:05:41.370 number would be 3. 00:05:41.370 --> 00:05:43.150 x equals 3. 00:05:43.150 --> 00:05:51.060 So we could write log base 3 of 27 is equal to 3. 00:05:54.100 --> 00:05:55.830 Let me think of another example. 00:05:55.830 --> 00:05:57.750 I'm only doing relatively small numbers because I don't have 00:05:57.750 --> 00:06:00.050 a calculator with me and I have to do them in my head. 00:06:00.050 --> 00:06:07.710 So what is log-- let me think about this. 00:06:07.710 --> 00:06:14.440 What is log base 100 of 1? 00:06:14.440 --> 00:06:16.690 This is a trick problem. 00:06:16.690 --> 00:06:18.380 So once again, let's just say that this is equal 00:06:18.380 --> 00:06:22.440 to question mark. 00:06:22.440 --> 00:06:25.330 So remember this is log base 100 hundred of 1. 00:06:25.330 --> 00:06:30.250 So this says 100 to the question mark power 00:06:30.250 --> 00:06:32.720 is equal to 1. 00:06:32.720 --> 00:06:34.960 Well, what do we have to raise-- if we have any number 00:06:34.960 --> 00:06:37.530 and we raise it to what power, when do we get 1? 00:06:37.530 --> 00:06:39.790 Well, if you remember from the exponent rules, or actually not 00:06:39.790 --> 00:06:42.470 the exponent rules, from the exponent modules, anything to 00:06:42.470 --> 00:06:44.880 the 0-th power is equal to 1. 00:06:44.880 --> 00:06:51.330 So we could say 100 to the 0 power equals 1. 00:06:51.330 --> 00:07:00.410 So we could say log base 100 hundred of 1 is equal to 0 00:07:00.410 --> 00:07:04.930 because 100 to the 0-th power is equal to 1. 00:07:04.930 --> 00:07:07.860 Let me ask another question. 00:07:07.860 --> 00:07:16.120 What if I were to ask you log, let's say base 2 of 0? 00:07:16.120 --> 00:07:18.060 So what is that equal to? 00:07:18.060 --> 00:07:20.330 Well, what I'm asking you, I'm saying 2-- let's 00:07:20.330 --> 00:07:22.160 say that equals x. 00:07:22.160 --> 00:07:25.770 2 to some power x is equal to 0. 00:07:25.770 --> 00:07:28.430 So what is x? 00:07:28.430 --> 00:07:30.580 Well, is there anything that I can raise 2 to 00:07:30.580 --> 00:07:32.850 the power of to get 0? 00:07:32.850 --> 00:07:33.790 No. 00:07:33.790 --> 00:07:35.830 So this is undefined. 00:07:35.830 --> 00:07:38.710 Undefined or no solution. 00:07:38.710 --> 00:07:41.990 There's no number that I can raise 2 to the 00:07:41.990 --> 00:07:44.440 power of and get 0. 00:07:44.440 --> 00:07:51.320 Similarly if I were to ask you log base 3 of 00:07:51.320 --> 00:07:54.210 let's say, negative 1. 00:07:54.210 --> 00:07:56.810 And we're assuming we're dealing with the real numbers, 00:07:56.810 --> 00:07:58.630 which are most of the numbers that I think at this point 00:07:58.630 --> 00:08:00.440 you have dealt with. 00:08:00.440 --> 00:08:02.660 There's nothing I can raise three 3 to the power of to 00:08:02.660 --> 00:08:04.240 get a negative number, so this is undefined. 00:08:10.510 --> 00:08:14.620 So as long as you have a positive base here, this 00:08:14.620 --> 00:08:21.380 number, in order to be defined, has to be greater than-- well, 00:08:21.380 --> 00:08:23.680 it has to be greater than or equal-- no. 00:08:23.680 --> 00:08:25.590 It has to be greater than 0. 00:08:25.590 --> 00:08:26.210 Not equal to. 00:08:26.210 --> 00:08:28.980 It cannot be 0 and it cannot be negative. 00:08:28.980 --> 00:08:30.020 Let's do a couple more problems. 00:08:30.020 --> 00:08:32.350 I think I have another minute and a half. 00:08:32.350 --> 00:08:36.390 You're already prepared to do the level 1 logarithms module, 00:08:36.390 --> 00:08:39.240 but let's do a couple of more. 00:08:39.240 --> 00:08:47.130 What is log base 8-- I'm going to do a slightly 00:08:47.130 --> 00:08:52.510 tricky one-- of 1/64. 00:08:52.510 --> 00:08:53.940 Interesting. 00:08:53.940 --> 00:09:00.010 We know that log base 8 of 64 would equal 2, right? 00:09:00.010 --> 00:09:02.800 Because 8 squared is equal to 64. 00:09:02.800 --> 00:09:06.240 But 8 to what power equals 1/64? 00:09:06.240 --> 00:09:09.320 Well, we learned from the negative exponent module that 00:09:09.320 --> 00:09:13.030 that is equal to negative 2. 00:09:13.030 --> 00:09:17.610 If you remember, 8 to the negative 2 power is the same 00:09:17.610 --> 00:09:20.230 thing as 1/8 to the 2 power. 00:09:20.230 --> 00:09:24.960 8 squared, which is equal to 1/64. 00:09:24.960 --> 00:09:26.960 Interesting. 00:09:26.960 --> 00:09:29.590 I'll leave this for you to think about. 00:09:29.590 --> 00:09:31.590 When you take the inverse of whatever you're taking the 00:09:31.590 --> 00:09:33.830 logarithm of, it turns the answer negative. 00:09:33.830 --> 00:09:36.260 And we'll do a lot more logarithm problems and explore 00:09:36.260 --> 00:09:38.880 a lot more of the properties of logarithms in future modules. 00:09:38.880 --> 00:09:43.120 But I think you're ready at this point to do the level 1 00:09:43.120 --> 00:09:45.770 logarithm set of exercises. 00:09:45.770 --> 00:09:47.600 See you in the next module.

30-60-90 Triangles II

https://www.youtube.com/watch?v=3mLUJSoh6i0

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en

WEBVTT Kind: captions Language: en 00:00:00.860 --> 00:00:03.250 Let's continue with the 30, 60, 90 triangles. 00:00:06.480 --> 00:00:09.640 So just review what we just learned, or hopefully learned-- 00:00:09.640 --> 00:00:15.910 at minimum what we just saw, --is if we have a 30, 60, 90 -- 00:00:15.910 --> 00:00:18.380 and once again, remember: this is only applies to 30, 60, 90 00:00:18.380 --> 00:00:26.560 triangles --and if I were to say the hypotenuse is of length 00:00:26.560 --> 00:00:31.320 h, we learned that the side opposite the 30-degree angle, 00:00:31.320 --> 00:00:34.340 and this is the shortest side of the triangle, is going to be 00:00:34.340 --> 00:00:37.270 h over 2, or 1/2 times the hypotenuse. 00:00:37.270 --> 00:00:40.240 And we also learned that the longer side, or the side 00:00:40.240 --> 00:00:42.810 opposite the 60-degree side, is equal to the square 00:00:42.810 --> 00:00:46.840 root of 3 over 2 times h. 00:00:46.840 --> 00:00:50.640 So let's do a problem where we use this information. 00:00:50.640 --> 00:00:56.370 Let's say I had this triangle right here. 00:00:56.370 --> 00:00:58.010 It's a 90-degree triangle; let's say that this 00:00:58.010 --> 00:01:00.690 is 30 degrees. 00:01:00.690 --> 00:01:02.750 And we could also figure out obviously if that's 30, this 00:01:02.750 --> 00:01:07.040 is 90, that this is also 60 degrees. 00:01:07.040 --> 00:01:10.510 And let's say that the hypotenuse is 12. 00:01:10.510 --> 00:01:12.300 The length is 12 and we know that this is the hypotenuse 00:01:12.300 --> 00:01:14.980 because it's opposite the right angle. 00:01:14.980 --> 00:01:18.630 What is the side right here? 00:01:18.630 --> 00:01:21.840 Well, is the side opposite the 60-degree angle, or is it 00:01:21.840 --> 00:01:23.910 opposite the 30-degree angle? 00:01:23.910 --> 00:01:26.460 It's the 30-degree angle that opens into it, right? 00:01:26.460 --> 00:01:28.650 I drew this triangle a little bit different on purpose. 00:01:28.650 --> 00:01:32.050 The 30-degree angle opens up into this side, and it's 00:01:32.050 --> 00:01:34.060 also the shortest side. 00:01:34.060 --> 00:01:37.360 We learned that the side opposite the 30-degree angle is 00:01:37.360 --> 00:01:40.680 half the hypotenuse, so the hypotenuse is 12; 00:01:40.680 --> 00:01:42.860 this would be 6. 00:01:42.860 --> 00:01:46.310 And this side, which is opposite the 60-degree side, is 00:01:46.310 --> 00:01:49.730 equal to the square root of 3 over 2 times the hypotenuse. 00:01:49.730 --> 00:01:54.690 So it's the square root of 3 over 2 times 12, or it's just 00:01:54.690 --> 00:01:58.150 equal to 6 square roots of 3. 00:01:58.150 --> 00:02:01.150 Another interesting thing is, of course, the longer 00:02:01.150 --> 00:02:04.600 non-hypotenuse side is square root of 3 times longer 00:02:04.600 --> 00:02:06.270 than the short side. 00:02:06.270 --> 00:02:07.810 I don't confuse you too much. 00:02:07.810 --> 00:02:08.660 Let's do another one. 00:02:15.010 --> 00:02:20.800 Let's say this is 30 degrees-- it's our right triangle --and I 00:02:20.800 --> 00:02:28.390 were to tell you that this side right here is 5, what is 00:02:28.390 --> 00:02:29.900 the length of this side? 00:02:33.970 --> 00:02:35.750 Well first of all let's figure out what we have. 00:02:35.750 --> 00:02:37.390 5 is which side? 00:02:37.390 --> 00:02:39.540 So if this is 30 degrees, we know that this is 00:02:39.540 --> 00:02:41.990 going to be 60 degrees. 00:02:41.990 --> 00:02:47.010 So 5 is opposite the 60-degree side, and x is the hypotenuse. 00:02:47.010 --> 00:02:49.840 Since x is opposite the 90-degree side, it's also 00:02:49.840 --> 00:02:53.010 the longest side of the right triangle. 00:02:53.010 --> 00:02:57.910 So we know from our formula that 5 is equal to the square 00:02:57.910 --> 00:03:00.940 root of 3 over 2 times the hypotenuse, which in 00:03:00.940 --> 00:03:02.850 this example is x. 00:03:02.850 --> 00:03:04.240 And now we just solve for x. 00:03:04.240 --> 00:03:06.770 We can multiply both sides by the inverse 00:03:06.770 --> 00:03:07.865 of this coefficient. 00:03:07.865 --> 00:03:19.710 So if you just multiply 2 times the square root of 3-- can 00:03:19.710 --> 00:03:25.030 ignore this --we get 10 over the square root of three here. 00:03:25.030 --> 00:03:27.140 And, of course, this 2 cancels out with this 2. 00:03:27.140 --> 00:03:28.667 This square root of 3 cancels out this square root 00:03:28.667 --> 00:03:30.970 of 3 is equal to x. 00:03:30.970 --> 00:03:33.510 And now if you watched the last couple of presentations, you 00:03:33.510 --> 00:03:36.690 realize that this could be the right answer, but we have a 00:03:36.690 --> 00:03:39.660 square root of 3 in the denominator, which people don't 00:03:39.660 --> 00:03:42.980 like because it's an irrational number in the denominator. 00:03:42.980 --> 00:03:44.690 And I guess we could have a debate as to 00:03:44.690 --> 00:03:46.010 why that might be bad. 00:03:46.010 --> 00:03:49.870 So let's rationalize this denominator. 00:03:49.870 --> 00:03:55.150 We say x is equal to 10 over the square to 3; to rationalize 00:03:55.150 --> 00:03:57.750 this denominator we can multiply the numerator and the 00:03:57.750 --> 00:03:59.910 denominator by the square root of 3. 00:03:59.910 --> 00:04:02.670 Because as long as we multiply the numerator and the 00:04:02.670 --> 00:04:05.280 denominator by the same thing, it's like multiplying by 1. 00:04:05.280 --> 00:04:09.790 So this is equal to 10 square roots of 3 over square root of 00:04:09.790 --> 00:04:12.996 3 times square of 3; well that's just 3. 00:04:12.996 --> 00:04:16.212 So x equals 10 square roots of 3 over 3. 00:04:16.212 --> 00:04:17.870 That's the hypotenuse. 00:04:17.870 --> 00:04:18.990 I know I confused you. 00:04:18.990 --> 00:04:22.920 And, of course, if this is 10 square root of 3 over 3-- 00:04:22.920 --> 00:04:26.600 that's the hypotenuse --we know that the 30-degree side-- this 00:04:26.600 --> 00:04:28.820 is 30 degrees --we know the 30-degree side is half of 00:04:28.820 --> 00:04:35.430 that, so it's 5 square root of 3 over 3. 00:04:35.430 --> 00:04:38.100 Anyway, I think that might give you a sense of the 00:04:38.100 --> 00:04:40.230 30, 60, 90 triangles. 00:04:40.230 --> 00:04:43.980 I think you might be ready now to try some of the level two 00:04:43.980 --> 00:04:46.080 Pythagorean Theorem problems. 00:04:46.080 --> 00:04:47.600 Have fun.

Intro to 30-60-90 triangles

https://www.youtube.com/watch?v=Qwet4cIpnCM

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en

WEBVTT Kind: captions Language: en 00:00:01.500 --> 00:00:03.430 Sorry for starting the presentation with a cough. 00:00:03.430 --> 00:00:06.220 I think I still have a little bit of a bug going around. 00:00:06.220 --> 00:00:10.980 But now I want to continue with the 45-45-90 triangles. 00:00:10.980 --> 00:00:15.190 So in the last presentation we learned that either side of a 00:00:15.190 --> 00:00:19.830 45-45-90 triangle that isn't the hypotenuse is equal to the 00:00:19.830 --> 00:00:25.600 square route of 2 over 2 times the hypotenuse. 00:00:25.600 --> 00:00:26.850 Let's do a couple of more problems. 00:00:26.850 --> 00:00:30.680 So if I were to tell you that the hypotenuse of this 00:00:30.680 --> 00:00:33.010 triangle-- once again, this only works for 00:00:33.010 --> 00:00:35.760 45-45-90 triangles. 00:00:35.760 --> 00:00:37.870 And if I just draw one 45 you know the other angle's 00:00:37.870 --> 00:00:39.780 got to be 45 as well. 00:00:39.780 --> 00:00:42.960 If I told you that the hypotenuse here is, 00:00:42.960 --> 00:00:44.690 let me say, 10. 00:00:44.690 --> 00:00:46.510 We know this is a hypotenuse because it's opposite 00:00:46.510 --> 00:00:48.340 the right angle. 00:00:48.340 --> 00:00:50.680 And then I would ask you what this side is, x. 00:00:50.680 --> 00:00:54.300 Well we know that x is equal to the square root of 2 over 00:00:54.300 --> 00:00:55.490 2 times the hypotenuse. 00:00:55.490 --> 00:01:01.440 So that's square root of 2 over 2 times 10. 00:01:01.440 --> 00:01:07.700 Or, x is equal to 5 square roots of 2. 00:01:07.700 --> 00:01:07.990 Right? 00:01:07.990 --> 00:01:08.910 10 divided by 2. 00:01:08.910 --> 00:01:12.160 So x is equal to 5 square roots of 2. 00:01:12.160 --> 00:01:15.630 And we know that this side and this side are equal. 00:01:15.630 --> 00:01:15.900 Right? 00:01:15.900 --> 00:01:18.490 I guess we know this is an isosceles triangle because 00:01:18.490 --> 00:01:20.280 these two angles are the same. 00:01:20.280 --> 00:01:23.770 So we also that this side is 5 over 2. 00:01:23.770 --> 00:01:25.830 And if you're not sure, try it out. 00:01:25.830 --> 00:01:27.460 Let's try the Pythagorean theorem. 00:01:27.460 --> 00:01:32.050 We know from the Pythagorean theorem that 5 root 2 squared, 00:01:32.050 --> 00:01:37.420 plus 5 root 2 squared is equal to the hypotenuse squared, 00:01:37.420 --> 00:01:39.090 where the hypotenuse is 10. 00:01:39.090 --> 00:01:41.130 Is equal to 100. 00:01:41.130 --> 00:01:43.170 Or this is just 25 times 2. 00:01:43.170 --> 00:01:43.855 So that's 50. 00:01:48.250 --> 00:01:49.590 But this is 100 up here. 00:01:49.590 --> 00:01:51.380 Is equal to 100. 00:01:51.380 --> 00:01:53.780 And we know, of course, that this is true. 00:01:53.780 --> 00:01:54.620 So it worked. 00:01:54.620 --> 00:01:56.290 We proved it using the Pythagorean theorem, and 00:01:56.290 --> 00:01:57.740 that's actually how we came up with this formula 00:01:57.740 --> 00:01:59.260 in the first place. 00:01:59.260 --> 00:02:00.820 Maybe you want to go back to one of those presentations 00:02:00.820 --> 00:02:03.590 if you forget how we came up with this. 00:02:03.590 --> 00:02:05.890 I'm actually now going to introduce another 00:02:05.890 --> 00:02:06.620 type of triangle. 00:02:06.620 --> 00:02:11.160 And I'm going to do it the same way, by just posing a problem 00:02:11.160 --> 00:02:14.490 to you and then using the Pythagorean theorem 00:02:14.490 --> 00:02:16.980 to figure it out. 00:02:16.980 --> 00:02:18.780 This is another type of triangle called a 00:02:18.780 --> 00:02:20.140 30-60-90 triangle. 00:02:25.550 --> 00:02:28.220 And if I don't have time for this I will do 00:02:28.220 --> 00:02:31.120 another presentation. 00:02:31.120 --> 00:02:33.965 Let's say I have a right triangle. 00:02:38.610 --> 00:02:42.710 That's not a pretty one, but we use what we have. 00:02:42.710 --> 00:02:43.920 That's a right angle. 00:02:43.920 --> 00:02:48.260 And if I were to tell you that this is a 30 degree angle. 00:02:48.260 --> 00:02:49.940 Well we know that the angles in a triangle 00:02:49.940 --> 00:02:51.730 have to add up to 180. 00:02:51.730 --> 00:02:56.570 So if this is 30, this is 90, and let's say that this is x. 00:02:56.570 --> 00:03:02.400 x plus 30 plus 90 is equal to 180, because the angles in 00:03:02.400 --> 00:03:04.310 a triangle add up to 180. 00:03:04.310 --> 00:03:07.770 We know that x is equal to 60. 00:03:07.770 --> 00:03:08.600 Right? 00:03:08.600 --> 00:03:10.870 So this angle is 60. 00:03:10.870 --> 00:03:14.370 And this is why it's called a 30-60-90 triangle-- because 00:03:14.370 --> 00:03:17.320 that's the names of the three angles in the triangle. 00:03:17.320 --> 00:03:24.320 And if I were to tell you that the hypotenuse is-- instead of 00:03:24.320 --> 00:03:27.130 calling it c, like we always do, let's call it h-- and I 00:03:27.130 --> 00:03:30.020 want to figure out the other sides, how do we do that? 00:03:30.020 --> 00:03:32.700 Well we can do that using pretty much the 00:03:32.700 --> 00:03:34.210 Pythagorean theorem. 00:03:34.210 --> 00:03:36.410 And here I'm going to do a little trick. 00:03:36.410 --> 00:03:42.780 Let's draw another copy of this triangle, but flip it over 00:03:42.780 --> 00:03:45.990 draw it the other side. 00:03:45.990 --> 00:03:47.950 And this is the same triangle, it's just facing the 00:03:47.950 --> 00:03:48.690 other direction. 00:03:48.690 --> 00:03:48.910 Right? 00:03:48.910 --> 00:03:51.040 If this is 90 degrees we know that these two 00:03:51.040 --> 00:03:53.140 angles are supplementary. 00:03:53.140 --> 00:03:55.890 You might want to review the angles module if you forget 00:03:55.890 --> 00:03:58.980 that two angles that share kind of this common line would 00:03:58.980 --> 00:04:00.000 add up to 180 degrees. 00:04:00.000 --> 00:04:01.680 So this is 90, this will also be 90. 00:04:01.680 --> 00:04:02.390 And you can eyeball it. 00:04:02.390 --> 00:04:04.010 It makes sense. 00:04:04.010 --> 00:04:06.040 And since we flip it, this triangle is the exact 00:04:06.040 --> 00:04:06.890 same triangle as this. 00:04:06.890 --> 00:04:09.130 It's just flipped over the other side. 00:04:09.130 --> 00:04:12.400 We also know that this angle is 30 degrees. 00:04:12.400 --> 00:04:16.510 And we also know that this angle is 60 degrees. 00:04:16.510 --> 00:04:18.190 Right? 00:04:18.190 --> 00:04:20.450 Well if this angle is 30 degrees and this angle is 30 00:04:20.450 --> 00:04:26.490 degrees, we also know that this larger angle-- goes all the way 00:04:26.490 --> 00:04:30.230 from here to here-- is 60 degrees. 00:04:30.230 --> 00:04:31.770 Right? 00:04:31.770 --> 00:04:34.760 Well if this angle is 60 degrees, this top angle is 60 00:04:34.760 --> 00:04:38.920 degrees, and this angle on the right is 60 degrees, then we 00:04:38.920 --> 00:04:43.910 know from the theorem that we learned when we did 45-45-90 00:04:43.910 --> 00:04:47.860 triangles that if these two angles are the same then the 00:04:47.860 --> 00:04:52.030 sides that they don't share have to be the same as well. 00:04:52.030 --> 00:04:53.440 So what are the sides they don't share? 00:04:53.440 --> 00:04:55.490 Well, it's this side and this side. 00:04:55.490 --> 00:04:58.720 So if this side is h then this side is h. 00:04:58.720 --> 00:05:01.200 Right? 00:05:01.200 --> 00:05:03.680 But this angle is also 60 degrees. 00:05:03.680 --> 00:05:07.600 So if we look at this 60 degrees and this 60 degrees, we 00:05:07.600 --> 00:05:10.760 know that the sides that they don't share are also equal. 00:05:10.760 --> 00:05:13.800 Well they share this side, so the sides that they don't share 00:05:13.800 --> 00:05:15.370 are this side and this side. 00:05:15.370 --> 00:05:19.460 So this side is h, we also know that this side is h. 00:05:19.460 --> 00:05:21.270 Right? 00:05:21.270 --> 00:05:23.470 So it turns out that if you have 60 degrees, 60 degrees, 00:05:23.470 --> 00:05:26.680 and 60 degrees that all the sides have the same lengths, or 00:05:26.680 --> 00:05:27.810 it's an equilateral triangle. 00:05:27.810 --> 00:05:29.670 And that's something to keep in mind. 00:05:29.670 --> 00:05:32.080 And that makes sense too, because an equilateral triangle 00:05:32.080 --> 00:05:33.830 is symmetric no matter how you look at it. 00:05:33.830 --> 00:05:36.030 So it makes sense that all of the angles would be the same 00:05:36.030 --> 00:05:39.370 and all of the sides would have the same length. 00:05:39.370 --> 00:05:40.420 But, hm. 00:05:40.420 --> 00:05:43.090 When we originally did this problem we only used half of 00:05:43.090 --> 00:05:44.050 this equilateral triangle. 00:05:44.050 --> 00:05:48.970 So we know this whole side right here is of length h. 00:05:48.970 --> 00:05:53.670 But if that whole side is of length h, well then this side 00:05:53.670 --> 00:05:56.530 right here, just the base of our original triangle-- and I'm 00:05:56.530 --> 00:05:58.480 trying to be messy on purpose. 00:05:58.480 --> 00:06:00.490 We tried another color. 00:06:00.490 --> 00:06:02.180 This is going to be half of that side. 00:06:02.180 --> 00:06:03.460 Right? 00:06:03.460 --> 00:06:07.890 Because that's h over 2, and this is also h over 2. 00:06:07.890 --> 00:06:08.770 Right over here. 00:06:12.380 --> 00:06:14.990 So if we go back to our original triangle, and we said 00:06:14.990 --> 00:06:17.730 that this is 30 degrees and that this is the hypotenuse, 00:06:17.730 --> 00:06:21.540 because it's opposite the right angle, we know that the side 00:06:21.540 --> 00:06:26.350 opposite the 30 degree side is 1/2 of the hypotenuse. 00:06:26.350 --> 00:06:28.140 And just a reminder, how did we do that? 00:06:28.140 --> 00:06:29.840 Well we doubled the triangle. 00:06:29.840 --> 00:06:31.570 Turned it into an equilateral triangle. 00:06:31.570 --> 00:06:33.490 Figured out this whole side has to be the same 00:06:33.490 --> 00:06:34.490 as the hypotenuse. 00:06:34.490 --> 00:06:36.760 And this is 1/2 of that whole side. 00:06:36.760 --> 00:06:38.420 So it's 1/2 of the hypotenuse. 00:06:38.420 --> 00:06:39.090 So let's remember that. 00:06:39.090 --> 00:06:43.060 The side opposite the 30 degree side is 1/2 of the hypotenuse. 00:06:43.060 --> 00:06:46.530 Let me redraw that on another page, because I think 00:06:46.530 --> 00:06:48.120 this is getting messy. 00:06:48.120 --> 00:06:49.880 So going back to what I had originally. 00:06:54.630 --> 00:06:56.570 This is a right angle. 00:06:56.570 --> 00:06:59.700 This is the hypotenuse-- this side right here. 00:06:59.700 --> 00:07:05.080 If this is 30 degrees, we just derived that the side opposite 00:07:05.080 --> 00:07:09.830 the 30 degrees-- it's like what the angle is opening into-- 00:07:09.830 --> 00:07:12.180 that this is equal to 1/2 the hypotenuse. 00:07:15.190 --> 00:07:17.300 If this is equal to 1/2 the hypotenuse then what 00:07:17.300 --> 00:07:19.450 is this side equal to? 00:07:19.450 --> 00:07:22.660 Well, here we can use the Pythagorean theorem again. 00:07:22.660 --> 00:07:25.685 We know that this side squared plus this side squared-- let's 00:07:25.685 --> 00:07:31.470 call this side A-- is equal to h squared. 00:07:31.470 --> 00:07:43.330 So we have 1/2 h squared plus A squared is equal to h squared. 00:07:43.330 --> 00:07:48.370 This is equal to h squared over 4 plus A squared, 00:07:48.370 --> 00:07:51.690 is equal to h squared. 00:07:51.690 --> 00:07:53.630 Well, we subtract h squared from both sides. 00:07:53.630 --> 00:08:01.270 We get A squared is equal to h squared minus h squared over 4. 00:08:01.270 --> 00:08:07.930 So this equals h squared times 1 minus 1/4. 00:08:07.930 --> 00:08:14.150 This is equal to 3/4 h squared. 00:08:14.150 --> 00:08:17.110 And once going that's equal to A squared. 00:08:17.110 --> 00:08:19.710 I'm running out of space, so I'm going to go all 00:08:19.710 --> 00:08:21.730 the way over here. 00:08:21.730 --> 00:08:27.170 So take the square root of both sides, and we get A is equal 00:08:27.170 --> 00:08:30.920 to-- the square root of 3/4 is the same thing as the 00:08:30.920 --> 00:08:36.270 square root of 3 over 2. 00:08:36.270 --> 00:08:40.510 And then the square root of h squared is just h. 00:08:40.510 --> 00:08:42.350 And this A-- remember, this is an area. 00:08:42.350 --> 00:08:43.990 This is what decides the length of the side. 00:08:43.990 --> 00:08:45.630 I probably shouldn't have used A. 00:08:45.630 --> 00:08:53.070 But this is equal to the square root of 3 over 2, times h. 00:08:53.070 --> 00:08:53.670 So there. 00:08:53.670 --> 00:08:56.320 We've derived what all the sides relative to the 00:08:56.320 --> 00:08:59.320 hypotenuse are of a 30-60-90 triangle. 00:08:59.320 --> 00:09:01.360 So if this is a 60 degree side. 00:09:01.360 --> 00:09:04.750 So if we know the hypotenuse and we know this is a 30-60-90 00:09:04.750 --> 00:09:08.080 triangle, we know the side opposite the 30 degree side 00:09:08.080 --> 00:09:10.500 is 1/2 the hypotenuse. 00:09:10.500 --> 00:09:14.010 And we know the side opposite the 60 degree side is the 00:09:14.010 --> 00:09:18.410 square root of 3 over 2, times the hypotenuse. 00:09:18.410 --> 00:09:22.250 In the next module I'll show you how using this information, 00:09:22.250 --> 00:09:24.120 which you may or may not want to memorize-- it's probably 00:09:24.120 --> 00:09:26.950 good to memorize and practice with, because it'll make you 00:09:26.950 --> 00:09:30.850 very fast on standardized tests-- how we can use this 00:09:30.850 --> 00:09:34.740 information to solve the sides of a 30-60-90 triangle 00:09:34.740 --> 00:09:35.900 very quickly. 00:09:35.900 --> 00:09:37.780 See you in the next presentation.

45-45-90 triangles

https://www.youtube.com/watch?v=tSHitjFIjd8

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en

WEBVTT Kind: captions Language: en 00:00:01.710 --> 00:00:05.420 Welcome to the presentation on 45-45-90 triangles. 00:00:05.420 --> 00:00:07.200 Let me write that down. 00:00:07.200 --> 00:00:08.300 How come the pen-- oh, there you go. 00:00:08.300 --> 00:00:15.770 45-45-90 triangles. 00:00:15.770 --> 00:00:19.050 Or we could say 45-45-90 right triangles, but that might be 00:00:19.050 --> 00:00:21.630 redundant, because we know any angle that has a 90 degree 00:00:21.630 --> 00:00:24.110 measure in it is a right triangle. 00:00:24.110 --> 00:00:27.790 And as you can imagine, the 45-45-90, these are actually 00:00:27.790 --> 00:00:30.910 the degrees of the angles of the triangle. 00:00:30.910 --> 00:00:33.220 So why are these triangles special? 00:00:33.220 --> 00:00:35.720 Well, if you saw the last presentation I gave you a 00:00:35.720 --> 00:00:43.950 little theorem that told you that if two of the base angles 00:00:43.950 --> 00:00:49.000 of a triangle are equal-- and it's I guess only a base angle 00:00:49.000 --> 00:00:49.800 if you draw it like this. 00:00:49.800 --> 00:00:51.830 You could draw it like this, in which case it's maybe not so 00:00:51.830 --> 00:00:55.410 obviously a base angle, but it would still be true. 00:00:55.410 --> 00:00:58.520 If these two angles are equal, then the sides that they don't 00:00:58.520 --> 00:01:02.000 share-- so this side and this side in this example, or this 00:01:02.000 --> 00:01:05.280 side and this side in this example-- then the two sides 00:01:05.280 --> 00:01:07.050 are going to be equal. 00:01:07.050 --> 00:01:11.140 So what's interesting about a 45-45-90 triangle is that 00:01:11.140 --> 00:01:13.900 it is a right triangle that has this property. 00:01:13.900 --> 00:01:16.400 And how do we know that it's the only right triangle 00:01:16.400 --> 00:01:17.690 that has this property? 00:01:17.690 --> 00:01:20.790 Well, you could imagine a world where I told you that 00:01:20.790 --> 00:01:24.140 this is a right triangle. 00:01:24.140 --> 00:01:28.030 This is 90 degrees, so this is the hypotenuse. 00:01:28.030 --> 00:01:32.140 Right, it's the side opposite the 90 degree angle. 00:01:32.140 --> 00:01:36.780 And if I were to tell you that these two angles are equal to 00:01:36.780 --> 00:01:39.640 each other, what do those two angles have to be? 00:01:39.640 --> 00:01:42.840 Well if we call these two angles x, we know that the 00:01:42.840 --> 00:01:44.410 angles in a triangle add up to 180. 00:01:44.410 --> 00:01:49.220 So we'd say x plus x plus-- this is 90-- plus 00:01:49.220 --> 00:01:52.650 90 is equal to 180. 00:01:52.650 --> 00:01:57.950 Or 2x plus 90 is equal to 180. 00:01:57.950 --> 00:02:01.260 Or 2x is equal to 90. 00:02:01.260 --> 00:02:05.500 Or x is equal to 45 degrees. 00:02:05.500 --> 00:02:10.180 So the only right triangle in which the other two angles are 00:02:10.180 --> 00:02:17.990 equal is a 45-45-90 triangle. 00:02:17.990 --> 00:02:22.680 So what's interesting about a 45-45-90 triangle? 00:02:22.680 --> 00:02:27.160 Well other than what I just told you-- let me redraw it. 00:02:27.160 --> 00:02:29.180 I'll redraw it like this. 00:02:29.180 --> 00:02:35.190 So we already know this is 90 degrees, this is 45 degrees, 00:02:35.190 --> 00:02:37.320 this is 45 degrees. 00:02:37.320 --> 00:02:40.370 And based on what I just told you, we also know that the 00:02:40.370 --> 00:02:45.850 sides that the 45 degree angles don't share are equal. 00:02:45.850 --> 00:02:49.560 So this side is equal to this side. 00:02:49.560 --> 00:02:52.080 And if we're viewing it from a Pythagorean theorem point of 00:02:52.080 --> 00:02:55.240 view, this tells us that the two sides that are not the 00:02:55.240 --> 00:02:57.710 hypotenuse are equal. 00:02:57.710 --> 00:02:58.400 So this is a hypotenuse. 00:03:03.660 --> 00:03:09.500 So let's call this side A and this side B. 00:03:09.500 --> 00:03:11.360 We know from the Pythagorean theorem-- let's say the 00:03:11.360 --> 00:03:14.880 hypotenuse is equal to C-- the Pythagorean theorem tells us 00:03:14.880 --> 00:03:21.380 that A squared plus B squared is equal to C squared. 00:03:21.380 --> 00:03:21.863 Right? 00:03:24.720 --> 00:03:26.620 Well we know that A equals B, because this is a 00:03:26.620 --> 00:03:30.070 45-45-90 triangle. 00:03:30.070 --> 00:03:32.010 So we could substitute A for B or B for A. 00:03:32.010 --> 00:03:34.580 But let's just substitute B for A. 00:03:34.580 --> 00:03:38.960 So we could say B squared plus B squared is 00:03:38.960 --> 00:03:41.530 equal to C squared. 00:03:41.530 --> 00:03:47.490 Or 2B squared is equal to C squared. 00:03:47.490 --> 00:03:54.940 Or B squared is equal to C squared over 2. 00:03:54.940 --> 00:04:03.640 Or B is equal to the square root of C squared over 2. 00:04:03.640 --> 00:04:06.530 Which is equal to C-- because we just took the square root of 00:04:06.530 --> 00:04:09.130 the numerator and the square root of the denominator-- C 00:04:09.130 --> 00:04:10.570 over the square root of 2. 00:04:10.570 --> 00:04:15.250 And actually, even though this is a presentation on triangles, 00:04:15.250 --> 00:04:17.630 I'm going to give you a little bit of actually information 00:04:17.630 --> 00:04:19.930 on something called rationalizing denominators. 00:04:19.930 --> 00:04:21.270 So this is perfectly correct. 00:04:21.270 --> 00:04:25.950 We just arrived at B-- and we also know that A equals B-- but 00:04:25.950 --> 00:04:29.510 that B is equal to C divided by the square root of 2. 00:04:29.510 --> 00:04:31.820 It turns out that in most of mathematics, and I never 00:04:31.820 --> 00:04:34.780 understood quite exactly why this was the case, people 00:04:34.780 --> 00:04:37.870 don't like square root of 2s in the denominator. 00:04:37.870 --> 00:04:40.720 Or in general they don't like irrational numbers 00:04:40.720 --> 00:04:41.140 in the denominator. 00:04:41.140 --> 00:04:45.030 Irrational numbers are numbers that have decimal places that 00:04:45.030 --> 00:04:46.920 never repeat and never end. 00:04:46.920 --> 00:04:49.870 So the way that they get rid of irrational numbers in the 00:04:49.870 --> 00:04:52.230 denominator is that you do something called rationalizing 00:04:52.230 --> 00:04:53.570 the denominator. 00:04:53.570 --> 00:04:55.456 And the way you rationalize a denominator-- let's take 00:04:55.456 --> 00:04:56.110 our example right now. 00:04:56.110 --> 00:05:00.640 If we had C over the square root of 2, we just multiply 00:05:00.640 --> 00:05:03.200 both the numerator and the denominator by the 00:05:03.200 --> 00:05:05.130 same number, right? 00:05:05.130 --> 00:05:08.120 Because when you multiply the numerator and the denominator 00:05:08.120 --> 00:05:11.280 by the same number, that's just like multiplying it by 1. 00:05:11.280 --> 00:05:13.680 The square root of 2 over the square root of 2 is 1. 00:05:13.680 --> 00:05:15.530 And as you see, the reason we're doing this is because 00:05:15.530 --> 00:05:17.020 square root of 2 times square root of 2, what's the 00:05:17.020 --> 00:05:19.040 square root of 2 times square root of 2? 00:05:19.040 --> 00:05:20.220 Right, it's 2. 00:05:20.220 --> 00:05:21.030 Right? 00:05:21.030 --> 00:05:23.930 We just said, something times something is 2, well the square 00:05:23.930 --> 00:05:25.990 root of 2 times square root of 2, that's going to be 2. 00:05:25.990 --> 00:05:31.010 And then the numerator is C times the square root of 2. 00:05:31.010 --> 00:05:34.420 So notice, C times the square root of 2 over 2 is the same 00:05:34.420 --> 00:05:37.150 thing as C over the square root of 2. 00:05:37.150 --> 00:05:39.520 And this is important to realize, because sometimes 00:05:39.520 --> 00:05:41.090 while you're taking a standardized test or you're 00:05:41.090 --> 00:05:44.190 doing a test in class, you might get an answer that looks 00:05:44.190 --> 00:05:46.320 like this, has a square root of 2, or maybe even a square root 00:05:46.320 --> 00:05:49.550 of 3 or whatever, in the denominator. 00:05:49.550 --> 00:05:51.420 And you might not see your answer if it's a multiple 00:05:51.420 --> 00:05:52.750 choice question. 00:05:52.750 --> 00:05:55.710 What you ned to do in that case is rationalize the denominator. 00:05:55.710 --> 00:05:57.990 So multiply the numerator and the denominator by square 00:05:57.990 --> 00:06:01.470 root of 2 and you'll get square root of 2 over 2. 00:06:01.470 --> 00:06:03.250 But anyway, back to the problem. 00:06:03.250 --> 00:06:04.450 So what did we learn? 00:06:04.450 --> 00:06:06.880 This is equal to B, right? 00:06:06.880 --> 00:06:11.240 So turns out that B is equal to C times the square 00:06:11.240 --> 00:06:13.420 root of 2 over 2. 00:06:13.420 --> 00:06:14.410 So let me write that. 00:06:14.410 --> 00:06:18.760 So we know that A equals B, right? 00:06:18.760 --> 00:06:27.610 And that equals the square root of 2 over 2 times C. 00:06:27.610 --> 00:06:29.680 Now you might want to memorize this, though you can always 00:06:29.680 --> 00:06:32.440 derive it if you use the Pythagorean theorem and 00:06:32.440 --> 00:06:35.720 remember that the sides that aren't the hypotenuse in a 00:06:35.720 --> 00:06:40.110 45-45-90 triangle are equal to each other. 00:06:40.110 --> 00:06:41.370 But this is very good to know. 00:06:41.370 --> 00:06:44.645 Because if, say, you're taking the SAT and you need to solve a 00:06:44.645 --> 00:06:48.180 problem really fast, and if you have this memorized and someone 00:06:48.180 --> 00:06:49.943 gives you the hypotenuse, you can figure out what are the 00:06:49.943 --> 00:06:51.890 sides very fast, or i8f someone gives you one of the sides, 00:06:51.890 --> 00:06:54.100 you can figure out the hypotenuse very fast. 00:06:54.100 --> 00:06:56.290 Let's try that out. 00:06:56.290 --> 00:06:59.250 I'm going to erase everything. 00:06:59.250 --> 00:07:06.060 So we learned just now that A is equal to B is equal to the 00:07:06.060 --> 00:07:10.210 square root of 2 over 2 times C. 00:07:10.210 --> 00:07:16.220 So if I were to give you a right triangle, and I were to 00:07:16.220 --> 00:07:23.760 tell you that this angle is 90 and this angle is 45, and that 00:07:23.760 --> 00:07:28.570 this side is, let's say this side is 8. 00:07:28.570 --> 00:07:32.670 I want to figure out what this side is. 00:07:32.670 --> 00:07:34.590 Well first of all, let's figure out what side 00:07:34.590 --> 00:07:35.500 is the hypotenuse. 00:07:35.500 --> 00:07:39.620 Well the hypotenuse is the side opposite the right angle. 00:07:39.620 --> 00:07:42.060 So we're trying to actually figure out the hypotenuse. 00:07:42.060 --> 00:07:44.640 Let's call the hypotenuse C. 00:07:44.640 --> 00:07:47.560 And we also know this is a 45-45-90 triangle, right? 00:07:47.560 --> 00:07:50.180 Because this angle is 45, so this one also has to be 45, 00:07:50.180 --> 00:07:54.620 because 45 plus 90 plus 90 is equal to 180. 00:07:54.620 --> 00:07:58.840 So this is a 45-45-90 triangle, and we know one of the sides-- 00:07:58.840 --> 00:08:05.880 this side could be A or B-- we know that 8 is equal to the 00:08:05.880 --> 00:08:10.030 square root of 2 over 2 times C. 00:08:10.030 --> 00:08:12.160 C is what we're trying to figure out. 00:08:12.160 --> 00:08:16.400 So if we multiply both sides of this equation by 2 times the 00:08:16.400 --> 00:08:22.010 square root of 2-- I'm just multiplying it by the inverse 00:08:22.010 --> 00:08:23.600 of the coefficient on C. 00:08:23.600 --> 00:08:25.750 Because the square root of 2 cancels out that square root 00:08:25.750 --> 00:08:28.430 of 2, this 2 cancels out with this 2. 00:08:28.430 --> 00:08:37.640 We get 2 times 8, 16 over the square root of 2 equals C. 00:08:37.640 --> 00:08:40.200 Which would be correct, but as I just showed you, people don't 00:08:40.200 --> 00:08:42.120 like having radicals in the denominator. 00:08:42.120 --> 00:08:46.250 So we can just say C is equal to 16 over the square root of 00:08:46.250 --> 00:08:51.290 2 times the square root of 2 over the square root of 2. 00:08:51.290 --> 00:08:58.790 So this equals 16 square roots of 2 over 2. 00:08:58.790 --> 00:09:04.330 Which is the same thing as 8 square roots of 2. 00:09:04.330 --> 00:09:10.170 So C in this example is 8 square roots of 2. 00:09:10.170 --> 00:09:13.790 And we also knows, since this is a 45-45-90 triangle, 00:09:13.790 --> 00:09:16.700 that this side is 8. 00:09:16.700 --> 00:09:17.940 Hope that makes sense. 00:09:17.940 --> 00:09:19.740 In the next presentation I'm going to show you a 00:09:19.740 --> 00:09:20.680 different type of triangle. 00:09:20.680 --> 00:09:22.900 Actually, I might even start off with a couple more examples 00:09:22.900 --> 00:09:25.080 of this, because I feel I might have rushed it a bit. 00:09:25.080 --> 00:09:28.450 But anyway, I'll see you soon in the next presentation.

Pythagorean theorem II

https://www.youtube.com/watch?v=nMhJLn5ives

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WEBVTT Kind: captions Language: en 00:00:01.090 --> 00:00:02.690 I promised you that I'd give you some more Pythagorean 00:00:02.690 --> 00:00:05.720 theorem problems, so I will now give you more Pythagorean 00:00:05.720 --> 00:00:06.780 theorem problems. 00:00:09.790 --> 00:00:12.382 And once again, this is all about practice. 00:00:12.382 --> 00:00:28.020 Let's say I had a triangle-- that's an ugly looking right 00:00:28.020 --> 00:00:35.030 triangle, let me draw another one --and if I were to tell 00:00:35.030 --> 00:00:40.750 you that that side is 7, the side is 6, and I want to 00:00:40.750 --> 00:00:42.250 figure out this side. 00:00:42.250 --> 00:00:45.510 Well, we learned in the last presentation: which of these 00:00:45.510 --> 00:00:46.990 sides is the hypotenuse? 00:00:46.990 --> 00:00:49.470 Well, here's the right angle, so the side opposite the right 00:00:49.470 --> 00:00:51.600 angle is the hypotenuse. 00:00:51.600 --> 00:00:53.120 So what we want to do is actually figure 00:00:53.120 --> 00:00:54.730 out the hypotenuse. 00:00:54.730 --> 00:01:00.730 So we know that 6 squared plus 7 squared is equal to 00:01:00.730 --> 00:01:01.700 the hypotenuse squared. 00:01:01.700 --> 00:01:03.800 And in the Pythagorean theorem they use C to represent the 00:01:03.800 --> 00:01:05.470 hypotenuse, so we'll use C here as well. 00:01:10.930 --> 00:01:16.030 And 36 plus 49 is equal to C squared. 00:01:21.150 --> 00:01:25.510 85 is equal to C squared. 00:01:25.510 --> 00:01:30.760 Or C is equal to the square root of 85. 00:01:30.760 --> 00:01:32.490 And this is the part that most people have trouble with, is 00:01:32.490 --> 00:01:34.650 actually simplifying the radical. 00:01:34.650 --> 00:01:40.290 So the square root of 85: can I factor 85 so it's a product of 00:01:40.290 --> 00:01:42.820 a perfect square and another number? 00:01:42.820 --> 00:01:45.920 85 isn't divisible by 4. 00:01:45.920 --> 00:01:48.350 So it won't be divisible by 16 or any of the multiples of 4. 00:01:52.400 --> 00:01:55.940 5 goes into 85 how many times? 00:01:55.940 --> 00:01:58.340 No, that's not perfect square, either. 00:01:58.340 --> 00:02:02.030 I don't think 85 can be factored further as a 00:02:02.030 --> 00:02:04.230 product of a perfect square and another number. 00:02:04.230 --> 00:02:06.980 So you might correct me; I might be wrong. 00:02:06.980 --> 00:02:09.570 This might be good exercise for you to do later, but as far as 00:02:09.570 --> 00:02:12.670 I can tell we have gotten our answer. 00:02:12.670 --> 00:02:15.070 The answer here is the square root of 85. 00:02:15.070 --> 00:02:17.250 And if we actually wanted to estimate what that is, let's 00:02:17.250 --> 00:02:21.810 think about it: the square root of 81 is 9, and the square root 00:02:21.810 --> 00:02:25.010 of 100 is 10 , so it's some place in between 9 and 10, and 00:02:25.010 --> 00:02:26.445 it's probably a little bit closer to 9. 00:02:26.445 --> 00:02:28.245 So it's 9 point something, something, something. 00:02:28.245 --> 00:02:30.260 And that's a good reality check; that makes sense. 00:02:30.260 --> 00:02:33.080 If this side is 6, this side is 7, 9 point something, 00:02:33.080 --> 00:02:36.270 something, something makes sense for that length. 00:02:36.270 --> 00:02:37.260 Let me give you another problem. 00:02:37.260 --> 00:02:44.790 [DRAWING] 00:02:44.790 --> 00:02:49.250 Let's say that this is 10 . 00:02:49.250 --> 00:02:51.300 This is 3. 00:02:51.300 --> 00:02:53.090 What is this side? 00:02:53.090 --> 00:02:55.060 First, let's identify our hypotenuse. 00:02:55.060 --> 00:02:57.680 We have our right angle here, so the side opposite the right 00:02:57.680 --> 00:03:00.230 angle is the hypotenuse and it's also the longest side. 00:03:00.230 --> 00:03:01.116 So it's 10. 00:03:01.116 --> 00:03:05.390 So 10 squared is equal to the sum of the squares 00:03:05.390 --> 00:03:06.640 of the other two sides. 00:03:06.640 --> 00:03:10.256 This is equal to 3 squared-- let's call this A. 00:03:10.256 --> 00:03:11.890 Pick it arbitrarily. 00:03:11.890 --> 00:03:14.380 --plus A squared. 00:03:14.380 --> 00:03:23.860 Well, this is 100, is equal to 9 plus A squared, or A squared 00:03:23.860 --> 00:03:29.720 is equal to 100 minus 9. 00:03:29.720 --> 00:03:32.560 A squared is equal to 91. 00:03:38.390 --> 00:03:40.390 I don't think that can be simplified further, either. 00:03:40.390 --> 00:03:41.710 3 doesn't go into it. 00:03:41.710 --> 00:03:43.950 I wonder, is 91 a prime number? 00:03:43.950 --> 00:03:44.880 I'm not sure. 00:03:44.880 --> 00:03:49.200 As far as I know, we're done with this problem. 00:03:49.200 --> 00:03:51.890 Let me give you another problem, And actually, this 00:03:51.890 --> 00:03:56.500 time I'm going to include one extra step just to confuse you 00:03:56.500 --> 00:04:00.240 because I think you're getting this a little bit too easily. 00:04:00.240 --> 00:04:01.805 Let's say I have a triangle. 00:04:05.130 --> 00:04:07.990 And once again, we're dealing all with right triangles now. 00:04:07.990 --> 00:04:10.130 And never are you going to attempt to use the Pythagorean 00:04:10.130 --> 00:04:12.780 theorem unless you know for a fact that's all right triangle. 00:04:16.130 --> 00:04:19.810 But this example, we know that this is right triangle. 00:04:19.810 --> 00:04:25.050 If I would tell you the length of this side is 5, and if our 00:04:25.050 --> 00:04:32.810 tell you that this angle is 45 degrees, can we figure out the 00:04:32.810 --> 00:04:36.410 other two sides of this triangle? 00:04:36.410 --> 00:04:38.220 Well, we can't use the Pythagorean theorem directly 00:04:38.220 --> 00:04:40.830 because the Pythagorean theorem tells us that if have a right 00:04:40.830 --> 00:04:43.750 triangle and we know two of the sides that we can figure 00:04:43.750 --> 00:04:45.140 out the third side. 00:04:45.140 --> 00:04:47.320 Here we have a right triangle and we only 00:04:47.320 --> 00:04:48.870 know one of the sides. 00:04:48.870 --> 00:04:51.080 So we can't figure out the other two just yet. 00:04:51.080 --> 00:04:54.330 But maybe we can use this extra information right here, this 45 00:04:54.330 --> 00:04:57.120 degrees, to figure out another side, and then we'd be able 00:04:57.120 --> 00:04:59.280 use the Pythagorean theorem. 00:04:59.280 --> 00:05:01.810 Well, we know that the angles in a triangle 00:05:01.810 --> 00:05:03.860 add up to 180 degrees. 00:05:03.860 --> 00:05:05.610 Well, hopefully you know the angles in a triangle 00:05:05.610 --> 00:05:06.630 add up to 180 degrees. 00:05:06.630 --> 00:05:08.320 If you don't it's my fault because I haven't taught 00:05:08.320 --> 00:05:09.720 you that already. 00:05:09.720 --> 00:05:14.310 So let's figure out what the angles of this 00:05:14.310 --> 00:05:15.080 triangle add up to. 00:05:15.080 --> 00:05:17.410 Well, I mean we know they add up to 180, but using that 00:05:17.410 --> 00:05:20.790 information, we could figure out what this angle is. 00:05:20.790 --> 00:05:23.590 Because we know that this angle is 90, this angle is 45. 00:05:23.590 --> 00:05:30.340 So we say 45-- lets call this angle x; I'm trying to make it 00:05:30.340 --> 00:05:35.870 messy --45 plus 90-- this [INAUDIBLE] 00:05:35.870 --> 00:05:40.720 is a 90 degree angle --plus is equal to 180 degrees. 00:05:40.720 --> 00:05:43.520 And that's because the angles in a triangle always 00:05:43.520 --> 00:05:46.740 add up to 180 degrees. 00:05:46.740 --> 00:05:55.970 So if we just solve for x, we get 135 plus x is equal to 180. 00:05:55.970 --> 00:05:57.550 Subtract 135 from both sides. 00:05:57.550 --> 00:06:01.190 We get x is equal to 45 degrees. 00:06:01.190 --> 00:06:02.680 Interesting. 00:06:02.680 --> 00:06:06.800 x is also 45 degrees. 00:06:06.800 --> 00:06:11.380 So we have a 90 degree angle and two 45 degree angles. 00:06:11.380 --> 00:06:13.710 Now I'm going to give you another theorem that's not 00:06:13.710 --> 00:06:16.920 named after the head of a religion or the 00:06:16.920 --> 00:06:17.560 founder of religion. 00:06:17.560 --> 00:06:19.730 I actually don't think this theorem doesn't have a name at. 00:06:19.730 --> 00:06:26.920 All It's the fact that if I have another triangle --I'm 00:06:26.920 --> 00:06:31.980 going to draw another triangle out here --where two of the 00:06:31.980 --> 00:06:34.840 base angles are the same-- and when I say base angle, I just 00:06:34.840 --> 00:06:39.890 mean if these two angles are the same, let's call it a. 00:06:39.890 --> 00:06:44.770 They're both a --then the sides that they don't share-- these 00:06:44.770 --> 00:06:46.610 angles share this side, right? 00:06:46.610 --> 00:06:49.560 --but if we look at the sides that they don't share, we know 00:06:49.560 --> 00:06:53.240 that these sides are equal. 00:06:53.240 --> 00:06:54.810 I forgot what we call this in geometry class. 00:06:54.810 --> 00:06:57.270 Maybe I'll look it up in another presentation; 00:06:57.270 --> 00:06:57.960 I'll let you know. 00:06:57.960 --> 00:07:00.040 But I got this far without knowing what the name 00:07:00.040 --> 00:07:01.370 of the theorem is. 00:07:01.370 --> 00:07:04.170 And it makes sense; you don't even need me to tell you that. 00:07:07.080 --> 00:07:10.480 If I were to change one of these angles, the length 00:07:10.480 --> 00:07:11.660 would also change. 00:07:11.660 --> 00:07:14.310 Or another way to think about it, the only way-- no, I 00:07:14.310 --> 00:07:15.350 don't confuse you too much. 00:07:15.350 --> 00:07:18.820 But you can visually see that if these two sides are the 00:07:18.820 --> 00:07:21.670 same, then these two angles are going to be the same. 00:07:21.670 --> 00:07:25.430 If you changed one of these sides' lengths, then the angles 00:07:25.430 --> 00:07:28.660 will also change, or the angles will not be equal anymore. 00:07:28.660 --> 00:07:31.120 But I'll leave that for you to think about. 00:07:31.120 --> 00:07:34.320 But just take my word for it right now that if two angles in 00:07:34.320 --> 00:07:39.400 a triangle are equivalent, then the sides that they don't share 00:07:39.400 --> 00:07:41.690 are also equal in length. 00:07:41.690 --> 00:07:43.820 Make sure you remember: not the side that they share-- because 00:07:43.820 --> 00:07:46.920 that can't be equal to anything --it's the side that they don't 00:07:46.920 --> 00:07:49.410 share are equal in length. 00:07:49.410 --> 00:07:52.990 So here we have an example where we have to equal angles. 00:07:52.990 --> 00:07:55.020 They're both 45 degrees. 00:07:55.020 --> 00:07:58.910 So that means that the sides that they don't share-- this is 00:07:58.910 --> 00:08:00.230 the side they share, right? 00:08:00.230 --> 00:08:03.210 Both angle share this side --so that means that the side that 00:08:03.210 --> 00:08:05.080 they don't share are equal. 00:08:05.080 --> 00:08:08.460 So this side is equal to this side. 00:08:08.460 --> 00:08:10.520 And I think you might be experiencing an ah-hah 00:08:10.520 --> 00:08:12.020 moment that right now. 00:08:12.020 --> 00:08:15.380 Well this side is equal to this side-- I gave you at the 00:08:15.380 --> 00:08:18.050 beginning of this problem that this side is equal to 5 --so 00:08:18.050 --> 00:08:20.320 then we know that this side is equal to 5. 00:08:20.320 --> 00:08:23.920 And now we can do the Pythagorean theorem. 00:08:23.920 --> 00:08:25.750 We know this is the hypotenuse, right? 00:08:28.940 --> 00:08:35.180 So we can say 5 squared plus 5 squared is equal to-- let's say 00:08:35.180 --> 00:08:38.950 C squared, where C is the length of the hypotenuse --5 00:08:38.950 --> 00:08:42.010 squared plus 5 squared-- that's just 50 --is equal 00:08:42.010 --> 00:08:44.110 to C squared. 00:08:44.110 --> 00:08:48.370 And then we get C is equal to the square root of 50. 00:08:48.370 --> 00:08:56.250 And 50 is 2 times 25, so C is equal to 5 square roots of 2. 00:08:56.250 --> 00:08:57.220 Interesting. 00:08:57.220 --> 00:09:00.110 So I think I might have given you a lot of information there. 00:09:00.110 --> 00:09:02.840 If you get confused, maybe you want to re-watch this video. 00:09:02.840 --> 00:09:05.630 But on the next video I'm actually going to give you more 00:09:05.630 --> 00:09:08.095 information about this type of triangle, which is actually a 00:09:08.095 --> 00:09:11.550 very common type of triangle you'll see in geometry and 00:09:11.550 --> 00:09:14.470 trigonometry 45, 45, 90 triangle. 00:09:14.470 --> 00:09:15.930 And it makes sense why it's called that because it has 00:09:15.930 --> 00:09:19.930 45 degrees, 45 degrees, and a 90 degree angle. 00:09:19.930 --> 00:09:22.460 And I'll actually show you a quick way of using that 00:09:22.460 --> 00:09:25.920 information that it is a 45, 45, 90 degree triangle to 00:09:25.920 --> 00:09:29.520 figure out the size if you're given even one of the sides. 00:09:29.520 --> 00:09:31.870 I hope I haven't confused you too much, and I look forward 00:09:31.870 --> 00:09:33.195 to seeing you in the next presentation. 00:09:33.195 --> 00:09:35.120 See you later.

Introduction to the Pythagorean theorem

https://www.youtube.com/watch?v=s9t7rNhaBp8

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WEBVTT Kind: captions Language: en 00:00:01.030 --> 00:00:04.470 Welcome to the presentation on the Pythagorean theorem. 00:00:04.470 --> 00:00:06.820 I apologize if my voice sounds a little horsie. 00:00:06.820 --> 00:00:08.112 A little hoarse, not horsie. 00:00:08.112 --> 00:00:10.070 I was singing a little bit too much last night, 00:00:10.070 --> 00:00:11.460 so please forgive me. 00:00:11.460 --> 00:00:13.710 Well, anyway, we will now teach you 00:00:13.710 --> 00:00:15.330 about the Pythagorean theorem. 00:00:15.330 --> 00:00:17.730 And you might have heard of this before. 00:00:17.730 --> 00:00:21.020 As far as I know, it is the only mathematical theorem 00:00:21.020 --> 00:00:23.681 named after the founder of a religion. 00:00:23.681 --> 00:00:25.680 Pythagoras, actually, I think his whole religion 00:00:25.680 --> 00:00:26.790 was based on mathematics. 00:00:26.790 --> 00:00:28.040 But I'm no historian here. 00:00:28.040 --> 00:00:30.340 So I'll leave that to the historians. 00:00:30.340 --> 00:00:33.070 But anyway, let's get started on what the Pythagorean theorem is 00:00:33.070 --> 00:00:34.540 all about. 00:00:34.540 --> 00:00:37.790 If I were to give you a triangle-- 00:00:37.790 --> 00:00:44.030 let me give you a triangle-- and I 00:00:44.030 --> 00:00:46.820 were to tell you that it's not a normal triangle. 00:00:46.820 --> 00:00:50.150 It is a right triangle. 00:00:50.150 --> 00:00:53.060 And all a right triangle is is a triangle 00:00:53.060 --> 00:00:58.060 that has one side equal to 90 degrees. 00:00:58.060 --> 00:00:59.640 And I'll leave you to think about 00:00:59.640 --> 00:01:01.490 whether it's ever possible for a triangle 00:01:01.490 --> 00:01:04.330 to have more than one side that's 90 degrees. 00:01:04.330 --> 00:01:06.570 But anyway, just granted that a right triangle 00:01:06.570 --> 00:01:08.270 is a side that has at least-- well, 00:01:08.270 --> 00:01:10.410 let me say a right triangle is a triangle that 00:01:10.410 --> 00:01:13.980 has only one side that's at 90 degrees. 00:01:13.980 --> 00:01:16.740 And if you have a right triangle, what the Pythagorean 00:01:16.740 --> 00:01:21.160 theorem allows you to do is if I give you a right triangle 00:01:21.160 --> 00:01:22.960 and I give you two of the sides, we 00:01:22.960 --> 00:01:25.209 can figure out the third side. 00:01:25.209 --> 00:01:26.750 So before I throw the theorem at you, 00:01:26.750 --> 00:01:29.320 let me actually give you a couple of more definitions. 00:01:29.320 --> 00:01:30.950 Actually, just one more. 00:01:30.950 --> 00:01:34.740 So if this is the right angle in a right triangle-- 00:01:34.740 --> 00:01:36.130 it's at 90 degrees. 00:01:36.130 --> 00:01:38.511 And we symbolize that by drawing the angles like this, 00:01:38.511 --> 00:01:40.760 kind of like a box instead of drawing it like a curve, 00:01:40.760 --> 00:01:41.472 like that. 00:01:41.472 --> 00:01:43.430 I hope I'm not messing up the drawing too much. 00:01:48.650 --> 00:01:53.216 The side opposite the right angle is called the hypotenuse. 00:01:55.740 --> 00:01:58.490 And I really should look up where this word comes from. 00:01:58.490 --> 00:02:01.830 Because I think it's a large and unwieldy word, 00:02:01.830 --> 00:02:04.527 and it's a little daunting at first. 00:02:04.527 --> 00:02:06.610 My sister told me that she had a math teacher once 00:02:06.610 --> 00:02:10.389 who made people memorize it's a high pot that is in use. 00:02:10.389 --> 00:02:12.200 So I don't know if that helps you or not. 00:02:12.200 --> 00:02:14.170 But over time, you'll use the term hypotenuse 00:02:14.170 --> 00:02:16.170 so much that it'll seem just like a normal word. 00:02:16.170 --> 00:02:18.003 Although when you look at it, it really does 00:02:18.003 --> 00:02:19.300 look kind of strange. 00:02:19.300 --> 00:02:21.320 Anyway, going back to definitions, 00:02:21.320 --> 00:02:26.310 the hypotenuse is the side opposite the 90-degree angle. 00:02:26.310 --> 00:02:28.400 And if you look at any right triangle, 00:02:28.400 --> 00:02:31.800 you'll also quickly realize that the hypotenuse is the longest 00:02:31.800 --> 00:02:33.930 side of the right triangle. 00:02:33.930 --> 00:02:35.900 So I think we're done now with definitions. 00:02:35.900 --> 00:02:39.760 So what does the Pythagorean theorem tell us? 00:02:39.760 --> 00:02:44.775 Well, let's call C is equal to the length of the hypotenuse. 00:02:48.040 --> 00:02:51.860 And let A be the length of this side. 00:02:51.860 --> 00:02:55.000 And let B equal the length of this side. 00:02:55.000 --> 00:02:56.540 What the Pythagorean theorem tells 00:02:56.540 --> 00:03:09.760 us is that A squared plus B squared is equal to C squared. 00:03:09.760 --> 00:03:11.650 Now, that very simple formula might 00:03:11.650 --> 00:03:14.200 be one of the most powerful formulas in mathematics. 00:03:14.200 --> 00:03:16.680 From this, you go into Euclidean geometry. 00:03:16.680 --> 00:03:18.622 You go into trigonometry. 00:03:18.622 --> 00:03:20.205 You can do anything with this formula, 00:03:20.205 --> 00:03:23.620 but we'll leave that to future lectures. 00:03:23.620 --> 00:03:25.610 Let's actually test this formula. 00:03:25.610 --> 00:03:27.370 Or not test it-- let's use the formula. 00:03:27.370 --> 00:03:28.828 Maybe in another presentation, I'll 00:03:28.828 --> 00:03:32.750 actually do a proof or, at minimum, a visual proof of it. 00:03:32.750 --> 00:03:35.470 I apologize ahead of time that I'm a bit scatterbrained today. 00:03:35.470 --> 00:03:36.990 It's been a while since I last did a video. 00:03:36.990 --> 00:03:39.050 And once again, I told you I sang a little bit too much 00:03:39.050 --> 00:03:39.560 last night. 00:03:39.560 --> 00:03:41.870 So my throat is sore. 00:03:41.870 --> 00:03:43.836 OK, so we have a triangle. 00:03:47.960 --> 00:03:50.446 And remember, it has to be a right triangle. 00:03:50.446 --> 00:03:52.570 So let's say that this is a right angle right here. 00:03:52.570 --> 00:03:54.390 It's 90 degrees. 00:03:54.390 --> 00:04:01.010 And if I were to tell you that this side is of length 4. 00:04:01.010 --> 00:04:02.490 And actually, let me change that. 00:04:02.490 --> 00:04:04.320 This side is of length 3. 00:04:04.320 --> 00:04:06.310 This side is of length 4. 00:04:06.310 --> 00:04:08.720 And we want to figure out the side of this length. 00:04:08.720 --> 00:04:11.650 So the first thing I do when I look at a right triangle 00:04:11.650 --> 00:04:13.440 is I figure out what the hypotenuse is. 00:04:13.440 --> 00:04:15.360 Which side is the hypotenuse? 00:04:15.360 --> 00:04:16.760 Well, there's two ways to do it. 00:04:16.760 --> 00:04:17.801 There's actually one way. 00:04:17.801 --> 00:04:19.380 You look at where the right angle is. 00:04:19.380 --> 00:04:21.420 And it's the side opposite to that. 00:04:21.420 --> 00:04:23.590 So this is the hypotenuse. 00:04:23.590 --> 00:04:27.134 This would be C in our formula, the Pythagorean theorem. 00:04:27.134 --> 00:04:28.550 We could call it whatever we want. 00:04:28.550 --> 00:04:31.680 But just for simplicity, remember A squared plus B 00:04:31.680 --> 00:04:33.490 squared is equal to C squared. 00:04:33.490 --> 00:04:37.170 So in this case, we see that the other two sides, each of them 00:04:37.170 --> 00:04:39.520 squared, when added together will equal C squared. 00:04:39.520 --> 00:04:44.860 So we get 3 squared plus 4 squared 00:04:44.860 --> 00:04:50.390 is equal to C squared, where C is our hypotenuse. 00:04:50.390 --> 00:04:57.120 So 3 squared is 9, plus 16 is equal to C squared. 00:04:57.120 --> 00:05:02.360 25 is equal to C squared. 00:05:02.360 --> 00:05:05.490 And C could be plus or minus 5. 00:05:05.490 --> 00:05:10.730 But we know that you can't have a minus 5 length in geometry. 00:05:10.730 --> 00:05:16.690 So we know that C is equal to 5. 00:05:16.690 --> 00:05:18.310 So using the Pythagorean theorem, 00:05:18.310 --> 00:05:21.887 we just figured out that if we know the sides-- if one side is 00:05:21.887 --> 00:05:23.345 3, the other side is 4, then we can 00:05:23.345 --> 00:05:24.886 use Pythagorean theorem to figure out 00:05:24.886 --> 00:05:30.034 that the hypotenuse of this triangle has the length 5. 00:05:30.034 --> 00:05:31.075 Let's do another example. 00:05:35.140 --> 00:05:42.170 Let's say, once again, this is the right angle. 00:05:42.170 --> 00:05:45.260 This side is of length 12. 00:05:45.260 --> 00:05:48.720 This slide is of length 6. 00:05:48.720 --> 00:05:51.610 And I want to figure out what this side is. 00:05:51.610 --> 00:05:53.830 So let's write down the Pythagorean theorem. 00:05:53.830 --> 00:05:59.010 A squared plus B squared is equal to C squared, where 00:05:59.010 --> 00:06:01.602 C is that length of the hypotenuse. 00:06:01.602 --> 00:06:04.060 So the first thing I want to do when I look at our triangle 00:06:04.060 --> 00:06:07.380 that I just drew is which side is the hypotenuse. 00:06:07.380 --> 00:06:10.710 Well, this right here is the right angle. 00:06:10.710 --> 00:06:14.615 So the hypotenuse is this side right here. 00:06:14.615 --> 00:06:16.240 And we can also eyeball it and say, oh, 00:06:16.240 --> 00:06:18.750 that's definitely the longest side of this triangle. 00:06:18.750 --> 00:06:23.170 So we know that A squared plus B squared 00:06:23.170 --> 00:06:27.110 is equal to 12 squared, which is 144. 00:06:27.110 --> 00:06:30.960 Now we know we have one side, but we 00:06:30.960 --> 00:06:32.220 don't have the other side. 00:06:32.220 --> 00:06:33.636 So I've got to ask you a question. 00:06:33.636 --> 00:06:36.210 Does it matter which side we substitute for A or B? 00:06:36.210 --> 00:06:38.491 Well, no, just because A or B-- they 00:06:38.491 --> 00:06:40.240 kind of do the same thing in this formula. 00:06:40.240 --> 00:06:43.087 So we could pick any side to be A other than the hypotenuse. 00:06:43.087 --> 00:06:44.670 And we'll pick the other side to be B. 00:06:44.670 --> 00:06:48.320 So let's just say that this side is B, 00:06:48.320 --> 00:06:51.700 and let's say that this side is A. 00:06:51.700 --> 00:06:52.990 So we know what A is. 00:06:52.990 --> 00:07:00.490 So we get 6 squared plus B squared is equal to 144. 00:07:00.490 --> 00:07:08.790 So we get 36 plus B squared is equal to 144. 00:07:08.790 --> 00:07:16.320 B squared is equal to 144 minus 36. 00:07:16.320 --> 00:07:20.870 B squared is equal to 112. 00:07:20.870 --> 00:07:23.690 Now we've got to simplify what the square root of 112 is. 00:07:23.690 --> 00:07:25.820 And what we did in those radical modules 00:07:25.820 --> 00:07:27.459 probably is helpful here. 00:07:27.459 --> 00:07:28.000 So let's see. 00:07:28.000 --> 00:07:32.246 B is equal to the square root of 112. 00:07:32.246 --> 00:07:33.120 Let's think about it. 00:07:33.120 --> 00:07:34.940 How many times does 4 go into 112? 00:07:34.940 --> 00:07:39.780 4 goes into 120 five times, so it'll go into it 28 times. 00:07:39.780 --> 00:07:42.420 And then 4 goes into 28 seven times. 00:07:42.420 --> 00:07:48.780 So I actually think that this is equal to 16 times 7. 00:07:48.780 --> 00:07:49.400 Am I right? 00:07:49.400 --> 00:07:53.600 7 times 10 is 70, plus 42 is 112. 00:07:53.600 --> 00:07:54.350 Right. 00:07:54.350 --> 00:07:56.770 So B equals the square root of 16 times 7. 00:07:56.770 --> 00:07:59.190 See, I just factored that as a product 00:07:59.190 --> 00:08:01.239 of a perfect square and a prime number. 00:08:01.239 --> 00:08:03.530 Or actually, it doesn't have to be a prime number, just 00:08:03.530 --> 00:08:05.630 a non-perfect square. 00:08:05.630 --> 00:08:12.680 And then I get B is equal to 4 square roots of 7. 00:08:12.680 --> 00:08:14.120 So there we go. 00:08:14.120 --> 00:08:23.360 If this is 12, this is 6, this is 4 square roots of 7. 00:08:23.360 --> 00:08:26.590 I think that's all the time I have now for this presentation. 00:08:26.590 --> 00:08:28.590 Right after this, I'll do one more presentation 00:08:28.590 --> 00:08:32.460 where I give a couple of more Pythagorean theorem problems. 00:08:32.460 --> 00:08:34.030 See you soon.

More percent problems

https://www.youtube.com/watch?v=4oeoIOan_h4

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en

WEBVTT Kind: captions Language: en 00:00:01.210 --> 00:00:07.750 Let's say I go to a store and I have $50 in my pocket. 00:00:07.750 --> 00:00:11.070 $50 in my wallet. 00:00:14.110 --> 00:00:18.670 And at the store that day they say it is a 25% 00:00:18.670 --> 00:00:24.680 off marked price sale. 00:00:24.680 --> 00:00:28.880 So 25% off marked price means that if the marked price is 00:00:28.880 --> 00:00:31.670 $100 the price I'm going to pay is going to be 00:00:31.670 --> 00:00:34.390 25% less than $100. 00:00:34.390 --> 00:00:40.830 So my question to you is if I have $50, what is the highest 00:00:40.830 --> 00:00:42.600 marked price I can afford? 00:00:42.600 --> 00:00:45.270 Because I need to know that before I go finding something 00:00:45.270 --> 00:00:47.270 that I might like. 00:00:47.270 --> 00:00:49.920 So let's do a little bit of algebra. 00:00:49.920 --> 00:01:05.240 So let x be the highest marked price that I can afford. 00:01:05.240 --> 00:01:10.290 So if the sale is 25% off of x, we could say that the new 00:01:10.290 --> 00:01:20.680 price, the sale price will be x minus 25% of x is equal 00:01:20.680 --> 00:01:22.790 to the sale price. 00:01:22.790 --> 00:01:26.730 And I'm assuming that I'm in a state without sales tax. 00:01:26.730 --> 00:01:29.760 Whatever the sale price is, is what I have to pay in cash. 00:01:29.760 --> 00:01:33.820 So x minus 25% x is equal to the sale price. 00:01:33.820 --> 00:01:35.860 The discount is going to be 25% of x. 00:01:35.860 --> 00:01:44.490 But we know that this is the same thing as x minus 0.25x. 00:01:44.490 --> 00:01:46.530 And we know that that's the same thing as-- well, because 00:01:46.530 --> 00:01:49.600 we know this is 1x, x is the same thing is 1x. 00:01:49.600 --> 00:01:50.910 1x minus 0.25x. 00:01:50.910 --> 00:02:00.050 Well, that means that 0.75x is equal to the sale price, right? 00:02:00.050 --> 00:02:07.180 All I did is I rewrote x minus 25% of x as 1x minus 0.25x. 00:02:07.180 --> 00:02:09.300 And that's the same thing as 0.75x. 00:02:09.300 --> 00:02:12.180 Because 1 minus 0.25 is 0.75. 00:02:12.180 --> 00:02:15.290 So 0.75x is going to be the sale price. 00:02:15.290 --> 00:02:17.980 Well, what's the sale price that I can afford? 00:02:17.980 --> 00:02:22.440 Well, the sale price I can afford is $50. 00:02:22.440 --> 00:02:28.725 So 0.75x is going to equal $50. 00:02:31.430 --> 00:02:34.820 If x is any larger number than the number I'm solving for, 00:02:34.820 --> 00:02:37.330 then the sale price is going to be more than $50 and I 00:02:37.330 --> 00:02:38.360 won't be able to afford it. 00:02:38.360 --> 00:02:41.520 So that's how we set the-- the highest I can pay is $50 00:02:41.520 --> 00:02:42.950 and that's the sale price. 00:02:42.950 --> 00:02:45.170 So going back to how we did these problems before. 00:02:45.170 --> 00:02:47.710 We just divide both sides by 0.75. 00:02:47.710 --> 00:02:52.120 And we say that the highest marked price that I can afford 00:02:52.120 --> 00:02:58.010 is $50 divided by 0.75. 00:02:58.010 --> 00:03:00.410 And let's figure out what that is. 00:03:00.410 --> 00:03:07.490 0.75 goes into 50-- let's add some 0's in the back. 00:03:07.490 --> 00:03:09.815 If I take this decimal 2 to the right. 00:03:09.815 --> 00:03:15.220 Take this decimal, move it 2 to the right, goes right there. 00:03:15.220 --> 00:03:18.930 So 0.75 goes into 50 the same number of times 00:03:18.930 --> 00:03:22.950 that 75 goes into 5,000. 00:03:22.950 --> 00:03:23.790 So let's do this. 00:03:23.790 --> 00:03:25.870 75 goes into 50 zero times. 00:03:25.870 --> 00:03:29.710 75 goes into 500-- so let me think about that. 00:03:29.710 --> 00:03:32.065 I think it goes into it six times. 00:03:34.730 --> 00:03:36.110 Because seven times is going to be too much. 00:03:36.110 --> 00:03:40.040 So it goes into it six times. 00:03:40.040 --> 00:03:44.840 6 times 5 is 30. 00:03:44.840 --> 00:03:46.890 6 times 7 is 42. 00:03:46.890 --> 00:03:50.040 Plus 3 is 45. 00:03:50.040 --> 00:03:53.150 So the remainder is 50. 00:03:53.150 --> 00:03:54.270 I see a pattern. 00:03:54.270 --> 00:03:55.640 Bring down the 0. 00:03:55.640 --> 00:03:56.640 Well, same thing again. 00:03:56.640 --> 00:04:00.620 75 goes into 500 six times. 00:04:00.620 --> 00:04:03.670 6 times 75 is going to be 450 again. 00:04:03.670 --> 00:04:05.730 We're going to keep having that same pattern over 00:04:05.730 --> 00:04:06.600 and over and over again. 00:04:06.600 --> 00:04:12.100 It's actually 66.666-- I hope you don't think I'm an evil 00:04:12.100 --> 00:04:15.500 person because of this number that happened to show up. 00:04:15.500 --> 00:04:19.090 But anyway, so the highest sale price that I can afford or the 00:04:19.090 --> 00:04:24.270 highest marked price I can afford is $66 dollars. 00:04:24.270 --> 00:04:28.420 And if I were to around up, and $0.67 if I were to 00:04:28.420 --> 00:04:31.070 round to the nearest penny. 00:04:31.070 --> 00:04:34.350 If I were to write this kind of as a repeating decimal, I could 00:04:34.350 --> 00:04:38.800 write this as 66.66 repeating. 00:04:38.800 --> 00:04:42.150 Or I also know that 0.6666 going on forever is 00:04:42.150 --> 00:04:43.230 the same thing as 2/3. 00:04:43.230 --> 00:04:46.580 So it's 66 and 2/3. 00:04:46.580 --> 00:04:48.290 But since we're working with money and we're working with 00:04:48.290 --> 00:04:50.280 dollars, we should just round to the nearest penny. 00:04:50.280 --> 00:04:55.320 So the highest marked price that I can afford is $66.67. 00:04:55.320 --> 00:04:58.830 So if I go and I see a nice pair of shoes for 00:04:58.830 --> 00:05:02.360 $55, I can afford it. 00:05:02.360 --> 00:05:05.580 If I see a nice tie for $70, I can't afford it with 00:05:05.580 --> 00:05:07.350 the $50 in my pocket. 00:05:07.350 --> 00:05:10.620 So hopefully not only will this teach you a little bit of math, 00:05:10.620 --> 00:05:13.680 but it'll help you do a little bit of shopping. 00:05:13.680 --> 00:05:15.430 So let me ask you another problem, a very 00:05:15.430 --> 00:05:17.760 interesting problem. 00:05:17.760 --> 00:05:22.480 Let's say I start with an arbitrary-- let's put 00:05:22.480 --> 00:05:23.210 a fixed number on it. 00:05:23.210 --> 00:05:27.260 Let's say I start with $100. 00:05:27.260 --> 00:05:35.630 And after one year it grows by 25%. 00:05:38.690 --> 00:05:42.000 And then the next year, let's call that year 00:05:42.000 --> 00:05:45.300 two, it shrinks by 25%. 00:05:45.300 --> 00:05:47.100 So this could have happened in the stock market. 00:05:47.100 --> 00:05:49.370 The first year I have a good year, my portfolio 00:05:49.370 --> 00:05:51.070 grows by 25%. 00:05:51.070 --> 00:05:52.990 The second year I have a bad year and my portfolio 00:05:52.990 --> 00:05:54.730 shrinks by 25%. 00:05:54.730 --> 00:05:57.540 So my question is how much money do I have at the 00:05:57.540 --> 00:05:59.930 end of the two years? 00:05:59.930 --> 00:06:02.070 Well a lot of people might say, oh, this is easy, Sal. 00:06:02.070 --> 00:06:06.270 If I grow by 25% and then I shrink by 25% I'll end up with 00:06:06.270 --> 00:06:08.320 the same amount of money. 00:06:08.320 --> 00:06:13.780 But I'll show you it's actually not that simple because the 25% 00:06:13.780 --> 00:06:16.520 in either case or in both cases is actually a different 00:06:16.520 --> 00:06:18.100 amount of money. 00:06:18.100 --> 00:06:19.590 So let's figure this out. 00:06:19.590 --> 00:06:27.100 If I start with $100 and I grow it by 25%-- 25% of $100 is $25. 00:06:27.100 --> 00:06:28.260 So I grew it by $25. 00:06:28.260 --> 00:06:34.300 So I go to $125. 00:06:34.300 --> 00:06:38.590 So after one year of growing by 25% I end up with $125. 00:06:38.590 --> 00:06:44.670 And now this $125 is going to shrink by 25$. 00:06:44.670 --> 00:06:48.320 So if something shrinks by 25%, that means it's just going to 00:06:48.320 --> 00:06:52.680 be 0.75 or 75% of what it was before, right? 00:06:52.680 --> 00:06:55.850 1 minus 25%. 00:06:55.850 --> 00:06:59.500 0.75 times $125. 00:06:59.500 --> 00:07:01.830 So let's work that out here. 00:07:01.830 --> 00:07:07.325 $125 times 0.75. 00:07:07.325 --> 00:07:10.770 And just in case you're confused, I don't want to 00:07:10.770 --> 00:07:13.800 repeat it too much, but if something shrinks by 25% it is 00:07:13.800 --> 00:07:16.790 now 75% of its original value. 00:07:16.790 --> 00:07:23.140 So if $125 shrinks by 25% it's now 75% of $125 or 0.75. 00:07:23.140 --> 00:07:24.440 Let's do the math. 00:07:24.440 --> 00:07:27.980 5 times 5 is 25. 00:07:27.980 --> 00:07:31.810 2 times 5 is 10 plus 2 is 12. 00:07:31.810 --> 00:07:34.800 1 times 5-- 7. 00:07:34.800 --> 00:07:38.510 7 times 5 is 35. 00:07:38.510 --> 00:07:40.100 7 times 2 is 14. 00:07:40.100 --> 00:07:43.750 Plus 3 is 17. 00:07:43.750 --> 00:07:45.560 Sorry. 00:07:45.560 --> 00:07:46.880 7 times 1 is 7. 00:07:46.880 --> 00:07:48.980 Plus 1 is 8. 00:07:48.980 --> 00:07:53.800 So it's 5, 7, and then this is 7 actually. 00:07:53.800 --> 00:07:55.130 14. 00:07:55.130 --> 00:07:56.490 9. 00:07:56.490 --> 00:07:58.260 94.75, right? 00:07:58.260 --> 00:08:00.230 Two decimal points. 00:08:00.230 --> 00:08:03.410 94.75. 00:08:03.410 --> 00:08:05.750 So it's interesting. 00:08:05.750 --> 00:08:11.390 If I start with $100 and it grows by 25%, and then it 00:08:11.390 --> 00:08:16.450 shrinks by 25% I end up with less than I started with. 00:08:16.450 --> 00:08:18.890 And I want you to think about why that happens. 00:08:18.890 --> 00:08:24.400 Because 25% on $100 is the amount that I'm gaining. 00:08:24.400 --> 00:08:26.790 That's a smaller number than the amount that I'm losing. 00:08:26.790 --> 00:08:31.330 I'm losing 25% on $125. 00:08:31.330 --> 00:08:33.280 That's pretty interesting, don't you think? 00:08:33.280 --> 00:08:35.100 That's actually very interesting when a lot of 00:08:35.100 --> 00:08:39.480 people compare-- well, actually I won't go into stock 00:08:39.480 --> 00:08:40.340 returns and things. 00:08:40.340 --> 00:08:41.825 But I think that should be a pretty interesting thing. 00:08:41.825 --> 00:08:43.290 You should try that out with other examples. 00:08:43.290 --> 00:08:46.700 Another interesting thing is for any percentage gain, you 00:08:46.700 --> 00:08:49.050 should think about how much you would have to lose-- what 00:08:49.050 --> 00:08:51.260 percentage you would have to lose to end 00:08:51.260 --> 00:08:52.050 up where you started. 00:08:52.050 --> 00:08:54.080 That's another interesting project. 00:08:54.080 --> 00:08:56.700 Maybe I'll do that in a future presentation. 00:08:56.700 --> 00:08:59.430 Anyway, I think you're now ready to do some of those 00:08:59.430 --> 00:09:02.200 percent madness problems. 00:09:02.200 --> 00:09:03.330 Hope you have fun. 00:09:03.330 --> 00:09:04.630 Bye.

Taking percentages

https://www.youtube.com/watch?v=_SpE4hQ8D_o

vtt

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en

WEBVTT Kind: captions Language: en 00:00:00.900 --> 00:00:03.820 Let's get started with some problems. 00:00:03.820 --> 00:00:04.620 Let's see. 00:00:04.620 --> 00:00:19.806 First problem: what is 15% of 40? 00:00:19.806 --> 00:00:22.610 The way I do percent problems is I just convert the 00:00:22.610 --> 00:00:25.630 percentage to a decimal and then I multiply it times the 00:00:25.630 --> 00:00:27.720 number that I'm trying to get the percentage of. 00:00:27.720 --> 00:00:32.820 So 15% as a decimal is 0.15. 00:00:32.820 --> 00:00:36.360 You learned that from the percent to decimal conversion 00:00:36.360 --> 00:00:37.640 video, hopefully. 00:00:37.640 --> 00:00:39.550 And we just multiply this times 40. 00:00:39.550 --> 00:00:47.990 So let's say 40 times 0.15. 00:00:47.990 --> 00:00:49.510 5 times 0 is 0. 00:00:49.510 --> 00:00:53.100 5 times 4 is 20. 00:00:53.100 --> 00:00:54.620 Put a 0 there. 00:00:54.620 --> 00:00:56.660 And then 1 times 0 is 0. 00:00:56.660 --> 00:00:59.266 1 times 4 is 4. 00:00:59.266 --> 00:01:02.720 And you get 6 0 0. 00:01:02.720 --> 00:01:04.490 Then you count the decimal spots. 00:01:04.490 --> 00:01:05.670 1, 2. 00:01:05.670 --> 00:01:08.740 No decimals up there, so you go 1, 2 and you 00:01:08.740 --> 00:01:10.230 put the decimal there. 00:01:10.230 --> 00:01:18.870 So 15% of 40 is equal to 0.15 times 40, which equals 6.00. 00:01:18.870 --> 00:01:22.480 Well, that's just the same thing as 6. 00:01:22.480 --> 00:01:23.640 Let's do another problem. 00:01:23.640 --> 00:01:25.710 Hopefully, that didn't confuse you too much. 00:01:25.710 --> 00:01:27.960 And I'm going to try to confuse you this time just 00:01:27.960 --> 00:01:31.050 in case you weren't properly confused the last time. 00:01:31.050 --> 00:01:47.860 What is 0.2% of-- let me think of a number-- of 7. 00:01:47.860 --> 00:01:50.060 So a lot of people's inclinations would just 00:01:50.060 --> 00:01:55.010 say, oh, 0.2%, that's the same thing as 0.2. 00:01:55.010 --> 00:01:58.950 And if that was your inclination you would be wrong. 00:01:58.950 --> 00:02:00.610 Because remember, this isn't 0.2. 00:02:00.610 --> 00:02:03.290 This is 0.2%. 00:02:03.290 --> 00:02:04.760 So there's two ways of thinking about this. 00:02:04.760 --> 00:02:10.350 You could say that this is 0.2/100, which is, if you 00:02:10.350 --> 00:02:13.270 multiply the numerator and denominator by 10, is the 00:02:13.270 --> 00:02:17.830 same thing as 2/1,000. 00:02:17.830 --> 00:02:19.260 Or you can just do the technique where you 00:02:19.260 --> 00:02:22.260 move the decimal space over 2 to the left. 00:02:22.260 --> 00:02:26.360 In which case, if you're starting with 0.2 and you 00:02:26.360 --> 00:02:29.620 move the decimal space 2 to the left, you go bam. 00:02:29.620 --> 00:02:30.300 Whoops! 00:02:30.300 --> 00:02:32.090 Bam, bam. 00:02:32.090 --> 00:02:33.170 That's where the decimal goes. 00:02:33.170 --> 00:02:37.370 So it's 0.002. 00:02:37.370 --> 00:02:38.410 This is key. 00:02:38.410 --> 00:02:43.370 0.2% is the same thing as 0.002. 00:02:43.370 --> 00:02:46.110 This can always trip you up and I've made this careless mistake 00:02:46.110 --> 00:02:47.890 all the time, so don't feel bad if you ever do it. 00:02:47.890 --> 00:02:49.460 But just always pay careful attention if you see a 00:02:49.460 --> 00:02:52.050 decimal and a percentage at the same time. 00:02:52.050 --> 00:02:55.400 So now that we've figured out how to write this percentage 00:02:55.400 --> 00:02:58.210 as a decimal we just have to multiply it times the number 00:02:58.210 --> 00:03:00.000 that we want to take the percentage of. 00:03:00.000 --> 00:03:09.190 So we say 0.002 times 7. 00:03:09.190 --> 00:03:10.490 Well, this is pretty straightforward. 00:03:10.490 --> 00:03:15.070 7 times 2 is 14. 00:03:15.070 --> 00:03:17.710 And how many total numbers do we have or how many total 00:03:17.710 --> 00:03:19.830 digits do we have behind the decimal point? 00:03:19.830 --> 00:03:20.240 Let's see. 00:03:20.240 --> 00:03:22.930 It's 1, 2, 3. 00:03:22.930 --> 00:03:28.600 So we need 1, 2, 3 digits behind the decimal point. 00:03:28.600 --> 00:03:36.175 So 0.2% of 7 is equal to 0.014. 00:03:36.175 --> 00:03:38.200 And you're probably thinking, boy, that's a really, 00:03:38.200 --> 00:03:39.360 really small number. 00:03:39.360 --> 00:03:44.545 And it makes sense because 0.2%, if you want to 00:03:44.545 --> 00:03:46.860 think about it, that's smaller than even 1%. 00:03:46.860 --> 00:03:48.585 So that's even smaller than 1/100. 00:03:48.585 --> 00:03:52.450 And actually, if you think about it, 0.2% is 1/500. 00:03:52.450 --> 00:03:55.760 And if you do the math, 1/500 of 7 will turn 00:03:55.760 --> 00:03:58.190 out to be this number. 00:03:58.190 --> 00:03:59.500 And that's an important thing to do. 00:03:59.500 --> 00:04:01.240 It's always good to do a reality check because when 00:04:01.240 --> 00:04:04.560 you're doing these decimal and these percent problems, it's 00:04:04.560 --> 00:04:09.280 very easy to kind of lose a factor of 10 here or there. 00:04:09.280 --> 00:04:10.240 Or gain a factor of 10. 00:04:10.240 --> 00:04:14.620 So always do a reality check to see if your answer makes sense. 00:04:14.620 --> 00:04:18.630 So now I'm going to confuse you even further. 00:04:18.630 --> 00:04:27.940 What if I were to ask you 4 is 20% of what number? 00:04:31.600 --> 00:04:33.730 So a lot of people's reflex might just be, 00:04:33.730 --> 00:04:34.850 oh, let me take 20%. 00:04:34.850 --> 00:04:36.360 It becomes 0.20. 00:04:36.360 --> 00:04:37.850 And multiply it times 4. 00:04:37.850 --> 00:04:41.570 And in that case, again, you may be wrong. 00:04:41.570 --> 00:04:42.080 Because think about it. 00:04:42.080 --> 00:04:44.700 I'm not saying what is 20% of 4? 00:04:44.700 --> 00:04:49.560 I'm saying that 20% of some number is 4. 00:04:49.560 --> 00:04:51.470 So now we're going to be doing a little bit of algebra. 00:04:51.470 --> 00:04:54.090 I bet you didn't expect that in the percent module. 00:04:54.090 --> 00:04:59.080 So let x equal the number. 00:05:05.760 --> 00:05:14.630 And this problem says that 20% of x is equal to 4. 00:05:14.630 --> 00:05:17.500 I think now it's in a form that you might recognize. 00:05:17.500 --> 00:05:19.240 So how do we write 20% as a decimal? 00:05:19.240 --> 00:05:21.970 Well, that's just 0.20 or 0.2. 00:05:21.970 --> 00:05:23.890 And we just multiply it by x to get 4. 00:05:23.890 --> 00:05:28.080 So 20%, that's the same thing as 0.2. 00:05:28.080 --> 00:05:30.450 It's the same thing as 0.20, but that last trailing 00:05:30.450 --> 00:05:31.720 0 doesn't mean much. 00:05:31.720 --> 00:05:36.160 0.2 times x is equal to 4. 00:05:36.160 --> 00:05:38.080 And now we have a level one linear equation. 00:05:38.080 --> 00:05:39.670 I bet you didn't expect to see that. 00:05:39.670 --> 00:05:42.540 So what do we do? 00:05:42.540 --> 00:05:43.550 Well there's two ways to view it. 00:05:43.550 --> 00:05:45.820 You can just divide both sides of this equation 00:05:45.820 --> 00:05:47.700 by the coefficient on x. 00:05:47.700 --> 00:05:54.010 So if you divide 0.2 here and you divide by 0.2 here. 00:05:54.010 --> 00:05:59.460 So you get x is equal to 4 divided by 0.2. 00:05:59.460 --> 00:06:03.350 So let's figure out what 4 divided by 0.2 is. 00:06:03.350 --> 00:06:05.120 I hope I have enough space. 00:06:05.120 --> 00:06:12.500 0.2 goes into 4-- I'm going to put a decimal point here. 00:06:12.500 --> 00:06:14.300 And the way we do these problems, we move the 00:06:14.300 --> 00:06:16.310 decimal point here one over to the right. 00:06:16.310 --> 00:06:18.520 So we just get a 2 and then we can move the decimal point 00:06:18.520 --> 00:06:20.420 here one over to the right. 00:06:20.420 --> 00:06:23.680 So this 0.2 goes into 4 the same number of times 00:06:23.680 --> 00:06:26.012 that 2 goes into 40. 00:06:26.012 --> 00:06:27.530 And this is easy. 00:06:27.530 --> 00:06:29.670 2 goes into 40 how many times? 00:06:29.670 --> 00:06:32.152 Well, 2 goes into 4 two times and then 2 goes 00:06:32.152 --> 00:06:32.980 into 0, zero times. 00:06:32.980 --> 00:06:34.050 You could've done that in your head. 00:06:34.050 --> 00:06:36.530 2 into 40 is twenty times. 00:06:36.530 --> 00:06:40.580 So 4 divided by 0.2 is 20. 00:06:40.580 --> 00:06:44.570 So the answer is 4 is 20% of 20. 00:06:47.120 --> 00:06:48.280 And does that make sense? 00:06:48.280 --> 00:06:49.640 Well, there's a couple of ways to think about it. 00:06:49.640 --> 00:06:53.610 20% is exactly 1/5. 00:06:53.610 --> 00:06:55.810 And 4 times 5 is 20. 00:06:55.810 --> 00:06:56.810 That makes sense. 00:06:56.810 --> 00:06:59.430 If you're still not sure we can check the problem. 00:06:59.430 --> 00:07:02.630 Let's take 20% of 20. 00:07:02.630 --> 00:07:12.880 So 20% of 20 is equal to 0.2 times 20. 00:07:12.880 --> 00:07:16.182 And if you do the math that also will equal 4. 00:07:16.182 --> 00:07:18.410 So you made sure you got the right answer. 00:07:18.410 --> 00:07:19.490 Let's do another one like that. 00:07:24.000 --> 00:07:25.810 I'm picking numbers randomly. 00:07:25.810 --> 00:07:36.140 Let's say 3 is 9% of what? 00:07:40.440 --> 00:07:52.320 Once again, let's let x equal the number that 3 is 9% of. 00:07:52.320 --> 00:07:53.680 You didn't have to write all that. 00:07:53.680 --> 00:08:00.980 Well, in that case we know that 0.09x-- 0.09, that's the same 00:08:00.980 --> 00:08:04.430 thing as 9% of x-- is equal to 3. 00:08:04.430 --> 00:08:10.770 Or that x is equal to 3 divided by 0.09. 00:08:10.770 --> 00:08:15.440 Well, if we do the decimal division, 0.09 goes into 3. 00:08:15.440 --> 00:08:17.290 Let's put a decimal point here. 00:08:17.290 --> 00:08:19.260 I don't know how many 0's I'm going to need. 00:08:19.260 --> 00:08:22.220 So if I move this decimal over to the right twice, then I'll 00:08:22.220 --> 00:08:24.980 move this decimal over to the right twice. 00:08:24.980 --> 00:08:27.580 So 0.09 goes into 3 the same number of times 00:08:27.580 --> 00:08:29.960 that 9 goes into 300. 00:08:29.960 --> 00:08:32.070 So 9 goes into 30 three times. 00:08:32.070 --> 00:08:33.820 3 times nine is 27. 00:08:33.820 --> 00:08:36.050 I think I see a pattern here already. 00:08:36.050 --> 00:08:41.310 30, 3, 3 times 9 is 27. 00:08:41.310 --> 00:08:45.010 You're going to keep getting 33-- the 3's are just 00:08:45.010 --> 00:08:46.490 going to go on forever. 00:08:46.490 --> 00:08:50.960 So it turns out that 3 is 9% of-- you can either write it as 00:08:50.960 --> 00:08:58.870 33.3 repeating or we all know that 0.3 forever is the 00:08:58.870 --> 00:09:00.310 same thing as 1/3. 00:09:00.310 --> 00:09:05.380 So 3 is 9% of 33 and 1/3. 00:09:05.380 --> 00:09:08.220 Either one of those would be an acceptable answer. 00:09:08.220 --> 00:09:09.700 And a lot of times when you're doing percentages you're 00:09:09.700 --> 00:09:11.885 actually just trying to get a ballpark. 00:09:11.885 --> 00:09:15.790 The precision might not always be the most important thing, 00:09:15.790 --> 00:09:18.060 but in this case we will be precise. 00:09:18.060 --> 00:09:19.390 And obviously, on tests and things you need to 00:09:19.390 --> 00:09:20.940 be precise as well. 00:09:20.940 --> 00:09:23.020 Hopefully, I didn't go too fast and you have a good 00:09:23.020 --> 00:09:24.340 sense of percentage. 00:09:24.340 --> 00:09:26.770 The important thing for these type of problems is pay 00:09:26.770 --> 00:09:28.740 attention to how the problem is written. 00:09:28.740 --> 00:09:32.480 If it says find 10% of 100. 00:09:32.480 --> 00:09:33.050 That's easy. 00:09:33.050 --> 00:09:36.010 You just convert 10% to a decimal and multiply it by 100. 00:09:36.010 --> 00:09:39.570 But if I were to ask you 100 is 10% of what? 00:09:39.570 --> 00:09:41.340 You have to remember that that's a different problem. 00:09:41.340 --> 00:09:45.440 In which case, 100 is 10% of-- and if you did the math, 00:09:45.440 --> 00:09:46.850 it would be 1,000. 00:09:46.850 --> 00:09:50.730 I think I spoke very quickly on this problem on this module, so 00:09:50.730 --> 00:09:51.980 I hope you didn't get too confused. 00:09:51.980 --> 00:09:54.440 But I will record more.

Growing by a percentage

https://www.youtube.com/watch?v=X2jVap1YgwI

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https://www.youtube.com/api/timedtext?v=X2jVap1YgwI&ei=fmeUZc-TMNGIp-oP97K-mAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=ADC501151C6DBB87C8E100883BFCA0F00EDB2C9A.7D6682D1BF825555A1428F239A5EB4F2E79DDAAA&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.920 --> 00:00:03.540 Let's do some more percentage problems. 00:00:03.540 --> 00:00:08.130 Let's say that I start this year in my stock 00:00:08.130 --> 00:00:13.520 portfolio with $95.00. 00:00:13.520 --> 00:00:26.010 And I say that my portfolio grows by, let's say, 15%. 00:00:30.350 --> 00:00:31.480 How much do I have now? 00:00:40.430 --> 00:00:40.780 OK. 00:00:40.780 --> 00:00:42.900 I think you might be able to figure this out on your own, 00:00:42.900 --> 00:00:45.426 but of course we'll do some example problems, just in case 00:00:45.426 --> 00:00:46.710 it's a little confusing. 00:00:46.710 --> 00:00:49.440 So I'm starting with $95.00, and I'll get rid of 00:00:49.440 --> 00:00:50.090 the dollar sign. 00:00:50.090 --> 00:00:51.910 We know we're working with dollars. 00:00:51.910 --> 00:00:55.220 95 dollars, right? 00:00:55.220 --> 00:00:58.410 And I'm going to earn, or I'm going to grow just because 00:00:58.410 --> 00:01:02.170 I was an excellent stock investor, that 95 dollars 00:01:02.170 --> 00:01:04.730 is going to grow by 15%. 00:01:04.730 --> 00:01:11.090 So to that 95 dollars, I'm going to add another 15% of 95. 00:01:11.090 --> 00:01:18.880 So we know we write 15% as a decimal, as 0.15, so 95 plus 00:01:18.880 --> 00:01:23.540 0.15 of 95, so this is times 95-- that dot 00:01:23.540 --> 00:01:24.780 is just a times sign. 00:01:24.780 --> 00:01:26.890 It's not a decimal, it's a times, it's a little higher 00:01:26.890 --> 00:01:31.790 than a decimal-- So 95 plus 0.15 times 95 is what 00:01:31.790 --> 00:01:33.390 we have now, right? 00:01:33.390 --> 00:01:36.410 Because we started with 95 dollars, and then we made 00:01:36.410 --> 00:01:40.630 another 15% times what we started with. 00:01:40.630 --> 00:01:41.940 Hopefully that make sense. 00:01:41.940 --> 00:01:46.330 Another way to say it, the 95 dollars has grown by 15%. 00:01:46.330 --> 00:01:47.940 So let's just work this out. 00:01:47.940 --> 00:01:55.460 This is the same thing as 95 plus-- what's 0.15 times 95? 00:01:55.460 --> 00:01:56.480 Let's see. 00:01:56.480 --> 00:01:58.420 So let me do this, hopefully I'll have enough space here. 00:01:58.420 --> 00:02:03.640 95 times 0.15-- I don't want to run out of space. 00:02:03.640 --> 00:02:05.130 Actually, let me do it up here, I think I'm about to run out 00:02:05.130 --> 00:02:11.780 of space-- 95 times 0.15. 00:02:11.780 --> 00:02:21.730 5 times 5 is 25, 9 times 5 is 45 plus 2 is 47, 1 times 95 is 00:02:21.730 --> 00:02:29.580 95, bring down the 5, 12, carry the 1, 15. 00:02:29.580 --> 00:02:30.880 And how many decimals do we have? 00:02:30.880 --> 00:02:32.300 1, 2. 00:02:32.300 --> 00:02:35.666 15.25. 00:02:35.666 --> 00:02:37.230 Actually, is that right? 00:02:37.230 --> 00:02:39.250 I think I made a mistake here. 00:02:39.250 --> 00:02:44.100 See 5 times 5 is 25. 00:02:44.100 --> 00:02:49.045 5 times 9 is 45, plus 2 is 47. 00:02:49.045 --> 00:02:55.500 And we bring the 0 here, it's 95, 1 times 5, 1 times 9, then 00:02:55.500 --> 00:03:00.500 we add 5 plus 0 is 5, 7 plus 5 is 12-- oh. 00:03:00.500 --> 00:03:00.790 See? 00:03:00.790 --> 00:03:01.465 I made a mistake. 00:03:01.465 --> 00:03:06.260 It's 14.25, not 15.25. 00:03:06.260 --> 00:03:08.100 So I'll ask you an interesting question? 00:03:08.100 --> 00:03:12.580 How did I know that 15.25 was a mistake? 00:03:12.580 --> 00:03:14.310 Well, I did a reality check. 00:03:14.310 --> 00:03:18.830 I said, well, I know in my head that 15% of 100 is 15, so if 00:03:18.830 --> 00:03:25.800 15% of 100 is 15, how can 15% of 95 be more than 15? 00:03:25.800 --> 00:03:27.500 I think that might have made sense. 00:03:27.500 --> 00:03:29.790 The bottom line is 95 is less than 100. 00:03:29.790 --> 00:03:33.460 So 15% of 95 had to be less than 15, so I knew my 00:03:33.460 --> 00:03:35.560 answer of 15.25 was wrong. 00:03:35.560 --> 00:03:37.940 And so it turns out that I actually made an addition 00:03:37.940 --> 00:03:40.750 error, and the answer is 14.25. 00:03:40.750 --> 00:03:44.980 So the answer is going to be 95 plus 15% of 95, which is the 00:03:44.980 --> 00:03:57.570 same thing as 95 plus 14.25, well, that equals what? 00:03:57.570 --> 00:04:05.350 109.25. 00:04:05.350 --> 00:04:07.560 Notice how easy I made this for you to read, 00:04:07.560 --> 00:04:08.920 especially this 2 here. 00:04:08.920 --> 00:04:10.490 109.25. 00:04:10.490 --> 00:04:13.990 So if I start off with $95.00 and my portfolio grows-- or the 00:04:13.990 --> 00:04:17.430 amount of money I have-- grows by 15%, I'll end 00:04:17.430 --> 00:04:21.432 up with $109.25. 00:04:21.432 --> 00:04:22.660 Let's do another problem. 00:04:25.560 --> 00:04:31.490 Let's say I start off with some amount of money, and after a 00:04:31.490 --> 00:04:47.260 year, let's says my portfolio grows 25%, and after growing 00:04:47.260 --> 00:04:56.320 25%, I now have $100. 00:04:56.320 --> 00:04:58.810 How much did I originally have? 00:04:58.810 --> 00:05:02.750 Notice I'm not saying that the $100 is growing by 25%. 00:05:02.750 --> 00:05:07.280 I'm saying that I start with some amount of money, it grows 00:05:07.280 --> 00:05:13.780 by 25%, and I end up with $100 after it grew by 25%. 00:05:13.780 --> 00:05:16.010 To solve this one, we might have to break out 00:05:16.010 --> 00:05:17.640 a little bit of algebra. 00:05:17.640 --> 00:05:22.480 So let x equal what I start with. 00:05:29.610 --> 00:05:34.070 So just like the last problem, I start with x and it grows by 00:05:34.070 --> 00:05:43.450 25%, so x plus 25% of x is equal to 100, and we know this 00:05:43.450 --> 00:05:52.190 25% of x we can just rewrite as x plus 0.25 of x is equal to 00:05:52.190 --> 00:05:56.900 100, and now actually we have a level-- actually this might be 00:05:56.900 --> 00:06:01.820 level 3 system, level 3 linear equation-- but the bottom 00:06:01.820 --> 00:06:05.075 line, we can just add the coefficients on the x. 00:06:05.075 --> 00:06:07.350 x is the same thing as 1x, right? 00:06:07.350 --> 00:06:12.790 So 1x plus 0.25x, well that's just the same thing as 1 plus 00:06:12.790 --> 00:06:16.310 0.25, plus x-- we're just doing the distributive property 00:06:16.310 --> 00:06:18.960 in reverse-- equals 100. 00:06:18.960 --> 00:06:21.090 And what's 1 plus 0.25? 00:06:21.090 --> 00:06:23.360 That's easy, it's 1.25. 00:06:23.360 --> 00:06:31.960 So we say 1.25x is equal to 100. 00:06:31.960 --> 00:06:32.650 Not too hard. 00:06:32.650 --> 00:06:34.770 And after you do a lot of these problems, you're going to 00:06:34.770 --> 00:06:39.370 intuitively say, oh, if some number grows by 25%, and it 00:06:39.370 --> 00:06:42.615 becomes 100, that means that 1.25 times that number 00:06:42.615 --> 00:06:44.480 is equal to 100. 00:06:44.480 --> 00:06:46.640 And if this doesn't make sense, sit and think about it a little 00:06:46.640 --> 00:06:50.110 bit, maybe rewatch the video, and hopefully it'll, over time, 00:06:50.110 --> 00:06:51.610 start to make a lot of sense to you. 00:06:51.610 --> 00:06:53.740 This type of math is very very useful. 00:06:53.740 --> 00:06:55.900 I actually work at a hedge fund, and I'm doing 00:06:55.900 --> 00:06:58.960 this type of math in my head day and night. 00:06:58.960 --> 00:07:06.660 So 1.25 times x is equal to 100, so x would equal 00:07:06.660 --> 00:07:10.680 100 divided by 1.25. 00:07:10.680 --> 00:07:11.730 I just realized you probably don't know 00:07:11.730 --> 00:07:12.910 what a hedge fund is. 00:07:12.910 --> 00:07:14.950 I invest in stocks for a living. 00:07:14.950 --> 00:07:16.640 Anyway, back to the math. 00:07:16.640 --> 00:07:19.900 So x is equal to 100 divided by 1.25. 00:07:19.900 --> 00:07:25.570 So let me make some space here, just because I 00:07:25.570 --> 00:07:28.430 used up too much space. 00:07:28.430 --> 00:07:31.060 Let me get rid of my little let x statement. 00:07:31.060 --> 00:07:34.500 Actually I think we know what x is and we know 00:07:34.500 --> 00:07:36.620 how we got to there. 00:07:36.620 --> 00:07:38.570 If you forgot how we got there, you can I guess 00:07:38.570 --> 00:07:39.590 rewatch the video. 00:07:42.150 --> 00:07:43.110 Let's see. 00:07:43.110 --> 00:07:47.600 Let me make the pen thin again, and go back to 00:07:47.600 --> 00:07:49.600 the orange color, OK. 00:07:49.600 --> 00:07:55.590 X equals 100 divided by 1.25, so we say 1.25 goes into 00:07:55.590 --> 00:07:59.410 100.00-- I'm going to add a couple of 0's, I don't know how 00:07:59.410 --> 00:08:01.720 many I'm going to need, probably added too many-- if I 00:08:01.720 --> 00:08:06.150 move this decimal over two to the right, I need to move this 00:08:06.150 --> 00:08:07.450 one over two to the right. 00:08:10.390 --> 00:08:14.410 And I say how many times does 100 go into 100-- how many 00:08:14.410 --> 00:08:16.660 times does 125 go into 100? 00:08:16.660 --> 00:08:17.530 None. 00:08:17.530 --> 00:08:19.420 How many times does it go into 1000? 00:08:19.420 --> 00:08:21.010 It goes into it eight times. 00:08:21.010 --> 00:08:24.130 I happen to know that in my head, but you could do trial 00:08:24.130 --> 00:08:25.580 and error and think about it. 00:08:25.580 --> 00:08:28.490 8 times-- if you want to think about it, 8 times 100 is 00:08:28.490 --> 00:08:32.600 800, and then 8 times 25 is 200, so it becomes 1000. 00:08:32.600 --> 00:08:34.480 You could work out if you like, but I think I'm running out of 00:08:34.480 --> 00:08:36.370 time, so I'm going to do this fast. 00:08:36.370 --> 00:08:39.920 8 times 125 is 1000. 00:08:39.920 --> 00:08:42.600 Remember this thing isn't here. 00:08:42.600 --> 00:08:48.370 1000, so 1000 minus 1000 is 0, so you can bring down the 0. 00:08:48.370 --> 00:08:52.900 125 goes into 0 zero times, and we just keep getting 0's. 00:08:52.900 --> 00:08:55.680 This is just a decimal division problem. 00:08:55.680 --> 00:08:59.500 So it turns out that if your portfolio grew by 25% and 00:08:59.500 --> 00:09:05.120 you ended up with $100.00 you started with $80.00. 00:09:05.120 --> 00:09:09.650 And that makes sense, because 25% is roughly 1/4, right? 00:09:09.650 --> 00:09:13.590 So if I started with $80.00 and I grow by 1/4, that means I 00:09:13.590 --> 00:09:17.660 grew by $20, because 25% of 80 is 20. 00:09:17.660 --> 00:09:21.020 So if I start with 80 and I grow by 20, 00:09:21.020 --> 00:09:22.740 that gets me to 100. 00:09:22.740 --> 00:09:24.020 Makes sense. 00:09:24.020 --> 00:09:28.070 So remember, all you have to say is, well, some number times 00:09:28.070 --> 00:09:32.950 1.25-- because I'm growing it by 25%-- is equal to 100. 00:09:32.950 --> 00:09:36.240 Don't worry, if you're still confused, I'm going to add at 00:09:36.240 --> 00:09:38.500 least one more presentation on a couple of more 00:09:38.500 --> 00:09:40.440 examples like this.

Percent and decimals

https://www.youtube.com/watch?v=RvtdJnYFNhc

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https://www.youtube.com/api/timedtext?v=RvtdJnYFNhc&ei=fWeUZdyDBdGwp-oPybGbiAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249837&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=9DA94F22AB06171006342E5AB648465DCECF0709.1A868C98FECFD0148539C073228846269605B62B&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.170 --> 00:00:03.080 Let's get started and learn how to convert 00:00:03.080 --> 00:00:04.430 percentages to decimals. 00:00:04.430 --> 00:00:07.000 And if we have time maybe we'll also learn how to convert 00:00:07.000 --> 00:00:08.840 decimals into percentages. 00:00:08.840 --> 00:00:11.700 So let's get started with what I think is a problem you 00:00:11.700 --> 00:00:13.060 probably already know how to do. 00:00:13.060 --> 00:00:14.930 If I said I have 50%. 00:00:18.270 --> 00:00:19.520 I don't know if I wanted to write that thick, 00:00:19.520 --> 00:00:20.630 but we'll go with it. 00:00:20.630 --> 00:00:22.420 Actually, let me change it to a thinner one. 00:00:22.420 --> 00:00:24.450 And I want to turn that into a decimal. 00:00:24.450 --> 00:00:26.580 Well you probably already have a sense of what 00:00:26.580 --> 00:00:28.940 decimal represents 50%. 00:00:28.940 --> 00:00:31.790 If I told you we're having a sale and it's 50% off you know 00:00:31.790 --> 00:00:34.270 that's roughly half off, or another way, how do you 00:00:34.270 --> 00:00:35.870 say half as a decimal? 00:00:35.870 --> 00:00:44.240 Well that's the same thing as 0.5 So you might have known 00:00:44.240 --> 00:00:48.150 that in your head, but is there a system for being able to 00:00:48.150 --> 00:00:49.990 convert this 50% to 0.5? 00:00:49.990 --> 00:00:52.770 Well, it turns out it's pretty straightforward. 00:00:52.770 --> 00:00:56.590 All you do is you say-- whatever percentage it is, 00:00:56.590 --> 00:01:03.760 that's the same thing as the number over 100. 00:01:03.760 --> 00:01:13.772 And 50/100 is the same thing as 5/10 or 0.5. 00:01:13.772 --> 00:01:17.250 Now even a simpler way of converting percentage to 00:01:17.250 --> 00:01:19.620 decimals and, I think, you're going to realize converting a 00:01:19.620 --> 00:01:22.170 percentage to a decimal or the other way around, you can 00:01:22.170 --> 00:01:23.800 pretty much do it in your head. 00:01:23.800 --> 00:01:28.180 If I say, let's say 50-- and I'm just going to add one 00:01:28.180 --> 00:01:34.170 decimal of accuracy here just to show you a point-- 50.0%. 00:01:34.170 --> 00:01:38.640 If I want to convert that 50.0% into a decimal, all I do is I 00:01:38.640 --> 00:01:40.900 get rid of the percent sign. 00:01:40.900 --> 00:01:42.320 So I'll do it here. 00:01:42.320 --> 00:01:45.500 50.0%. 00:01:45.500 --> 00:01:49.100 I get rid of the percent sign and I take the decimal point 00:01:49.100 --> 00:01:51.960 and I move it over two spaces to the left. 00:01:51.960 --> 00:01:55.590 So I say 1, 2. 00:01:55.590 --> 00:01:57.580 So this is where the new decimal is. 00:01:57.580 --> 00:02:05.260 So this equals we could say 0.500. 00:02:05.260 --> 00:02:10.560 So 50.0% is equal to 0.500. 00:02:10.560 --> 00:02:14.170 And of course, these last two 0's really don't mean anything 00:02:14.170 --> 00:02:16.640 for our purposes, so we'll get rid of them. 00:02:16.640 --> 00:02:22.310 So that's the same thing as 0.5 or just 0.5. 00:02:22.310 --> 00:02:24.290 50% equals 0.5. 00:02:24.290 --> 00:02:26.810 So you're probably saying, well, sure. 00:02:26.810 --> 00:02:28.200 That looks easy, but what if the problem gets 00:02:28.200 --> 00:02:28.700 a little harder? 00:02:28.700 --> 00:02:30.110 50% I could have done in my head. 00:02:30.110 --> 00:02:35.080 So let's try some, I would say, slightly harder problems. 00:02:35.080 --> 00:02:46.240 If I were to tell you that-- let's say 16.32%. 00:02:46.240 --> 00:02:49.100 Well, let's just do it the way I just showed you. 00:02:49.100 --> 00:02:50.200 I'll rewrite it down here. 00:02:50.200 --> 00:02:56.520 16.32%. 00:02:56.520 --> 00:02:59.610 So if we get rid of the percent sign, scratch it out, we just 00:02:59.610 --> 00:03:03.740 have to move the decimal over two spaces to the left. 00:03:03.740 --> 00:03:07.830 So 1, 2. 00:03:07.830 --> 00:03:09.250 This is the new place for the decimal. 00:03:09.250 --> 00:03:10.580 That decimal goes away. 00:03:10.580 --> 00:03:23.340 So it's 0.1632 is equal to 16.32%. 00:03:23.340 --> 00:03:24.850 I think you might be getting the idea now. 00:03:24.850 --> 00:03:28.310 Let me do another one in green. 00:03:28.310 --> 00:03:31.910 Let's say I had-- and this one actually confuses 00:03:31.910 --> 00:03:32.540 a lot of people. 00:03:32.540 --> 00:03:41.830 Let's say I had 0.25%. 00:03:41.830 --> 00:03:45.950 So the important thing to remember is, I'll rewrite here. 00:03:45.950 --> 00:03:53.170 0.25-- and maybe I'll write 00.-- And you're probably 00:03:53.170 --> 00:03:55.260 wondering why I'm doing this, but I think you'll see in a 00:03:55.260 --> 00:03:58.100 second why I wrote that leading 0 there even though it doesn't 00:03:58.100 --> 00:04:00.020 seem to add much to it. 00:04:00.020 --> 00:04:01.630 00.25%. 00:04:01.630 --> 00:04:03.250 Well, what's the system I just showed you? 00:04:03.250 --> 00:04:05.570 You get rid of the percent sign and you move the 00:04:05.570 --> 00:04:10.000 decimal over 1, 2 spaces. 00:04:10.000 --> 00:04:15.600 So that equals 0.0025. 00:04:15.600 --> 00:04:20.380 So 0.25% is equal to 0.0025. 00:04:20.380 --> 00:04:24.480 And you could put a leading 0 here if you want. 00:04:24.480 --> 00:04:27.090 Actually, I should probably tell you to always do that 00:04:27.090 --> 00:04:29.260 because it makes it easier to read. 00:04:29.260 --> 00:04:34.050 So 0.25% is equal to 0.0025. 00:04:34.050 --> 00:04:37.270 And I want to just contrast that with 25%. 00:04:41.200 --> 00:04:44.250 25%, what do you think that equals? 00:04:44.250 --> 00:04:44.920 Well. 00:04:44.920 --> 00:04:47.970 you do the same thing that we've been doing. 00:04:47.970 --> 00:04:51.680 You get rid of the percent sign and you move the decimal space. 00:04:51.680 --> 00:04:53.220 In this case-- actually, let me leave that there. 00:04:53.220 --> 00:04:53.800 I'll just rewrite it here. 00:04:53.800 --> 00:04:54.130 25%. 00:04:56.800 --> 00:04:58.320 And you're probably saying, where is the decimal in this? 00:04:58.320 --> 00:05:02.213 Well, the decimal is after the number because that 25% is 00:05:02.213 --> 00:05:04.560 the same thing as 25.0%. 00:05:04.560 --> 00:05:08.510 So if we get rid of the percent sign, we move the decimal over 00:05:08.510 --> 00:05:16.550 two spaces to the left, and that equals 0.25 as a decimal, 00:05:16.550 --> 00:05:19.220 or 0.250, but we can ignore that last 0. 00:05:19.220 --> 00:05:29.280 So 25% equals 0.25 while 0.25% is equal to 0.0025. 00:05:29.280 --> 00:05:31.970 And I want you to maybe sit and think about how small 00:05:31.970 --> 00:05:35.390 of a number 0.25% is. 00:05:35.390 --> 00:05:37.910 Let's do a couple more and maybe we'll convert going 00:05:37.910 --> 00:05:38.510 the other direction. 00:05:41.400 --> 00:05:48.820 Let's say I have the decimal 0.01 and I wanted to convert 00:05:48.820 --> 00:05:49.860 that into a percent. 00:05:56.090 --> 00:05:58.120 Well, here we just do it the opposite. 00:05:58.120 --> 00:06:00.550 We could look at it two ways. 00:06:00.550 --> 00:06:03.870 We could say well, whatever number this is, we multiply it 00:06:03.870 --> 00:06:06.180 by 100 and add a percent sign. 00:06:06.180 --> 00:06:12.770 So if you say 0.01 times 100, and then we'll 00:06:12.770 --> 00:06:15.590 add a percent sign. 00:06:15.590 --> 00:06:15.750 So a. 00:06:15.750 --> 00:06:17.150 0.01 times 100? 00:06:17.150 --> 00:06:18.740 Well that's just 1. 00:06:18.740 --> 00:06:20.420 You could do the math. 00:06:20.420 --> 00:06:21.380 You add the percent sign. 00:06:21.380 --> 00:06:23.550 Well, that equals 1%. 00:06:23.550 --> 00:06:27.930 Or an even easier way, when we go from a percent to a decimal, 00:06:27.930 --> 00:06:31.770 we move the decimal place over two to the left. 00:06:31.770 --> 00:06:33.930 So when you go from a decimal to a percent, 00:06:33.930 --> 00:06:35.160 we'll do the opposite. 00:06:35.160 --> 00:06:37.250 We move the decimal two to the right. 00:06:37.250 --> 00:06:40.680 So if we do that, let me just rewrite it. 00:06:40.680 --> 00:06:42.420 0.01. 00:06:42.420 --> 00:06:44.960 Just go 1, 2. 00:06:44.960 --> 00:06:46.180 The new decimal place is here. 00:06:48.860 --> 00:06:54.530 If I get rid of that decimals that's 01 decimal 00 whatever. 00:06:54.530 --> 00:06:56.610 Obviously, this leading 0 means nothing, so that's 00:06:56.610 --> 00:06:58.830 the same thing as 1.00. 00:06:58.830 --> 00:07:02.260 Which is the same thing as 1. 00:07:02.260 --> 00:07:07.080 And does it make sense that moving the decimal space two to 00:07:07.080 --> 00:07:09.580 the right-- that's really just the same thing as multiplying 00:07:09.580 --> 00:07:11.720 it by 100, right? 00:07:11.720 --> 00:07:15.070 If I multiply something by 10 it's like moving the decimal 00:07:15.070 --> 00:07:16.990 space one to the right. 00:07:16.990 --> 00:07:20.720 If I divide something by 10 it's like moving the 00:07:20.720 --> 00:07:22.230 decimal space one to left. 00:07:22.230 --> 00:07:24.320 Let's do a couple more while I have time. 00:07:24.320 --> 00:07:27.630 I think I have three more minutes. 00:07:27.630 --> 00:07:36.390 Let's say I had 1.25 and I wanted to convert 00:07:36.390 --> 00:07:37.810 that to a percent. 00:07:37.810 --> 00:07:41.380 Well, the easiest way is just to take-- I'll rewrite it here. 00:07:41.380 --> 00:07:43.200 1.25. 00:07:43.200 --> 00:07:47.120 Take the decimal point, move it two to the right. 00:07:47.120 --> 00:07:49.190 That's here. 00:07:49.190 --> 00:07:50.220 And then I'll add a percent. 00:07:50.220 --> 00:07:55.460 So that equals 125%. 00:07:55.460 --> 00:07:57.100 And if you think about it, the way people talk about 00:07:57.100 --> 00:07:58.260 percent it makes sense. 00:07:58.260 --> 00:08:04.700 If I told you that I'm going to pay 1.25 times the 00:08:04.700 --> 00:08:06.150 price of something. 00:08:06.150 --> 00:08:07.860 That makes sense that that's also I'm going to pay 00:08:07.860 --> 00:08:09.960 125% of the price. 00:08:09.960 --> 00:08:12.330 Or if it doesn't make sense hopefully if you do these 00:08:12.330 --> 00:08:14.630 problems enough it will start to make sense. 00:08:14.630 --> 00:08:15.690 Let's do a couple of more. 00:08:15.690 --> 00:08:17.940 And you can go back and pause this if you think I'm going too 00:08:17.940 --> 00:08:19.380 fast, which I might be doing. 00:08:26.300 --> 00:08:27.680 Let me think. 00:08:27.680 --> 00:08:35.920 If I were to say 0.003 and I want to write 00:08:35.920 --> 00:08:39.520 this as a percent. 00:08:39.520 --> 00:08:43.450 Well, once again, we can move the decimal space 00:08:43.450 --> 00:08:45.010 two to the right. 00:08:45.010 --> 00:08:46.510 So 1, 2. 00:08:46.510 --> 00:08:49.830 And that's analogous to multiplying it by 100. 00:08:49.830 --> 00:08:53.350 So if we multiply the decimal two to the right we get 00:08:53.350 --> 00:08:57.120 00 decimal point 3. 00:08:57.120 --> 00:09:00.080 And then we add the percent. 00:09:00.080 --> 00:09:01.860 At least this first leading 0 doesn't mean anything, so 00:09:01.860 --> 00:09:03.150 that's the same thing as 0.3%. 00:09:06.810 --> 00:09:09.180 The important thing to realize is when you're converting from 00:09:09.180 --> 00:09:12.550 a percent to a decimal or a decimal to a percent, you're 00:09:12.550 --> 00:09:15.000 really just moving where that decimal point is. 00:09:15.000 --> 00:09:18.490 And if you run out of spaces you just have to add or get 00:09:18.490 --> 00:09:20.460 rid of 0's accordingly. 00:09:20.460 --> 00:09:23.070 And the important thing to always have in your mind is, 00:09:23.070 --> 00:09:27.260 when I convert from a decimal to a percent, the number in 00:09:27.260 --> 00:09:29.230 front of percent signs going to get bigger. 00:09:29.230 --> 00:09:31.590 And when I go from a percent sign to a decimal, I'm going 00:09:31.590 --> 00:09:33.920 to get a smaller number. 00:09:33.920 --> 00:09:41.880 If I say 25%, that's the same thing as 0.25. 00:09:41.880 --> 00:09:44.430 So this is a percent, and this is a decimal. 00:09:47.120 --> 00:09:52.080 So I went from a bigger number, 25, to a smaller number, 0.25. 00:09:52.080 --> 00:09:54.570 25% is equal to 0.25. 00:09:54.570 --> 00:09:59.840 Similarly, if I had a decimal, let's say 0.1. 00:09:59.840 --> 00:10:01.660 When I convert it to a percentage it's going to 00:10:01.660 --> 00:10:03.300 be a larger percentage. 00:10:03.300 --> 00:10:08.060 So 0.1 is the same thing as 10%. 00:10:08.060 --> 00:10:10.430 And how did I do that again? 00:10:10.430 --> 00:10:14.310 Well I said 0.1, I added an extra 0 because I'm going to 00:10:14.310 --> 00:10:16.850 have to move the decimal space over to the right twice. 00:10:16.850 --> 00:10:20.790 So I go 1, 2, and I get a 10. 00:10:20.790 --> 00:10:22.730 10%. 00:10:22.730 --> 00:10:24.940 Hopefully that answers all your questions for now. 00:10:24.940 --> 00:10:26.450 Have fun.

Ordering numeric expressions

https://www.youtube.com/watch?v=Llt-KkHugRQ

vtt

https://www.youtube.com/api/timedtext?v=Llt-KkHugRQ&ei=fmeUZYiyMuO_mLAPxa6xgAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D7C7DA22616D1303603516D100007C031738E457.4A1FB5F6BE3891D8E634E4FD1897755BBA0752A1&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.910 --> 00:00:04.360 Welcome to the presentation on ordering numbers. 00:00:04.360 --> 00:00:06.940 Let's get started with some problems that I think, as you 00:00:06.940 --> 00:00:09.270 go through the examples hopefully, you'll understand 00:00:09.270 --> 00:00:10.910 how to do these problems. 00:00:10.910 --> 00:00:11.700 So let's see. 00:00:11.700 --> 00:00:23.200 The first set of numbers that we have to order is 35.7%, 00:00:23.200 --> 00:00:44.590 108.1% 0.5, 13/93, and 1 and 7/68. 00:00:44.590 --> 00:00:46.590 So let's do this problem. 00:00:46.590 --> 00:00:48.810 The important thing to remember whenever you're doing this type 00:00:48.810 --> 00:00:52.820 of ordering of numbers is to realize that these are all just 00:00:52.820 --> 00:00:56.940 different ways to represent-- these are all a percent or a 00:00:56.940 --> 00:01:00.270 decimal or a fraction or a mixed-- are all just different 00:01:00.270 --> 00:01:02.680 ways of representing numbers. 00:01:02.680 --> 00:01:05.110 It's very hard to compare when you just look at it like this, 00:01:05.110 --> 00:01:07.130 so what I like to do is I like to convert them 00:01:07.130 --> 00:01:08.190 all to decimals. 00:01:08.190 --> 00:01:11.100 But there could be someone who likes to convert them all to 00:01:11.100 --> 00:01:14.220 percentages or convert them all to fractions and then compare. 00:01:14.220 --> 00:01:16.920 But I always find decimals to be the easiest way to compare. 00:01:16.920 --> 00:01:19.370 So let's start with this 35.7%. 00:01:19.370 --> 00:01:21.940 Let's turn this into a decimal. 00:01:21.940 --> 00:01:25.090 Well, the easiest thing to remember is if you have a 00:01:25.090 --> 00:01:27.490 percent you just get rid of the percent sign and 00:01:27.490 --> 00:01:28.580 put it over 100. 00:01:28.580 --> 00:01:38.970 So 35.7% is the same thing as 35.7/100. 00:01:38.970 --> 00:01:43.020 Like 5%, that's the same thing as 5/100 or 50% is just 00:01:43.020 --> 00:01:45.050 the same thing as 50/100. 00:01:45.050 --> 00:01:53.990 So 35.7/100, well, that just equals 0.357. 00:01:53.990 --> 00:01:55.730 If this got you a little confused another way to think 00:01:55.730 --> 00:02:01.970 about percentage points is if I write 35.7%, all you have to do 00:02:01.970 --> 00:02:05.540 is get rid of the percent sign and move the decimal to the 00:02:05.540 --> 00:02:10.140 left two spaces and it becomes 0.357. 00:02:10.140 --> 00:02:11.870 Let me give you a couple of more examples down here. 00:02:11.870 --> 00:02:16.050 Let's say I had 5%. 00:02:16.050 --> 00:02:20.020 That is the same thing as 5/100. 00:02:20.020 --> 00:02:22.670 Or if you do the decimal technique, 5%, you could just 00:02:22.670 --> 00:02:24.730 move the decimal and you get rid of the percent. 00:02:24.730 --> 00:02:28.630 And you move the decimal over 1 and 2, and you put a 0 here. 00:02:28.630 --> 00:02:30.370 It's 0.05. 00:02:30.370 --> 00:02:33.280 And that's the same thing as 0.05. 00:02:33.280 --> 00:02:36.380 You also know that 0.05 and 5/100 are the same thing. 00:02:36.380 --> 00:02:37.620 So let's get back to the problem. 00:02:37.620 --> 00:02:40.772 I hope that distraction didn't distract you too much. 00:02:40.772 --> 00:02:43.190 Let me scratch out all this. 00:02:43.190 --> 00:02:49.050 So 35.7% is equal to 0.357. 00:02:49.050 --> 00:02:51.870 Similarly, 108.1%. 00:02:51.870 --> 00:02:54.080 Let's to the technique where we just get rid of the percent and 00:02:54.080 --> 00:02:59.350 move the decimal space over 1, 2 spaces to the left. 00:02:59.350 --> 00:03:08.600 So then that equals 1.081. 00:03:08.600 --> 00:03:11.570 See we already know that this is smaller than this. 00:03:11.570 --> 00:03:14.140 Well the next one is easy, it's already in decimal form. 00:03:14.140 --> 00:03:16.040 0.5 is just going to be equal to 0.5. 00:03:18.820 --> 00:03:21.050 Now 13/93. 00:03:21.050 --> 00:03:24.340 To convert a fraction into a decimal we just take the 00:03:24.340 --> 00:03:27.320 denominator and divide it into the numerator. 00:03:27.320 --> 00:03:29.350 So let's do that. 00:03:29.350 --> 00:03:33.020 93 goes into 13? 00:03:36.530 --> 00:03:39.760 Well, we know it goes into 13 zero times. 00:03:39.760 --> 00:03:43.990 So let's add a decimal point here. 00:03:43.990 --> 00:03:47.550 So how many times does 93 go into 130? 00:03:47.550 --> 00:03:49.530 Well, it goes into it one time. 00:03:49.530 --> 00:03:51.410 1 times 93 is 93. 00:03:55.061 --> 00:03:56.550 Becomes a 10. 00:03:56.550 --> 00:03:58.960 That becomes a 2. 00:03:58.960 --> 00:04:03.700 Then we're going to borrow, so get 37. 00:04:03.700 --> 00:04:06.590 Bring down a 0. 00:04:06.590 --> 00:04:10.010 So 93 goes into 370? 00:04:10.010 --> 00:04:10.470 Let's see. 00:04:10.470 --> 00:04:14.790 4 times 93 would be 372, so it actually goes into 00:04:14.790 --> 00:04:15.695 it only three times. 00:04:19.390 --> 00:04:22.880 3 times 3 is 9. 00:04:22.880 --> 00:04:25.270 3 times 9 is 27. 00:04:30.110 --> 00:04:31.605 So this equals? 00:04:31.605 --> 00:04:38.050 Let's see, this equals-- if we say that this 0 becomes a 10. 00:04:38.050 --> 00:04:39.620 This become a 16. 00:04:39.620 --> 00:04:42.440 This becomes a 2. 00:04:42.440 --> 00:04:45.210 81. 00:04:45.210 --> 00:04:48.120 And then we say, how many times does 93 go into 810? 00:04:48.120 --> 00:04:50.860 It goes roughly 8 times. 00:04:50.860 --> 00:04:52.860 And we could actually keep going, but for the sake of 00:04:52.860 --> 00:04:55.640 comparing these numbers, we've already gotten to a pretty 00:04:55.640 --> 00:04:57.580 good level of accuracy. 00:04:57.580 --> 00:05:00.740 So let's just stop this problem here because the decimal 00:05:00.740 --> 00:05:02.720 numbers could keep going on, but for the sake of comparison 00:05:02.720 --> 00:05:04.410 I think we've already got a good sense of what this 00:05:04.410 --> 00:05:05.360 decimal looks like. 00:05:05.360 --> 00:05:10.330 It's 0.138 and then it'll just keep going. 00:05:10.330 --> 00:05:13.010 So let's write that down. 00:05:13.010 --> 00:05:15.340 And then finally, we have this mixed number here. 00:05:15.340 --> 00:05:18.070 And let me erase some of my work because I don't 00:05:18.070 --> 00:05:18.840 want to confuse you. 00:05:18.840 --> 00:05:22.700 Actually, let me keep it the way it is right now. 00:05:22.700 --> 00:05:26.120 The easiest way to convert a mixed number into a decimal is 00:05:26.120 --> 00:05:29.630 to just say, OK, this is 1 and then some fraction 00:05:29.630 --> 00:05:32.920 that's less than 1. 00:05:32.920 --> 00:05:36.420 Or we could convert it to a fraction, an improper fraction 00:05:36.420 --> 00:05:38.790 like-- oh, actually there are no improper fractions here. 00:05:38.790 --> 00:05:39.640 Actually, let's do it that way. 00:05:39.640 --> 00:05:41.630 Let's convert to an improper fraction and then convert 00:05:41.630 --> 00:05:44.110 that into a decimal. 00:05:44.110 --> 00:05:46.060 Actually, I think I'm going to need more space, so let me 00:05:46.060 --> 00:05:48.740 clean up this a little bit. 00:05:58.240 --> 00:05:58.595 There. 00:05:58.595 --> 00:06:01.040 We have a little more space to work with now. 00:06:04.260 --> 00:06:08.570 So 1 and 7/68. 00:06:08.570 --> 00:06:13.700 So to go from a mixed number to an improper fraction, what you 00:06:13.700 --> 00:06:18.760 do is you take the 68 times 1 and add it to the 00:06:18.760 --> 00:06:19.720 numerator here. 00:06:19.720 --> 00:06:21.000 And why does this make sense? 00:06:21.000 --> 00:06:26.120 Because this is the same thing as 1 plus 7/68. 00:06:26.120 --> 00:06:29.680 1 and 7/68 is the same thing as 1 plus 7/68. 00:06:29.680 --> 00:06:32.800 And that's the same thing as you know from the fractions 00:06:32.800 --> 00:06:40.330 module, as 68/68 plus 7/68. 00:06:40.330 --> 00:06:47.650 And that's the same thing as 68 plus 7-- 75/68. 00:06:47.650 --> 00:06:51.790 So 1 and 7/68 is equal to 75/68. 00:06:51.790 --> 00:06:54.870 And now we convert this to a decimal using the technique 00:06:54.870 --> 00:06:56.350 we did for 13/93. 00:06:56.350 --> 00:06:58.570 So we say-- let me get some space. 00:06:58.570 --> 00:07:05.000 We say 68 goes into 75-- suspicion I'm going 00:07:05.000 --> 00:07:07.360 to run out of space. 00:07:07.360 --> 00:07:09.160 68 goes into 75 one time. 00:07:09.160 --> 00:07:13.290 1 times 68 is 68. 00:07:13.290 --> 00:07:16.460 75 minus 68 is 7. 00:07:16.460 --> 00:07:17.350 Bring down the 0. 00:07:17.350 --> 00:07:20.490 Actually, you don't have to write the decimal there. 00:07:20.490 --> 00:07:21.100 Ignore that decimal. 00:07:21.100 --> 00:07:24.400 68 goes into 70 one time. 00:07:24.400 --> 00:07:28.150 1 times 68 is 68. 00:07:28.150 --> 00:07:30.990 70 minus 68 is 2, bring down another 0. 00:07:30.990 --> 00:07:33.240 68 goes into 20 zero times. 00:07:33.240 --> 00:07:36.550 And the problem's going to keep going on, but I think we've 00:07:36.550 --> 00:07:38.990 already once again, gotten to enough accuracy that 00:07:38.990 --> 00:07:40.040 we can compare. 00:07:40.040 --> 00:07:48.320 So 1 and 7/68 we've now figured out is equal to 1.10-- and if 00:07:48.320 --> 00:07:51.000 we kept dividing we'll keep getting more decimals of 00:07:51.000 --> 00:07:53.510 accuracy, but I think we're now ready to compare. 00:07:53.510 --> 00:07:56.550 So all of these numbers I just rewrote them as decimals. 00:07:56.550 --> 00:08:00.550 So 35.7% is 0.357. 00:08:00.550 --> 00:08:05.000 108.1%-- ignore this for now because we just used 00:08:05.000 --> 00:08:05.720 that to do the work. 00:08:05.720 --> 00:08:09.660 It's 108.1% is equal to 1.081. 00:08:09.660 --> 00:08:11.260 0.5 is 0.5. 00:08:11.260 --> 00:08:15.770 13/93 is 0.138. 00:08:15.770 --> 00:08:20.850 And 1 and 7/68 is 1.10 and it'll keep going on. 00:08:20.850 --> 00:08:23.010 So what's the smallest? 00:08:23.010 --> 00:08:26.320 So the smallest is 0.-- actually, no. 00:08:26.320 --> 00:08:28.300 The smallest is right here. 00:08:28.300 --> 00:08:31.450 So I'm going to rank them from smallest to largest. 00:08:31.450 --> 00:08:36.250 So the smallest is 0.138. 00:08:36.250 --> 00:08:40.640 Then the next largest is going to be 0.357. 00:08:40.640 --> 00:08:44.470 Then the next largest is going to be 0.5. 00:08:44.470 --> 00:08:47.460 Then you're going to have 1.08. 00:08:47.460 --> 00:08:54.620 And then you're going to have 1 and 7/68. 00:08:54.620 --> 00:08:56.840 Well, actually, I'm going to do more examples of this, but for 00:08:56.840 --> 00:08:59.890 this video I think this is the only one I have time for. 00:08:59.890 --> 00:09:01.910 But hopefully this gives you a sense of doing these problems. 00:09:01.910 --> 00:09:04.370 I always find it easier to go into the decimal 00:09:04.370 --> 00:09:05.280 mode to compare. 00:09:05.280 --> 00:09:06.680 And actually, the hints on the module will 00:09:06.680 --> 00:09:08.670 be the same for you. 00:09:08.670 --> 00:09:11.040 But I think you're ready at least now to try the problems. 00:09:11.040 --> 00:09:13.170 If you're not, if you want to see other examples, you might 00:09:13.170 --> 00:09:16.830 just want to either re-watch this video and/or I might 00:09:16.830 --> 00:09:20.530 record some more videos with more examples right now. 00:09:20.530 --> 00:09:22.560 Anyway, have fun.

Mixed numbers and improper fractions

https://www.youtube.com/watch?v=1xuf6ZKF1_I

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https://www.youtube.com/api/timedtext?v=1xuf6ZKF1_I&ei=fmeUZfGxM-mAp-oPsaOUsAI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=B5AA9D92609DCA7C26E245E36A5EA8D4CB84652B.A134ADF7278523AEF92B9D044D8F203347743A6C&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.810 --> 00:00:03.610 We're now going to learn how to go from mixed numbers to 00:00:03.610 --> 00:00:05.680 improper fractions and vice versa. 00:00:05.680 --> 00:00:07.120 So first a little bit of terminology. 00:00:07.120 --> 00:00:08.490 What is a mixed number? 00:00:08.490 --> 00:00:10.350 Well, you've probably seen someone write, let's 00:00:10.350 --> 00:00:13.850 say, 2 and 1/2. 00:00:13.850 --> 00:00:15.470 This is a mixed number. 00:00:15.470 --> 00:00:16.950 You're saying why is it a mixed number? 00:00:16.950 --> 00:00:21.520 Well, because we're including a whole number and a fraction. 00:00:21.520 --> 00:00:22.630 So that's why it's a mixed number. 00:00:22.630 --> 00:00:24.510 It's a whole number mixed with a fraction. 00:00:24.510 --> 00:00:25.240 So 2 and 1/2. 00:00:25.240 --> 00:00:27.760 And I think you have a sense of what 2 and 1/2 is. 00:00:27.760 --> 00:00:31.050 It's some place halfway between 2 and 3. 00:00:31.050 --> 00:00:32.086 And what's an improper fractional? 00:00:32.086 --> 00:00:36.260 Well an Improper fraction is a fraction where the numerator is 00:00:36.260 --> 00:00:37.410 larger than the denominator. 00:00:37.410 --> 00:00:39.330 So let's give an example of an improper fraction. 00:00:39.330 --> 00:00:41.140 I'm just going to pick some random numbers. 00:00:41.140 --> 00:00:47.760 Let's say I had 23 over 5. 00:00:47.760 --> 00:00:49.390 This is an improper fraction. 00:00:49.390 --> 00:00:50.090 Why? 00:00:50.090 --> 00:00:52.290 Because 23 is larger than 5. 00:00:52.290 --> 00:00:54.030 It's that simple. 00:00:54.030 --> 00:00:57.450 It turns out that you can convert an improper fraction 00:00:57.450 --> 00:01:00.040 into a mixed number or a mixed number into an 00:01:00.040 --> 00:01:01.360 improper fraction. 00:01:01.360 --> 00:01:02.790 So let's start with the latter. 00:01:02.790 --> 00:01:05.380 Let's learn how to do a mixed number into an 00:01:05.380 --> 00:01:06.600 improper fraction. 00:01:06.600 --> 00:01:10.020 So first I'll just show you kind of just the basic 00:01:10.020 --> 00:01:11.390 systematic way of doing it. 00:01:11.390 --> 00:01:13.190 It'll always give you the right answer, and then hopefully I'll 00:01:13.190 --> 00:01:15.110 give you a little intuition for why it works. 00:01:15.110 --> 00:01:18.875 So if I wanted to convert 2 and 1/2 into an improper fraction 00:01:18.875 --> 00:01:22.900 or I want to unmix it you could say, all I do is I take the 00:01:22.900 --> 00:01:27.190 denominator in the fraction part, multiply it by the whole 00:01:27.190 --> 00:01:30.150 number, and add the numerator. 00:01:30.150 --> 00:01:31.130 So let's do that. 00:01:31.130 --> 00:01:33.580 I think if we do enough examples you'll 00:01:33.580 --> 00:01:34.600 get the pattern. 00:01:34.600 --> 00:01:40.340 So 2 times 2 is 4 plus 1 is 5. 00:01:40.340 --> 00:01:41.040 So let's write that. 00:01:41.040 --> 00:01:46.330 It's 2 times 2 plus 1, and that's going to 00:01:46.330 --> 00:01:47.600 be the new numerator. 00:01:47.600 --> 00:01:50.320 And it's going to be all of that over the old denominator. 00:01:50.320 --> 00:01:52.543 So that equals 5/2. 00:01:55.180 --> 00:01:58.410 So 2 and 1/2 is equal to 5/2. 00:02:01.150 --> 00:02:02.260 Let's do another one. 00:02:02.260 --> 00:02:08.050 Let's say I had 4 and 2/3. 00:02:08.050 --> 00:02:11.950 This is equal to -- so this is going to be all over 3. 00:02:11.950 --> 00:02:13.490 We keep the denominator the same. 00:02:13.490 --> 00:02:18.180 And then new numerator is going to be 3 times 4 plus 2. 00:02:18.180 --> 00:02:24.040 So it's going to be 3 times 4, and then you're going to add 2. 00:02:24.040 --> 00:02:26.640 Well that equals 3 times 4 -- order of operations, you always 00:02:26.640 --> 00:02:28.730 do multiplication first, and that's actually the way 00:02:28.730 --> 00:02:30.940 I taught it how to convert this anyway. 00:02:30.940 --> 00:02:34.240 3 times 4 is 12 plus 2 is 14. 00:02:34.240 --> 00:02:38.340 So that equals 14 over 3. 00:02:38.340 --> 00:02:39.030 Let's do another one. 00:02:39.030 --> 00:02:48.710 Let's say I had 6 and 17/18. 00:02:48.710 --> 00:02:50.520 I gave myself a hard problem. 00:02:50.520 --> 00:02:54.450 Well, we just keep the denominator the same. 00:02:54.450 --> 00:02:57.192 And then new numerator is going to be 18 times 6 00:02:57.192 --> 00:03:03.960 or 6 times 18 plus 17. 00:03:03.960 --> 00:03:05.200 Well 6 times 18. 00:03:05.200 --> 00:03:08.010 Let's see, that's 60 plus 48 it's 108, so that 00:03:08.010 --> 00:03:11.910 equals 108 plus 17. 00:03:11.910 --> 00:03:14.310 All that over 18. 00:03:14.310 --> 00:03:20.070 108 plus 17 is equal to 125 over 18. 00:03:20.070 --> 00:03:29.150 So, 6 and 17/18 is equal to 125 over 18. 00:03:29.150 --> 00:03:30.100 Let's do a couple more. 00:03:30.100 --> 00:03:32.810 And in a couple minutes I'm going to teach you how to go 00:03:32.810 --> 00:03:35.700 the other way, how to go from an improper fraction 00:03:35.700 --> 00:03:36.640 to a mixed number. 00:03:39.880 --> 00:03:41.680 And this one I'm going to try to give you a little bit of 00:03:41.680 --> 00:03:44.850 intuition for, why what I'm teaching you actually works. 00:03:44.850 --> 00:03:48.050 So let's say 2 and 1/4. 00:03:51.930 --> 00:03:54.920 If we use the -- I guess you'd call it a system that I just 00:03:54.920 --> 00:04:04.140 showed you -- that equals 4 times 2 plus 1 over 4. 00:04:04.140 --> 00:04:09.720 Well that equals, 4 times 2 is 8 plus 1 is 9, 9 over 4. 00:04:09.720 --> 00:04:14.110 I want to give you an intuition for why this actually works. 00:04:14.110 --> 00:04:17.150 So 2 and 1/4, let's actually draw that, see what 00:04:17.150 --> 00:04:18.280 it looks like. 00:04:18.280 --> 00:04:22.200 So let's put this back into kind of the pie analogy. 00:04:22.200 --> 00:04:26.130 So that's equal to one pie. 00:04:26.130 --> 00:04:28.040 Two pies. 00:04:28.040 --> 00:04:33.840 And then let's say a 1/4 of a pie. 00:04:33.840 --> 00:04:34.900 A 1/4 is like this. 00:04:38.350 --> 00:04:41.560 2 and 1/4, and ignore this, this is nothing. 00:04:41.560 --> 00:04:43.430 It's not a decimal point -- actually, let me erase it so it 00:04:43.430 --> 00:04:45.300 doesn't confuse you even more. 00:04:51.900 --> 00:04:53.930 So go back to the pieces of the pie. 00:04:53.930 --> 00:04:58.490 So there's 2 and 1/4 pieces of pie. 00:04:58.490 --> 00:05:03.000 And we want to re-write this as just how many 1/4s 00:05:03.000 --> 00:05:04.800 of pie are there total. 00:05:04.800 --> 00:05:08.770 Well if we take each of these pieces -- I need to change the 00:05:08.770 --> 00:05:13.225 color -- if we take each of these pieces and we divide it 00:05:13.225 --> 00:05:17.800 into 1/4s, we can now say how many total 1/4s of 00:05:17.800 --> 00:05:19.060 pie do we have? 00:05:19.060 --> 00:05:28.490 Well we have 1, 2, 3, 4, 5, 6, 7, 8, 9 fourths. 00:05:28.490 --> 00:05:29.760 Makes sense, right? 00:05:29.760 --> 00:05:33.620 2 and 1/4 is the same thing as 9/4. 00:05:33.620 --> 00:05:36.920 And this will work with any fraction. 00:05:36.920 --> 00:05:37.830 So let's go the other way. 00:05:37.830 --> 00:05:41.700 Let's figure out how to go from an improper fraction 00:05:41.700 --> 00:05:43.680 to a mixed number. 00:05:43.680 --> 00:05:49.260 Let's say I had 23 over 5. 00:05:49.260 --> 00:05:51.080 So here we go in the opposite direction. 00:05:51.080 --> 00:05:53.290 We actually take the denominator, we say how 00:05:53.290 --> 00:05:55.080 many times does it go into the numerator. 00:05:55.080 --> 00:05:57.650 And then we figure out the remainder. 00:05:57.650 --> 00:06:03.220 So let's say 5 goes into 23 -- well, 5 goes 00:06:03.220 --> 00:06:05.400 into 23 four times. 00:06:05.400 --> 00:06:08.890 4 times 5 is 20. 00:06:08.890 --> 00:06:11.240 And the remainder is 3. 00:06:11.240 --> 00:06:17.150 So 23 over 5, we can say that's equal to 4 and in 00:06:17.150 --> 00:06:19.970 the remainder 3 over 5. 00:06:19.970 --> 00:06:22.860 So it's 4 and 3/5. 00:06:25.450 --> 00:06:26.850 Let's review what we just did. 00:06:26.850 --> 00:06:29.220 We just took the denominator and divided it into 00:06:29.220 --> 00:06:30.140 the numerator. 00:06:30.140 --> 00:06:33.830 So 5 goes into 23 four times. 00:06:33.830 --> 00:06:38.050 And what's left over is 3. 00:06:38.050 --> 00:06:41.930 So, 5 goes into 23, 4 and 3/5 times. 00:06:41.930 --> 00:06:46.240 Or another way of saying that is 23 over 5 is 4 and 3/5. 00:06:46.240 --> 00:06:48.270 Let's do another example like that. 00:06:48.270 --> 00:06:51.850 Let's say 17 over 8. 00:06:51.850 --> 00:06:53.680 What does that equal as a mixed number? 00:06:53.680 --> 00:06:56.600 You can actually do this in your head, but I'll 00:06:56.600 --> 00:06:59.430 write it out just so you don't get confused. 00:06:59.430 --> 00:07:04.540 8 goes into 17 two times. 00:07:04.540 --> 00:07:07.550 2 times 8 is 16. 00:07:07.550 --> 00:07:09.380 17 minus 16 is 1. 00:07:09.380 --> 00:07:10.810 Remainder 1. 00:07:10.810 --> 00:07:19.060 So, 17 over 8 is equal to 2 -- that's this 2 -- and 1/8. 00:07:19.060 --> 00:07:22.610 Because we have one 8 left over. 00:07:22.610 --> 00:07:25.200 Let me show you kind of a visual way of representing this 00:07:25.200 --> 00:07:28.590 too, so it actually makes sense how this conversion is working. 00:07:28.590 --> 00:07:33.540 Let's say I had 5/2, right? 00:07:33.540 --> 00:07:37.780 So that literally means I have 5 halves, or if we go back to 00:07:37.780 --> 00:07:42.310 the pizza or the pie analogy, let's draw my five 00:07:42.310 --> 00:07:44.290 halves of pizza. 00:07:44.290 --> 00:07:49.880 So let's say I have one half of pizza here, and let's say I 00:07:49.880 --> 00:07:51.740 have another half of pizza here. 00:07:51.740 --> 00:07:54.530 I just flipped it over. 00:07:54.530 --> 00:07:55.340 So that's 2. 00:07:55.340 --> 00:07:57.720 So it's 1 half, 2 halves. 00:08:00.860 --> 00:08:03.570 So that's three halves. 00:08:03.570 --> 00:08:05.380 And then I have a fourth half here. 00:08:05.380 --> 00:08:07.900 These are halves of pizza, and then I have a fifth 00:08:07.900 --> 00:08:10.660 half here, right? 00:08:10.660 --> 00:08:12.910 So that's 5/2. 00:08:12.910 --> 00:08:17.150 Well, if we look at this, if we combine these two halves, this 00:08:17.150 --> 00:08:21.750 is equal to 1 piece, I have another piece, and then I 00:08:21.750 --> 00:08:23.910 have half of a piece, right? 00:08:23.910 --> 00:08:31.330 So that is equal to 2 and 1/2 pieces of pie. 00:08:31.330 --> 00:08:33.010 Hopefully that doesn't confuse you too much. 00:08:33.010 --> 00:08:37.290 And if we wanted to do this the systematic way, we could have 00:08:37.290 --> 00:08:43.580 said 2 goes into 5 -- well, 2 goes into 5 two times, and 00:08:43.580 --> 00:08:46.560 that 2 is right here. 00:08:46.560 --> 00:08:49.170 And then 2 times 2 is 4. 00:08:49.170 --> 00:08:51.870 5 minus 4 is 1, so the remainder is 1, and 00:08:51.870 --> 00:08:54.050 that's what we use here. 00:08:54.050 --> 00:08:56.710 And of course, we keep the denominator the same. 00:08:56.710 --> 00:08:59.090 So 5/2 equals 2 and 1/2. 00:08:59.090 --> 00:09:01.530 Hopefully that gives you a sense of how to go from one 00:09:01.530 --> 00:09:04.570 mixed number to an improper fraction, and vice versa, 00:09:04.570 --> 00:09:07.520 from an improper fraction to a mixed number. 00:09:07.520 --> 00:09:09.660 If you're still confused let me know and I might 00:09:09.660 --> 00:09:11.660 make some more modules. 00:09:11.660 --> 00:09:12.740 Have fun with the exercises.

Converting fractions to decimals

https://www.youtube.com/watch?v=Gn2pdkvdbGQ

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https://www.youtube.com/api/timedtext?v=Gn2pdkvdbGQ&ei=fmeUZbDQNP3CmLAPkOK3eA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=07A0A38C3B41119103827D188132261C8CC3027D.BDBB04A13403EC9CFC9E0656C376ED823A36E72F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:03.770 I'll now show you how to convert a fraction 00:00:03.770 --> 00:00:04.920 into a decimal. 00:00:04.920 --> 00:00:06.990 And if we have time, maybe we'll learn how to do a 00:00:06.990 --> 00:00:08.730 decimal into a fraction. 00:00:08.730 --> 00:00:11.420 So let's start with, what I would say, is a fairly 00:00:11.420 --> 00:00:12.480 straightforward example. 00:00:12.480 --> 00:00:15.210 Let's start with the fraction 1/2. 00:00:15.210 --> 00:00:17.390 And I want to convert that into a decimal. 00:00:17.390 --> 00:00:20.170 So the method I'm going to show you will always work. 00:00:20.170 --> 00:00:22.850 What you do is you take the denominator and you divide 00:00:22.850 --> 00:00:24.530 it into the numerator. 00:00:24.530 --> 00:00:25.510 Let's see how that works. 00:00:25.510 --> 00:00:29.110 So we take the denominator-- is 2-- and we're going to divide 00:00:29.110 --> 00:00:32.280 that into the numerator, 1. 00:00:32.280 --> 00:00:34.110 And you're probably saying, well, how do I divide 2 into 1? 00:00:34.110 --> 00:00:37.010 Well, if you remember from the dividing decimals module, we 00:00:37.010 --> 00:00:40.220 can just add a decimal point here and add some trailing 0's. 00:00:40.220 --> 00:00:42.880 We haven't actually changed the value of the number, but we're 00:00:42.880 --> 00:00:45.260 just getting some precision here. 00:00:45.260 --> 00:00:46.700 We put the decimal point here. 00:00:50.260 --> 00:00:50.650 Does 2 go into 1? 00:00:50.650 --> 00:00:51.280 No. 00:00:51.280 --> 00:00:56.180 2 goes into 10, so we go 2 goes into 10 five times. 00:00:56.180 --> 00:00:59.060 5 times 2 is 10. 00:00:59.060 --> 00:01:00.050 Remainder of 0. 00:01:00.050 --> 00:01:01.150 We're done. 00:01:01.150 --> 00:01:06.675 So 1/2 is equal to 0.5. 00:01:10.570 --> 00:01:12.050 Let's do a slightly harder one. 00:01:12.050 --> 00:01:15.000 Let's figure out 1/3. 00:01:15.000 --> 00:01:19.190 Well, once again, we take the denominator, 3, and we divide 00:01:19.190 --> 00:01:20.740 it into the numerator. 00:01:20.740 --> 00:01:25.470 And I'm just going to add a bunch of trailing 0's here. 00:01:25.470 --> 00:01:27.800 3 goes into-- well, 3 doesn't go into 1. 00:01:27.800 --> 00:01:30.150 3 goes into 10 three times. 00:01:30.150 --> 00:01:32.452 3 times 3 is 9. 00:01:32.452 --> 00:01:35.720 Let's subtract, get a 1, bring down the 0. 00:01:35.720 --> 00:01:37.700 3 goes into 10 three times. 00:01:37.700 --> 00:01:39.700 Actually, this decimal point is right here. 00:01:39.700 --> 00:01:42.710 3 times 3 is 9. 00:01:42.710 --> 00:01:43.930 Do you see a pattern here? 00:01:43.930 --> 00:01:45.070 We keep getting the same thing. 00:01:45.070 --> 00:01:47.350 As you see it's actually 0.3333. 00:01:47.350 --> 00:01:48.830 It goes on forever. 00:01:48.830 --> 00:01:52.160 And a way to actually represent this, obviously you can't write 00:01:52.160 --> 00:01:54.020 an infinite number of 3's. 00:01:54.020 --> 00:02:00.430 Is you could just write 0.-- well, you could write 0.33 00:02:00.430 --> 00:02:03.060 repeating, which means that the 0.33 will go on forever. 00:02:03.060 --> 00:02:06.960 Or you can actually even say 0.3 repeating. 00:02:06.960 --> 00:02:08.630 Although I tend to see this more often. 00:02:08.630 --> 00:02:09.840 Maybe I'm just mistaken. 00:02:09.840 --> 00:02:12.410 But in general, this line on top of the decimal means 00:02:12.410 --> 00:02:17.320 that this number pattern repeats indefinitely. 00:02:17.320 --> 00:02:25.210 So 1/3 is equal to 0.33333 and it goes on forever. 00:02:25.210 --> 00:02:29.770 Another way of writing that is 0.33 repeating. 00:02:29.770 --> 00:02:33.400 Let's do a couple of, maybe a little bit harder, but they 00:02:33.400 --> 00:02:35.060 all follow the same pattern. 00:02:35.060 --> 00:02:36.890 Let me pick some weird numbers. 00:02:40.470 --> 00:02:41.890 Let me actually do an improper fraction. 00:02:41.890 --> 00:02:49.050 Let me say 17/9. 00:02:49.050 --> 00:02:50.160 So here, it's interesting. 00:02:50.160 --> 00:02:52.260 The numerator is bigger than the denominator. 00:02:52.260 --> 00:02:54.200 So actually we're going to get a number larger than 1. 00:02:54.200 --> 00:02:55.270 But let's work it out. 00:02:55.270 --> 00:03:00.586 So we take 9 and we divide it into 17. 00:03:00.586 --> 00:03:06.000 And let's add some trailing 0's for the decimal point here. 00:03:06.000 --> 00:03:08.730 So 9 goes into 17 one time. 00:03:08.730 --> 00:03:11.260 1 times 9 is 9. 00:03:11.260 --> 00:03:14.040 17 minus 9 is 8. 00:03:14.040 --> 00:03:16.240 Bring down a 0. 00:03:16.240 --> 00:03:20.080 9 goes into 80-- well, we know that 9 times 9 is 81, so it has 00:03:20.080 --> 00:03:21.830 to go into it only eight times because it can't go 00:03:21.830 --> 00:03:23.230 into it nine times. 00:03:23.230 --> 00:03:27.010 8 times 9 is 72. 00:03:27.010 --> 00:03:29.560 80 minus 72 is 8. 00:03:29.560 --> 00:03:30.770 Bring down another 0. 00:03:30.770 --> 00:03:32.260 I think we see a pattern forming again. 00:03:32.260 --> 00:03:35.990 9 goes into 80 eight times. 00:03:35.990 --> 00:03:40.820 8 times 9 is 72. 00:03:40.820 --> 00:03:44.350 And clearly, I could keep doing this forever and 00:03:44.350 --> 00:03:46.790 we'd keep getting 8's. 00:03:46.790 --> 00:03:53.740 So we see 17 divided by 9 is equal to 1.88 where the 0.88 00:03:53.740 --> 00:03:56.080 actually repeats forever. 00:03:56.080 --> 00:03:59.200 Or, if we actually wanted to round this we could say that 00:03:59.200 --> 00:04:01.430 that is also equal to 1.-- depending where we wanted 00:04:01.430 --> 00:04:02.860 to round it, what place. 00:04:02.860 --> 00:04:05.990 We could say roughly 1.89. 00:04:05.990 --> 00:04:07.480 Or we could round in a different place. 00:04:07.480 --> 00:04:09.310 I rounded in the 100's place. 00:04:09.310 --> 00:04:11.350 But this is actually the exact answer. 00:04:11.350 --> 00:04:15.126 17/9 is equal to 1.88. 00:04:15.126 --> 00:04:17.380 I actually might do a separate module, but how would we write 00:04:17.380 --> 00:04:20.730 this as a mixed number? 00:04:20.730 --> 00:04:23.030 Well actually, I'm going to do that in a separate. 00:04:23.030 --> 00:04:24.390 I don't want to confuse you for now. 00:04:24.390 --> 00:04:25.380 Let's do a couple more problems. 00:04:28.560 --> 00:04:29.980 Let me do a real weird one. 00:04:29.980 --> 00:04:34.360 Let me do 17/93. 00:04:34.360 --> 00:04:36.710 What does that equal as a decimal? 00:04:36.710 --> 00:04:39.130 Well, we do the same thing. 00:04:39.130 --> 00:04:45.630 93 goes into-- I make a really long line up here because 00:04:45.630 --> 00:04:47.930 I don't know how many decimal places we'll do. 00:04:50.570 --> 00:04:53.220 And remember, it's always the denominator being divided 00:04:53.220 --> 00:04:54.930 into the numerator. 00:04:54.930 --> 00:04:56.950 This used to confuse me a lot of times because you're often 00:04:56.950 --> 00:04:59.630 dividing a larger number into a smaller number. 00:04:59.630 --> 00:05:02.580 So 93 goes into 17 zero times. 00:05:02.580 --> 00:05:04.080 There's a decimal. 00:05:04.080 --> 00:05:05.990 93 goes into 170? 00:05:05.990 --> 00:05:07.270 Goes into it one time. 00:05:07.270 --> 00:05:11.410 1 times 93 is 93. 00:05:11.410 --> 00:05:14.370 170 minus 93 is 77. 00:05:17.980 --> 00:05:20.360 Bring down the 0. 00:05:20.360 --> 00:05:23.700 93 goes into 770? 00:05:23.700 --> 00:05:24.660 Let's see. 00:05:24.660 --> 00:05:29.120 It will go into it, I think, roughly eight times. 00:05:29.120 --> 00:05:33.330 8 times 3 is 24. 00:05:33.330 --> 00:05:35.970 8 times 9 is 72. 00:05:35.970 --> 00:05:39.730 Plus 2 is 74. 00:05:39.730 --> 00:05:42.186 And then we subtract. 00:05:42.186 --> 00:05:43.990 10 and 6. 00:05:43.990 --> 00:05:46.710 It's equal to 26. 00:05:46.710 --> 00:05:47.760 Then we bring down another 0. 00:05:47.760 --> 00:05:52.800 93 goes into 26-- about two times. 00:05:52.800 --> 00:05:57.020 2 times 3 is 6. 00:05:57.020 --> 00:05:58.704 18. 00:05:58.704 --> 00:05:59.920 This is 74. 00:06:03.120 --> 00:06:03.930 0. 00:06:03.930 --> 00:06:06.380 So we could keep going. 00:06:06.380 --> 00:06:08.030 We could keep figuring out the decimal points. 00:06:08.030 --> 00:06:10.020 You could do this indefinitely. 00:06:10.020 --> 00:06:12.090 But if you wanted to at least get an approximation, you would 00:06:12.090 --> 00:06:23.490 say 17 goes into 93 0.-- or 17/93 is equal to 0.182 and 00:06:23.490 --> 00:06:25.020 then the decimals will keep going. 00:06:25.020 --> 00:06:27.170 And you can keep doing it if you want. 00:06:27.170 --> 00:06:28.650 If you actually saw this on exam they'd probably tell 00:06:28.650 --> 00:06:29.640 you to stop at some point. 00:06:29.640 --> 00:06:31.650 You know, round it to the nearest hundredths or 00:06:31.650 --> 00:06:33.610 thousandths place. 00:06:33.610 --> 00:06:36.550 And just so you know, let's try to convert it the other way, 00:06:36.550 --> 00:06:37.830 from decimals to fractions. 00:06:37.830 --> 00:06:40.090 Actually, this is, I think, you'll find a 00:06:40.090 --> 00:06:42.300 much easier thing to do. 00:06:42.300 --> 00:06:49.810 If I were to ask you what 0.035 is as a fraction? 00:06:49.810 --> 00:06:56.845 Well, all you do is you say, well, 0.035, we could write it 00:06:56.845 --> 00:07:05.130 this way-- we could write that's the same thing as 03-- 00:07:05.130 --> 00:07:06.300 well, I shouldn't write 035. 00:07:06.300 --> 00:07:10.700 That's the same thing as 35/1,000. 00:07:10.700 --> 00:07:11.580 And you're probably saying, Sal, how did 00:07:11.580 --> 00:07:14.120 you know it's 35/1000? 00:07:14.120 --> 00:07:18.590 Well because we went to 3-- this is the 10's place. 00:07:18.590 --> 00:07:20.230 Tenths not 10's. 00:07:20.230 --> 00:07:21.360 This is hundreths. 00:07:21.360 --> 00:07:23.230 This is the thousandths place. 00:07:23.230 --> 00:07:25.890 So we went to 3 decimals of significance. 00:07:25.890 --> 00:07:29.260 So this is 35 thousandths. 00:07:29.260 --> 00:07:38.650 If the decimal was let's say, if it was 0.030. 00:07:38.650 --> 00:07:40.140 There's a couple of ways we could say this. 00:07:40.140 --> 00:07:42.490 Well, we could say, oh well we got to 3-- we went to 00:07:42.490 --> 00:07:43.570 the thousandths Place. 00:07:43.570 --> 00:07:48.240 So this is the same thing as 30/1,000. 00:07:48.240 --> 00:07:48.610 or. 00:07:48.610 --> 00:07:55.550 We could have also said, well, 0.030 is the same thing as 00:07:55.550 --> 00:08:02.710 0.03 because this 0 really doesn't add any value. 00:08:02.710 --> 00:08:05.920 If we have 0.03 then we're only going to the hundredths place. 00:08:05.920 --> 00:08:11.100 So this is the same thing as 3/100. 00:08:11.100 --> 00:08:13.160 So let me ask you, are these two the same? 00:08:16.330 --> 00:08:16.670 Well, yeah. 00:08:16.670 --> 00:08:17.680 Sure they are. 00:08:17.680 --> 00:08:20.065 If we divide both the numerator and the denominator of both of 00:08:20.065 --> 00:08:24.890 these expressions by 10 we get 3/100. 00:08:24.890 --> 00:08:26.220 Let's go back to this case. 00:08:26.220 --> 00:08:27.550 Are we done with this? 00:08:27.550 --> 00:08:30.120 Is 35/1,000-- I mean, it's right. 00:08:30.120 --> 00:08:31.660 That is a fraction. 00:08:31.660 --> 00:08:32.584 35/1,000. 00:08:32.584 --> 00:08:35.440 But if we wanted to simplify it even more looks like we could 00:08:35.440 --> 00:08:38.530 divide both the numerator and the denominator by 5. 00:08:38.530 --> 00:08:40.860 And then, just to get it into simplest form, 00:08:40.860 --> 00:08:47.280 that equals 7/200. 00:08:47.280 --> 00:08:51.020 And if we wanted to convert 7/200 into a decimal using the 00:08:51.020 --> 00:08:54.150 technique we just did, so we would do 200 goes into 00:08:54.150 --> 00:08:56.120 7 and figure it out. 00:08:56.120 --> 00:09:00.170 We should get 0.035. 00:09:00.170 --> 00:09:02.650 I'll leave that up to you as an exercise. 00:09:02.650 --> 00:09:05.370 Hopefully now you get at least an initial understanding of how 00:09:05.370 --> 00:09:09.320 to convert a fraction into a decimal and maybe vice versa. 00:09:09.320 --> 00:09:11.840 And if you don't, just do some of the practices. 00:09:11.840 --> 00:09:16.990 And I will also try to record another module on this 00:09:16.990 --> 00:09:18.880 or another presentation. 00:09:18.880 --> 00:09:20.090 Have fun with the exercises.

Functions Part 2

https://www.youtube.com/watch?v=XEblO51pF5I

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en

WEBVTT Kind: captions Language: en 00:00:00.880 --> 00:00:03.040 Welcome to the second presentation on functions. 00:00:03.040 --> 00:00:05.370 So let's take off where we left off before. 00:00:05.370 --> 00:00:08.510 I still apologize -- in retrospect that that 00:00:08.510 --> 00:00:09.770 whole foul food example. 00:00:09.770 --> 00:00:12.460 Well maybe it was helpful, so I'm going to leave it there. 00:00:12.460 --> 00:00:15.050 Let's do some more problems. 00:00:15.050 --> 00:00:17.050 I think the best thing is to keep doing problems with you 00:00:17.050 --> 00:00:19.050 and I think you'll see the example, and hopefully 00:00:19.050 --> 00:00:21.960 you'll actually see that functions are kind of fun. 00:00:21.960 --> 00:00:24.230 Let's do some more problems. 00:00:24.230 --> 00:00:25.840 Let's start off with an example, not too different 00:00:25.840 --> 00:00:26.630 than what we saw before. 00:00:26.630 --> 00:00:47.210 Let's say that g of x is equal to 1 if x is even, and 00:00:47.210 --> 00:00:56.030 it equals 0 if x is odd. 00:00:56.030 --> 00:01:15.595 And let's say f of x is equal to x plus 3 times g of x. 00:01:18.240 --> 00:01:20.860 And let's say -- I'm going to make it really complicated -- 00:01:20.860 --> 00:01:22.130 well, actually I'm not going to make it any more 00:01:22.130 --> 00:01:22.990 complicated now. 00:01:22.990 --> 00:01:24.680 So let's try some problems. 00:01:24.680 --> 00:01:26.800 So let's give an example. 00:01:26.800 --> 00:01:32.080 What is f of 5. 00:01:32.080 --> 00:01:34.180 Well, it's really pretty straightforward. 00:01:34.180 --> 00:01:40.560 We take this 5 and we replace it for x in the function f. 00:01:40.560 --> 00:01:49.890 So f of 5 is equal to 5 plus 3 times g of 5, right? 00:01:49.890 --> 00:01:52.190 We just literally took this 5 and replace it everywhere 00:01:52.190 --> 00:01:53.360 where we see an x. 00:01:53.360 --> 00:01:57.070 If instead of a 5, I had like a dog here, it would be f of dog 00:01:57.070 --> 00:01:59.590 would equal dog plus 3 times g of dog. 00:01:59.590 --> 00:02:01.700 Not that that would necessarily make any sense, but 00:02:01.700 --> 00:02:02.970 you get the idea. 00:02:02.970 --> 00:02:06.530 So f of 5 equals 5 plus 3 times g of 5. 00:02:06.530 --> 00:02:07.290 But what does that equal? 00:02:07.290 --> 00:02:12.000 So the 5 stays the same, plus 3 times -- well what's g of 5? 00:02:12.000 --> 00:02:16.750 Well, if we put 5 here, if 5 is even we do 1, 00:02:16.750 --> 00:02:18.730 if five is odd we do 0. 00:02:18.730 --> 00:02:20.480 Well 5 is odd so it's a 0. 00:02:20.480 --> 00:02:22.270 So g of 5 is equal to 0. 00:02:22.270 --> 00:02:24.700 So this is 3 times 0. 00:02:24.700 --> 00:02:27.550 So this equals just 5, right, because 3 times 00:02:27.550 --> 00:02:29.400 0 is equal to 5. 00:02:29.400 --> 00:02:35.050 Well what would be f of 6? 00:02:35.050 --> 00:02:42.090 Well, f of 6 would equal 6 plus 3 times g of 6. 00:02:44.650 --> 00:02:48.540 And once again, that equals 6 plus -- well, this time g of 00:02:48.540 --> 00:02:51.560 6 is, well, 6 is even, so 1. 00:02:51.560 --> 00:02:53.370 So g of 6 is equal to 1. 00:02:53.370 --> 00:02:56.300 So this equals 6 plus 3 times 1. 00:02:56.300 --> 00:03:01.080 So this equals 6 plus 3 which equals 9. 00:03:01.080 --> 00:03:02.820 I think you might be getting the idea now. 00:03:02.820 --> 00:03:05.140 At first when you see a problem with a lot of these functions, 00:03:05.140 --> 00:03:05.770 it seems very confusing. 00:03:05.770 --> 00:03:08.910 But if you just keep taking what's inside of the 00:03:08.910 --> 00:03:11.960 parentheses and replacing that for x and just keep moving 00:03:11.960 --> 00:03:15.690 along that way, you make a lot of progress on these problems. 00:03:15.690 --> 00:03:18.260 Let's try a harder one. 00:03:18.260 --> 00:03:27.370 Let's say I said that f of x is equal to x squared plus 1. 00:03:27.370 --> 00:03:45.440 Let's say that g of x is equal to 2x plus f of x minus 3. 00:03:45.440 --> 00:03:53.325 And h of x is equal to 5x. 00:03:56.200 --> 00:03:58.440 Now I'm going to give you a tough problem. 00:03:58.440 --> 00:04:04.620 What is h of g of x? 00:04:04.620 --> 00:04:05.500 No. 00:04:05.500 --> 00:04:10.660 What is h of g of -- let's pick a number -- let's say 3? 00:04:10.660 --> 00:04:12.130 h of g of 3. 00:04:12.130 --> 00:04:14.310 Actually, we'll do examples in the future where we actually 00:04:14.310 --> 00:04:16.520 could leave the x there and we'll solve for it. 00:04:16.520 --> 00:04:21.810 But let's say this particular example, what is h of g of 3? 00:04:21.810 --> 00:04:24.050 At first you might say wow, this is crazy, Sal, I don't 00:04:24.050 --> 00:04:25.650 know how to even start here. 00:04:25.650 --> 00:04:27.200 But you just take it step-by-step. 00:04:27.200 --> 00:04:28.170 What can we figure out? 00:04:28.170 --> 00:04:30.350 Can we figure out what g of 3 is? 00:04:30.350 --> 00:04:31.540 Well sure. 00:04:31.540 --> 00:04:35.220 We could take the 3 and put it into the function g and 00:04:35.220 --> 00:04:37.000 see what it spits out. 00:04:37.000 --> 00:04:39.460 So let's work on g of 3 first. 00:04:39.460 --> 00:04:46.400 So, g of 3 equals -- well it's 2 times 3, right, we're just 00:04:46.400 --> 00:04:48.450 replacing wherever we see an x with a 3. 00:04:48.450 --> 00:04:56.970 So it's 2 times 3, so that's 6, plus f of -- what, we'll 00:04:56.970 --> 00:04:58.640 just replace the x again. 00:04:58.640 --> 00:05:02.140 3 minus 3, right? 00:05:02.140 --> 00:05:08.560 So this g of 3 is equal to 6 plus f of what? 00:05:08.560 --> 00:05:12.500 3 minus 3 is 0. 00:05:12.500 --> 00:05:15.200 Now we have to figure out f of 0 is. 00:05:15.200 --> 00:05:18.030 We have a definition here for f, so we just figure it out. 00:05:18.030 --> 00:05:24.040 f of 0 is equal to -- well, you replace the 0 here. 00:05:24.040 --> 00:05:26.810 So you get 0 squared, which is 0 plus 1. 00:05:26.810 --> 00:05:29.130 So it's f of 0 is 1. 00:05:29.130 --> 00:05:32.910 So you take that and you replace it for f of 0. 00:05:32.910 --> 00:05:39.360 So you get g of 3 is equal to 6 plus 1. 00:05:39.360 --> 00:05:44.820 So g of 3 is equal to 7, right? 00:05:44.820 --> 00:05:46.490 Now we know what g of 3 is equal to. 00:05:46.490 --> 00:05:49.470 We can substitute that back here. 00:05:49.470 --> 00:05:52.430 So that's the same thing -- we know g of 3 is equal to 7, 00:05:52.430 --> 00:05:56.920 so that's the same thing as h of 7. 00:05:56.920 --> 00:06:03.360 And h of 7 is just equal to 5 times 7 equals 35. 00:06:03.360 --> 00:06:06.060 So I think you're probably a little confused here, and I 00:06:06.060 --> 00:06:08.270 would have been if I was in your shoes. 00:06:08.270 --> 00:06:10.610 But the important thing is when you first see this problem 00:06:10.610 --> 00:06:13.730 you're like what can I tackle first? 00:06:13.730 --> 00:06:16.640 h of g of 3, it seems very confusing. 00:06:16.640 --> 00:06:18.390 Well, g of 3, can I tackle that? 00:06:18.390 --> 00:06:18.610 Sure. 00:06:18.610 --> 00:06:21.295 I have a definition of what the function g does when 00:06:21.295 --> 00:06:24.280 it's given an x, or in this case, was given a 3. 00:06:24.280 --> 00:06:25.190 And that's what we did. 00:06:25.190 --> 00:06:27.990 We figured out what g of 3 was first. 00:06:27.990 --> 00:06:30.410 And g of 3, we just have to do the 3, and we said well that's 00:06:30.410 --> 00:06:33.850 6 plus f of 3 minus 3, right? 00:06:33.850 --> 00:06:36.240 Because we just replaced that x with that 3. 00:06:36.240 --> 00:06:37.630 And we just kept solving. 00:06:37.630 --> 00:06:40.070 We figured out what f of 0 is up here. 00:06:40.070 --> 00:06:42.050 And we got g of 3 equals 7. 00:06:42.050 --> 00:06:44.290 Then we substituted that back in right here. 00:06:44.290 --> 00:06:49.200 We got h of 7 is equal to 35 because it was 5 times 7. 00:06:49.200 --> 00:06:50.700 Let's do some more problems. 00:06:50.700 --> 00:06:54.670 Actually, let's do another example with the same 00:06:54.670 --> 00:07:00.880 set of functions. 00:07:00.880 --> 00:07:02.980 I don't want to keep confusing you with new functions. 00:07:02.980 --> 00:07:08.260 Let me it erase this as fast as I can. 00:07:08.260 --> 00:07:10.925 I think I'm getting faster at this erasing business. 00:07:15.160 --> 00:07:17.210 You can sit and think a little bit about what we just 00:07:17.210 --> 00:07:18.110 did while I erase. 00:07:42.390 --> 00:07:44.530 So let's do another problem. 00:07:44.530 --> 00:07:56.530 What is f of h of 10? 00:07:59.450 --> 00:08:05.080 Well, first we want to figure out what h of 10 is, right? 00:08:05.080 --> 00:08:06.510 Well, we could do it in a different way 00:08:06.510 --> 00:08:07.170 as we'll see later. 00:08:07.170 --> 00:08:10.100 But we can figure out what h of 10 is pretty easily. 00:08:10.100 --> 00:08:12.460 We take the 10, substitute it in for x. 00:08:12.460 --> 00:08:14.830 h of 10 is equal to 5 times x. 00:08:14.830 --> 00:08:17.910 In this case x is 10 so it equals 50. 00:08:17.910 --> 00:08:22.910 So we know h of 10 equals 50. 00:08:22.910 --> 00:08:25.010 So we know h of 10 equals 50, so we substitute 00:08:25.010 --> 00:08:25.820 that back in here. 00:08:25.820 --> 00:08:29.080 So we say f of h of 10 is the same thing as f of 50. 00:08:32.380 --> 00:08:34.890 And then f of 50 is, I think pretty straightforward 00:08:34.890 --> 00:08:35.660 at this point. 00:08:35.660 --> 00:08:38.110 You just take that 50 and replace it back here. 00:08:38.110 --> 00:08:40.150 Well, it's 50 squared plus 1. 00:08:40.150 --> 00:08:42.690 Well, 50 squared is 2,500 plus 1. 00:08:45.750 --> 00:08:49.410 That equals 2,501. 00:08:49.410 --> 00:09:01.930 What is g of h of 1? 00:09:01.930 --> 00:09:08.920 Well, we take h of 1, h of 1 is 5, so this is equal to g of 5. 00:09:08.920 --> 00:09:12.840 And g of 5, we just replace the 5 here, so g of 5 is equal to 2 00:09:12.840 --> 00:09:16.810 times 5 plus f of 5 minus 3. 00:09:16.810 --> 00:09:19.230 We just take wherever we saw an x and replace it with a 5. 00:09:19.230 --> 00:09:25.520 Well, that's equal to 2 times 5 is 10, plus f of 5 minus 3. 00:09:25.520 --> 00:09:27.730 Well 5 minus 3 is 2. 00:09:27.730 --> 00:09:29.950 Plus f of 2. 00:09:29.950 --> 00:09:31.470 What's f of 2? 00:09:31.470 --> 00:09:34.902 Well, 2 squared plus 1 is 5, right? f of 2 is 5 00:09:34.902 --> 00:09:36.630 -- 2 squared plus 1. 00:09:36.630 --> 00:09:41.400 So that equals 10 plus 5 which equals 15. 00:09:41.400 --> 00:09:43.340 If you're still confused, don't worry. 00:09:43.340 --> 00:09:45.990 I'm about to record some more problems that will give you 00:09:45.990 --> 00:09:49.780 even more examples of function problems. 00:09:49.780 --> 00:09:51.680 See you in the next presentation. 00:09:51.680 --> 00:09:52.980 Bye.

Functions (Part III)

https://www.youtube.com/watch?v=5fcRSie63Hs

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WEBVTT Kind: captions Language: en 00:00:00.970 --> 00:00:01.220 Let's 00:00:01.220 --> 00:00:04.880 get going with more examples of function problems, and 00:00:04.880 --> 00:00:07.200 hopefully as we keep doing this, you're going to get 00:00:07.200 --> 00:00:08.665 the idea of how all this stuff works. 00:00:08.665 --> 00:00:12.740 So let's do another problem. 00:00:12.740 --> 00:00:14.720 I'll use green this time. 00:00:14.720 --> 00:00:17.580 Let me clear everything. 00:00:17.580 --> 00:00:19.650 So I'll show you-- I showed you that 1, you could define a 00:00:19.650 --> 00:00:22.920 function as just kind of a standard algebraic expression, 00:00:22.920 --> 00:00:25.710 you could also do it a kind of if number is odd, this is what 00:00:25.710 --> 00:00:27.980 you do, if a number is this, is what you do. 00:00:27.980 --> 00:00:30.030 You could also define a function visually. 00:00:30.030 --> 00:00:34.810 Let's say-- let me draw a graph, and I'll use the line 00:00:34.810 --> 00:00:40.060 tool so it's a reasonably neat graph-- that's 00:00:40.060 --> 00:00:43.480 the x-axis there. 00:00:43.480 --> 00:00:44.380 That's pretty good. 00:00:44.380 --> 00:00:48.830 And let's draw the f of x-axis, or you might be used to calling 00:00:48.830 --> 00:00:54.650 that the y-axis, but-- OK. 00:00:54.650 --> 00:00:57.492 I almost had it vertical, but let's see. 00:00:57.492 --> 00:00:59.490 Let's draw a few slashes here. 00:01:03.270 --> 00:01:05.740 And a couple here, like this. 00:01:09.970 --> 00:01:11.760 Sorry if you're getting bored while I draw this graph. 00:01:11.760 --> 00:01:13.750 I should really have some type of tool so that the 00:01:13.750 --> 00:01:15.410 graphs just show up. 00:01:15.410 --> 00:01:21.830 Let me draw a-- let's say that-- let me 00:01:21.830 --> 00:01:22.500 draw this function. 00:01:22.500 --> 00:01:23.070 So this is what? 00:01:23.070 --> 00:01:28.950 This is 1, 2, 3, 4, 5, this is negative 5, this is 5, this 00:01:28.950 --> 00:01:31.770 is 5, this is negative 5. 00:01:31.770 --> 00:01:38.410 And this is x-axis, and this is-- we'll call 00:01:38.410 --> 00:01:39.770 this the f of x-axis. 00:01:39.770 --> 00:01:43.460 Now that might not seem obvious to you at first, but all this 00:01:43.460 --> 00:01:47.210 is saying is let's say when x is equal to negative 5, this 00:01:47.210 --> 00:01:50.450 function-- I'm creating a function definition-- let's say 00:01:50.450 --> 00:01:59.436 it equals 2, that's negative 1, that stays the same, that stays 00:01:59.436 --> 00:02:10.140 the same, then it goes to here, and then it goes to here, to 00:02:10.140 --> 00:02:13.830 here, and then-- let's see. 00:02:13.830 --> 00:02:15.936 I hope I'm not boring you. 00:02:15.936 --> 00:02:18.450 And it just keeps moving up. 00:02:18.450 --> 00:02:19.910 Let me see, what would this look like-- this 00:02:19.910 --> 00:02:21.820 would look like this. 00:02:21.820 --> 00:02:25.240 So if I-- you might think I'm doing something very strange 00:02:25.240 --> 00:02:29.930 right now, but just bear with me while I draw this. 00:02:29.930 --> 00:02:32.983 I hope I don't mess up too much. 00:02:32.983 --> 00:02:35.640 And, see, one like that. 00:02:35.640 --> 00:02:37.220 See one like that. 00:02:37.220 --> 00:02:39.630 So we're like, Sal, this is a very strange looking graph. 00:02:39.630 --> 00:02:40.570 And it is. 00:02:40.570 --> 00:02:42.820 But what this is, is this is a function definition. 00:02:42.820 --> 00:02:46.200 This tells you whenever I input an x, at least for the x's that 00:02:46.200 --> 00:02:49.660 we can see on the graph, this graph tell me what 00:02:49.660 --> 00:02:51.030 f of x equals. 00:02:51.030 --> 00:02:56.830 So if x is equal to negative 5, f of x would equal plus 2. 00:02:56.830 --> 00:02:58.230 And we could draw a couple of examples. 00:02:58.230 --> 00:03:04.550 f of 0, well we go to 0 on the x-axis, and we say 00:03:04.550 --> 00:03:08.840 f of 0 is equal to 0. 00:03:08.840 --> 00:03:13.980 f of 1 is equal to-- well, we go to x equal to 1, and we 00:03:13.980 --> 00:03:17.800 just see where the chart is, well, it equals negative 1. 00:03:17.800 --> 00:03:18.620 I think you get the idea. 00:03:18.620 --> 00:03:21.990 This isn't too difficult, but this is a function definition. 00:03:21.990 --> 00:03:27.960 So we've defined this graph right here as f of x. 00:03:27.960 --> 00:03:31.330 So if that graph-- that's the graph of f of x, and let's say 00:03:31.330 --> 00:03:41.230 that we define g of x is equal to f of x-- let's say 00:03:41.230 --> 00:03:48.620 it's equal to f of x squared minus f of x. 00:03:48.620 --> 00:04:03.090 And let's say that h of x is equal to 3 minus x. 00:04:03.090 --> 00:04:18.640 So what if I were to ask you, what is h of g of negative 1? 00:04:18.640 --> 00:04:21.120 So just like we did in the previous problems, first we'll 00:04:21.120 --> 00:04:23.590 say, well, let's try to figure out what g of negative 1 is, 00:04:23.590 --> 00:04:27.020 and then we can substitute that into h of x. 00:04:27.020 --> 00:04:35.130 So g of negative 1 is equal to-- and this is how I do it. 00:04:35.130 --> 00:04:36.390 There's no trick to it. 00:04:36.390 --> 00:04:38.360 Wherever you see the x, you just substitute it with the 00:04:38.360 --> 00:04:40.940 number that you're saying is now the value for x. 00:04:40.940 --> 00:04:47.680 So you say, well, that's equal to f of negative 1 squared 00:04:47.680 --> 00:04:51.240 minus f of negative 1. 00:04:51.240 --> 00:04:53.203 All I did is at g of negative 1, I just substituted 00:04:53.203 --> 00:04:56.050 it wherever I saw an x. 00:04:56.050 --> 00:04:58.190 Well what's f of negative 1? 00:04:58.190 --> 00:05:01.740 Well, when x is equal to negative 1, f of 00:05:01.740 --> 00:05:03.620 x is equal to 1. 00:05:03.620 --> 00:05:07.300 So f of negative 1-- let's write that, f of negative 00:05:07.300 --> 00:05:09.660 1 is equal to 1. 00:05:09.660 --> 00:05:12.040 So g of negative 1 is equal to-- well, that's just 00:05:12.040 --> 00:05:16.940 1 squared minus 1, well that equals 0. 00:05:16.940 --> 00:05:20.050 Because f of negative 1 is 1, so it's 1 squared minus 00:05:20.050 --> 00:05:22.020 1 that equals 1 minus 1. 00:05:22.020 --> 00:05:23.500 0. 00:05:23.500 --> 00:05:25.940 So g of negative 1 is 0, so this is the 00:05:25.940 --> 00:05:29.410 same thing as h of 0. 00:05:29.410 --> 00:05:32.070 Because g of negative 1, we just figured out is 0. 00:05:32.070 --> 00:05:35.960 h of 0, we just take that 0 and substitute it here, so it's 3 00:05:35.960 --> 00:05:38.600 minus 0, so that just equals 3. 00:05:38.600 --> 00:05:40.660 And we solved the problem. 00:05:40.660 --> 00:05:42.710 Let's do another example, and I don't want to erase my graph 00:05:42.710 --> 00:05:47.470 since I took four minutes to actually draw it, let me 00:05:47.470 --> 00:05:51.230 erase what we just did here. 00:05:51.230 --> 00:05:53.500 And what you might want to do after you watch it the first 00:05:53.500 --> 00:05:55.710 time-- and this isn't true just of this video, actually of all 00:05:55.710 --> 00:05:58.080 the videos-- but especially the functions, after watching it 00:05:58.080 --> 00:06:01.890 once, you might want to rewatch it and pause it right after I 00:06:01.890 --> 00:06:04.380 give you the problem and try to do it yourself, and then see-- 00:06:04.380 --> 00:06:06.600 and if you get stuck, you can play it, or if you get an 00:06:06.600 --> 00:06:09.150 answer, just you can play the video and make sure that 00:06:09.150 --> 00:06:11.630 we did the same way. 00:06:11.630 --> 00:06:12.205 Let's see. 00:06:18.370 --> 00:06:19.750 I'm going to create another definition 00:06:19.750 --> 00:06:22.640 for g of x this time. 00:06:22.640 --> 00:06:26.565 Let's say that g of x-- oh whoops, I was trying to write 00:06:26.565 --> 00:06:38.440 in black-- let's say that g of x is equal to f of x 00:06:38.440 --> 00:06:51.000 squared plus f of x plus 2. 00:06:53.530 --> 00:07:01.380 So now, in this case, what is g of-- let's pick a random 00:07:01.380 --> 00:07:06.320 number-- what is g of minus-- no, let's pick a, let's 00:07:06.320 --> 00:07:08.760 say-- what is g of minus 2? 00:07:08.760 --> 00:07:10.290 After we try and pick a number that we could find 00:07:10.290 --> 00:07:11.820 an actual solution for. 00:07:11.820 --> 00:07:16.010 Well g of minus 2, wherever we see the x, x is not 00:07:16.010 --> 00:07:17.610 going to be minus 2. 00:07:17.610 --> 00:07:25.230 That is equal to f of minus 2 squared plus 00:07:25.230 --> 00:07:29.080 f of minus 2 plus 2. 00:07:29.080 --> 00:07:31.380 All we did is wherever we saw an x, we substituted 00:07:31.380 --> 00:07:33.090 it, minus 2 there. 00:07:33.090 --> 00:07:34.120 And let's simplify that. 00:07:34.120 --> 00:07:38.110 Well, f of minus 2 squared, we know what minus 2 squared is, 00:07:38.110 --> 00:07:45.270 that's the same thing as f of 4, plus f of minus 2 plus 2. 00:07:45.270 --> 00:07:46.670 That's 0. 00:07:46.670 --> 00:07:48.720 Plus f of 0. 00:07:48.720 --> 00:07:51.680 And now we just figure out what f of 4 and f of 0 is. 00:07:51.680 --> 00:07:55.120 Well, f of 4, we go where x equals r, it's right here, 00:07:55.120 --> 00:07:59.220 and when x equals 4, f of 4 is equal to 2. 00:07:59.220 --> 00:08:03.210 So this is equal to 2 plus f of 0. 00:08:03.210 --> 00:08:05.710 And just as a reminder, this is the definition of f. 00:08:05.710 --> 00:08:10.560 We didn't define it in terms of an algebraic expression, we 00:08:10.560 --> 00:08:13.320 defined in terms of an actual visual graph. 00:08:13.320 --> 00:08:16.290 So what's f of 0? f of 0 is 0. 00:08:16.290 --> 00:08:20.580 When x is equal to 0-- f of 0 is 0 so that's 2 plus 0-- so g 00:08:20.580 --> 00:08:23.880 of negative 2 is equal to 2. 00:08:23.880 --> 00:08:26.110 An interesting thing, you might want to make problems like this 00:08:26.110 --> 00:08:28.330 for yourself and keep experimenting with different 00:08:28.330 --> 00:08:30.420 types of functions, and a very interesting thing would 00:08:30.420 --> 00:08:33.870 actually be to graph g of x, and actually that's a 00:08:33.870 --> 00:08:34.670 good idea, I think. 00:08:34.670 --> 00:08:37.510 I think maybe we'll do that in the future modules to kind of 00:08:37.510 --> 00:08:39.330 play with functions and actually to try graph 00:08:39.330 --> 00:08:42.090 the functions and see how they turn out. 00:08:42.090 --> 00:08:45.580 I will-- I don't know if I have enough time-- actually, I'm 00:08:45.580 --> 00:08:48.320 going to wait until the next lecture to do a couple 00:08:48.320 --> 00:08:49.250 more examples. 00:08:49.250 --> 00:08:51.940 I want to do as many examples on the functions as I can with 00:08:51.940 --> 00:08:54.410 you, because I think as you keep watching and watching the 00:08:54.410 --> 00:08:58.220 function problems and seeing more and more variations on 00:08:58.220 --> 00:09:01.170 functions, you'll see both how general of a concept this is, 00:09:01.170 --> 00:09:03.900 and hopefully you'll get an idea of how the functions 00:09:03.900 --> 00:09:05.170 actually work. 00:09:05.170 --> 00:09:07.290 Well, I'll see you in the next lecture. 00:09:07.290 --> 00:09:08.780 Have fun.

Introduction to functions

https://www.youtube.com/watch?v=VhokQhjl5t0

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WEBVTT Kind: captions Language: en 00:00:01.200 --> 00:00:03.740 Welcome to the presentation on functions. 00:00:03.740 --> 00:00:06.330 Functions are something that, when I first learned it, it 00:00:06.330 --> 00:00:08.800 was kind of like I had a combination of I was 1, 00:00:08.800 --> 00:00:11.280 confused, and at the same time, I was like, well what's even 00:00:11.280 --> 00:00:12.710 the point of learning this? 00:00:12.710 --> 00:00:16.310 So hopefully, at least in this introduction lecture, we can 00:00:16.310 --> 00:00:19.070 get at least a very general sense of what a function is 00:00:19.070 --> 00:00:21.470 and why it might be useful. 00:00:21.470 --> 00:00:23.560 So let's just start off with just the overall 00:00:23.560 --> 00:00:24.570 concept of a function. 00:00:24.570 --> 00:00:27.450 A function is something that you can give it an input-- and 00:00:27.450 --> 00:00:29.350 we'll start with just one input, but actually you can 00:00:29.350 --> 00:00:32.160 give it multiple inputs-- you give a function an input, 00:00:32.160 --> 00:00:35.030 let's call that input x. 00:00:35.030 --> 00:00:39.420 And you can view a function as-- I guess a bunch of 00:00:39.420 --> 00:00:41.080 different ways you can view it. 00:00:41.080 --> 00:00:41.820 I don't know if you're familiar with the 00:00:41.820 --> 00:00:42.860 concept of a black box. 00:00:42.860 --> 00:00:45.060 A black box is kind of a box, you don't know what's inside of 00:00:45.060 --> 00:00:49.340 it, but if you put something into it like this x, and let's 00:00:49.340 --> 00:00:52.440 call that box-- let's say the function is called f, then 00:00:52.440 --> 00:00:56.760 it'll output what we call f of x. 00:00:56.760 --> 00:00:59.180 I know this terminology might seem a little confusing at 00:00:59.180 --> 00:01:03.820 first, but let's make some-- I guess, let's define what's 00:01:03.820 --> 00:01:05.670 inside the box in different ways. 00:01:05.670 --> 00:01:14.330 Let's say that the function was-- let's say that f of x is 00:01:14.330 --> 00:01:19.240 equal to x squared plus 1. 00:01:19.240 --> 00:01:22.280 Then, if I were to say what is f of-- let's 00:01:22.280 --> 00:01:25.870 say, what's f of 2? 00:01:25.870 --> 00:01:28.050 Well that means we're taking 2 and we're going to 00:01:28.050 --> 00:01:30.360 put it into the box. 00:01:30.360 --> 00:01:32.970 And I want to know what comes out of the box 00:01:32.970 --> 00:01:34.450 when I put 2 into it. 00:01:34.450 --> 00:01:38.430 Well inside the box, we know we do this to the input. 00:01:38.430 --> 00:01:42.060 We take the x, we square it, and we add 1, so f of 2 is 2 00:01:42.060 --> 00:01:44.500 squared, which is 4, plus 1. 00:01:44.500 --> 00:01:47.040 Which is equal to 5. 00:01:47.040 --> 00:01:47.900 I know what you're thinking. 00:01:47.900 --> 00:01:50.510 Probably like, well, Sal, this just seems like a very 00:01:50.510 --> 00:01:55.470 convoluted way of substituting x into an equation and just 00:01:55.470 --> 00:01:56.580 finding out the result. 00:01:56.580 --> 00:01:59.400 And I agree with you right now. 00:01:59.400 --> 00:02:02.770 But as you'll see, a function can become kind of a more 00:02:02.770 --> 00:02:05.420 general thing than just an equation. 00:02:05.420 --> 00:02:09.010 For example, let me say-- let me actually-- actually not, 00:02:09.010 --> 00:02:10.530 let me not erase this. 00:02:10.530 --> 00:02:13.150 Let me define a function as this. 00:02:13.150 --> 00:02:24.560 f of x is equal to x squared plus 1, if x is even, 00:02:24.560 --> 00:02:29.820 and it equals x squared minus 1 if x is odd. 00:02:32.800 --> 00:02:35.390 I know this would have been-- this is something that we've 00:02:35.390 --> 00:02:36.430 never really seen before. 00:02:36.430 --> 00:02:40.320 This isn't just what I would call an analytic expression, 00:02:40.320 --> 00:02:43.540 this isn't just x plus something squared. 00:02:43.540 --> 00:02:45.470 We're actually saying, depending on what type of x you 00:02:45.470 --> 00:02:47.850 put in, we're going to do a different thing to that x. 00:02:47.850 --> 00:02:49.480 So let me ask you a question. 00:02:49.480 --> 00:02:53.470 What's f of 2 in this example? 00:02:53.470 --> 00:02:57.930 Well if we put 2 here, it says if x is even you do this one, 00:02:57.930 --> 00:02:59.470 if x is odd you do this one. 00:02:59.470 --> 00:03:01.480 Well, 2 is even, so we do this top one. 00:03:01.480 --> 00:03:05.830 So we'd say 2 squared plus 1, well that equals 5. 00:03:05.830 --> 00:03:09.230 But then, what's f of 3? 00:03:09.230 --> 00:03:12.180 Well if we put the 3 in here, we'd use this 00:03:12.180 --> 00:03:13.930 case, because 3 is odd. 00:03:13.930 --> 00:03:18.410 So we do 3 squared minus 1. f of 3 is equal to 8. 00:03:18.410 --> 00:03:22.590 So notice, this was a little bit more I guess you could 00:03:22.590 --> 00:03:25.720 even say abstract or unusual in this case. 00:03:25.720 --> 00:03:27.527 I'm going to keep doing examples of functions and 00:03:27.527 --> 00:03:32.590 I'm going to show you how general this idea can be. 00:03:32.590 --> 00:03:35.380 And if you get confused, I'm going to show you that the 00:03:35.380 --> 00:03:37.250 actual function problems you're going to encounter are 00:03:37.250 --> 00:03:38.560 actually not that hard to do. 00:03:38.560 --> 00:03:41.000 I just want to make sure that you least get exposed to kind 00:03:41.000 --> 00:03:44.840 of the general idea of what a function is. 00:03:44.840 --> 00:03:49.920 You can view almost anything in the world as a function. 00:03:49.920 --> 00:03:53.740 Let's say that there is a function called Sal, because, 00:03:53.740 --> 00:03:54.835 you know, that's my name. 00:03:58.620 --> 00:04:00.950 And I'm a function. 00:04:00.950 --> 00:04:08.490 Let's say that if you were to-- let me think. 00:04:08.490 --> 00:04:14.190 If you were to give me food, what do I produce? 00:04:14.190 --> 00:04:17.740 So what is Sal of food? 00:04:17.740 --> 00:04:23.050 So if you input food into Sal, what will Sal produce? 00:04:23.050 --> 00:04:24.770 Well I won't go into some of the things that I would 00:04:24.770 --> 00:04:30.710 produce, but I would produce videos. 00:04:30.710 --> 00:04:33.690 I would produce math videos if you gave me food. 00:04:33.690 --> 00:04:36.370 Math videos. 00:04:36.370 --> 00:04:37.670 I'm just a function. 00:04:37.670 --> 00:04:40.900 You give me food and-- and maybe, actually, maybe 00:04:40.900 --> 00:04:41.920 I have multiple inputs. 00:04:41.920 --> 00:04:48.140 Maybe if you give me a food and a computer, and I would 00:04:48.140 --> 00:04:51.830 produce math videos for you. 00:04:51.830 --> 00:04:54.215 And maybe you are a function. 00:04:57.010 --> 00:04:57.880 I don't know your name. 00:04:57.880 --> 00:05:00.540 I would like to, but I don't know your name. 00:05:00.540 --> 00:05:09.440 And let's say if I were to input math videos into you, 00:05:09.440 --> 00:05:12.840 then you will produce-- let's see, what would you produce? 00:05:12.840 --> 00:05:17.080 If I gave you math videos, you would produce A's on tests. 00:05:23.580 --> 00:05:24.910 A's on your math test. 00:05:24.910 --> 00:05:28.150 Hopefully you're not taking someone else's math test. 00:05:28.150 --> 00:05:29.000 So it's interesting. 00:05:29.000 --> 00:05:31.690 If you give-- well, let's take the computer away. 00:05:31.690 --> 00:05:33.820 Let's say that all Sal needs is food. 00:05:33.820 --> 00:05:35.450 Which is kind of true. 00:05:35.450 --> 00:05:38.640 So if you put food into Sal, Sal of food, he 00:05:38.640 --> 00:05:42.050 produces math videos. 00:05:42.050 --> 00:05:45.930 And if I were to put math videos into you, then you 00:05:45.930 --> 00:05:49.120 produce A's on your math test. 00:05:49.120 --> 00:05:51.710 So let's think of an interesting problem. 00:05:51.710 --> 00:05:59.080 What is you of Sal of food? 00:06:02.720 --> 00:06:05.620 I know this seems very ridiculous, but I actually 00:06:05.620 --> 00:06:08.640 think we might be going someplace, so we might be 00:06:08.640 --> 00:06:10.510 getting somewhere with this kind of idea. 00:06:10.510 --> 00:06:14.000 Well, first we would try to figure out what is Sal of food. 00:06:14.000 --> 00:06:17.340 Well, we already figured out if you put food into Sal, Sal of 00:06:17.340 --> 00:06:19.080 food is equal to math videos. 00:06:19.080 --> 00:06:25.630 So this is the same thing as you of-- I'm trying to confuse 00:06:25.630 --> 00:06:32.120 you-- you of math videos. 00:06:32.120 --> 00:06:34.230 And I already determined, we already said, well, if you put 00:06:34.230 --> 00:06:37.830 math videos into the function called you, whatever your name 00:06:37.830 --> 00:06:42.130 might be, then it produces A's on your math test. 00:06:42.130 --> 00:06:47.480 So that you of math videos equals A's on your math test. 00:06:52.100 --> 00:06:57.180 So you of Sal of food will produce A's on your math test. 00:06:57.180 --> 00:06:59.000 And notice, we just said what happens when you 00:06:59.000 --> 00:07:02.150 put food into Sal. 00:07:02.150 --> 00:07:04.660 This could-- would be a very different outcome if you put, 00:07:04.660 --> 00:07:08.630 like, if you replaced food with let's say poison. 00:07:12.570 --> 00:07:19.810 Because if you put poison into Sal, Sal of poison-- not that I 00:07:19.810 --> 00:07:23.940 would recommend that you did this-- Sal of poison 00:07:23.940 --> 00:07:28.510 would equal death. 00:07:28.510 --> 00:07:31.610 No, no, I shouldn't say something, so no no no no. 00:07:31.610 --> 00:07:33.190 Well you get the idea. 00:07:33.190 --> 00:07:35.720 There wouldn't be math videos. 00:07:35.720 --> 00:07:36.600 Anyway. 00:07:36.600 --> 00:07:38.400 Let me move on. 00:07:38.400 --> 00:07:41.510 So with that kind of-- I'm not so clear whether that would be 00:07:41.510 --> 00:07:46.810 a useful example with the food and the math videos. 00:07:46.810 --> 00:07:50.700 Let's do some actual problems using functions. 00:07:50.700 --> 00:07:54.990 So if I were to tell you that I had one function, called f of x 00:07:54.990 --> 00:08:00.610 is equal to x plus 2, and I had another function that said g 00:08:00.610 --> 00:08:11.850 of x is equal to x squared minus 1. 00:08:11.850 --> 00:08:19.790 If I were to ask you what g of f of 3 is. 00:08:22.780 --> 00:08:27.440 Well the first thing we want to do is evaluate what f of 3 is. 00:08:27.440 --> 00:08:35.070 So if you-- the 3 would replace the x, so f of 3 is equal to 00:08:35.070 --> 00:08:39.210 3 plus 2, which equals 5. 00:08:39.210 --> 00:08:45.440 So g of f of 3 is the same thing as g of 5, because f 00:08:45.440 --> 00:08:46.390 of three is equal to 5. 00:08:46.390 --> 00:08:49.080 Sorry for the little bit of messiness. 00:08:49.080 --> 00:08:50.340 So then, what's g of 5? 00:08:50.340 --> 00:08:53.890 Well, then we take this 5, and we put it in in place of this 00:08:53.890 --> 00:09:02.760 x, so g of 5 is 5 squared, 25, minus 1, which equals 24. 00:09:02.760 --> 00:09:07.110 So g of f of 3 is equal to 24. 00:09:07.110 --> 00:09:08.950 Hopefully that gives you a taste of what a function is all 00:09:08.950 --> 00:09:12.000 about, and I really apologize if I have either confused or 00:09:12.000 --> 00:09:16.730 scared you with the Sal food/poison math video example. 00:09:16.730 --> 00:09:19.780 But in the next set of presentations, I'm going to do 00:09:19.780 --> 00:09:22.420 a lot more of these examples, and I think you'll get the idea 00:09:22.420 --> 00:09:25.110 of at least how to do these problems that you might see on 00:09:25.110 --> 00:09:27.350 your math tests, and maybe get a sense of what functions 00:09:27.350 --> 00:09:29.120 are all about. 00:09:29.120 --> 00:09:30.720 See you in the next video. 00:09:30.720 --> 00:09:32.020 Bye.

Integer sums

https://www.youtube.com/watch?v=W254ewkkMck

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WEBVTT Kind: captions Language: en 00:00:01.050 --> 00:00:03.900 Welcome to the presentation on finding sums of integers. 00:00:03.900 --> 00:00:06.670 You're probably wondering why are we doing this within 00:00:06.670 --> 00:00:08.150 the context of averages. 00:00:08.150 --> 00:00:10.240 Well, if you think about it, all an average is is you take 00:00:10.240 --> 00:00:13.030 a sum of a bunch of numbers and then you divide by the 00:00:13.030 --> 00:00:14.930 number of numbers you have. 00:00:14.930 --> 00:00:16.770 What we're going to do here is do a couple of algebra problems 00:00:16.770 --> 00:00:19.230 that involve just the sum parts first, and actually they can 00:00:19.230 --> 00:00:23.550 carry over into average problems as well. 00:00:23.550 --> 00:00:25.360 Let's get started with a problem. 00:00:25.360 --> 00:00:32.800 Let's say I told you that I had the sum of five consecutive 00:00:32.800 --> 00:00:48.580 integers is equal to 200. 00:00:48.580 --> 00:00:56.005 What is the smallest -- I apologize for my handwriting 00:00:56.005 --> 00:01:03.300 -- what is the smallest of the five integers? 00:01:03.300 --> 00:01:06.750 Well there's a couple of ways to do this, but I guess the 00:01:06.750 --> 00:01:09.760 most straightforward way is just to do it algebraically, 00:01:09.760 --> 00:01:11.080 I would say. 00:01:11.080 --> 00:01:15.315 So let's say that x is the smallest of the integers, 00:01:15.315 --> 00:01:16.650 right, so x is actually what we're going to 00:01:16.650 --> 00:01:17.610 want to figure out. 00:01:21.050 --> 00:01:24.810 Well if x is the smallest, what are the other four going to be? 00:01:24.810 --> 00:01:26.090 We have a total of five. 00:01:26.090 --> 00:01:27.580 Well, they're consecutive. 00:01:27.580 --> 00:01:29.620 Consecutive just means that they follow each other, 00:01:29.620 --> 00:01:32.100 like 5, 6, 7, 8, 9, 10. 00:01:32.100 --> 00:01:34.040 All of those are consecutive integers, right? 00:01:34.040 --> 00:01:36.140 And if you remember, integers are just whole numbers, so it 00:01:36.140 --> 00:01:38.110 can't be a fraction or a decimal. 00:01:38.110 --> 00:01:41.420 So if x is the smallest, so then the next integer is 00:01:41.420 --> 00:01:44.260 going to be x plus 1. 00:01:44.260 --> 00:01:47.360 And the one after that's going to be x plus 2. 00:01:47.360 --> 00:01:50.400 And the one after that's going to be x plus 3. 00:01:50.400 --> 00:01:54.050 And the one after that's going to be x plus 4, right? 00:01:54.050 --> 00:01:56.150 It might seem confusing I'm writing all of these x's. 00:01:56.150 --> 00:01:59.920 But if you think about it, if x was 5, then this would be 6, 00:01:59.920 --> 00:02:03.030 this would be 7, this would be 8, and this would be 9. 00:02:03.030 --> 00:02:05.150 And that's all I'm writing here, right? 00:02:05.150 --> 00:02:07.960 So these would be, assuming that x is the smallest of the 00:02:07.960 --> 00:02:12.550 integers, the five integers would be x, x plus 1, x plus 00:02:12.550 --> 00:02:15.320 2, x plus 3, and x plus 4. 00:02:15.320 --> 00:02:18.030 And we know that the sum of these five consecutive 00:02:18.030 --> 00:02:19.900 integers is 200. 00:02:19.900 --> 00:02:24.620 What is the sum of these five, I guess we could say 00:02:24.620 --> 00:02:27.110 numbers or expressions? 00:02:27.110 --> 00:02:30.670 Well let's see, we have five x's -- 1, 2, 3, 4, 5. 00:02:30.670 --> 00:02:36.662 So x plus x plus x plus x plus x is equal to just 5x. 00:02:36.662 --> 00:02:38.920 Or you could just say 5 times x. 00:02:38.920 --> 00:02:42.540 And then that's plus 1 plus 2 is 3, 3 plus 3 00:02:42.540 --> 00:02:45.600 is 6, 6 plus 4 is 10. 00:02:45.600 --> 00:02:48.490 So the sum of these five integers is going to be 5x plus 00:02:48.490 --> 00:02:51.520 10, and all I did is add up the x's and added up the constants. 00:02:51.520 --> 00:02:56.450 And we know that that is going to equal 200. 00:02:56.450 --> 00:02:58.330 Now this is just a level two linear equation. 00:02:58.330 --> 00:03:00.440 We can just solve for x. 00:03:00.440 --> 00:03:06.250 So we get 5x is equal to 190 -- I just subtracted 10 00:03:06.250 --> 00:03:08.020 from both sides, right? 00:03:08.020 --> 00:03:15.090 And then x is equal to -- let me divide 5 into 190. 00:03:15.090 --> 00:03:19.870 5 goes into 19 three times, 3 times 5 is 15. 00:03:19.870 --> 00:03:23.160 9 minus 5 is 4, bring down the 0. 00:03:23.160 --> 00:03:25.500 5 goes into 40, eight times. 00:03:25.500 --> 00:03:26.940 So x is equal to 38. 00:03:29.700 --> 00:03:32.010 Pretty straightforward problem, don't you think? 00:03:32.010 --> 00:03:36.790 Now what if I were to ask you what is the average of the 00:03:36.790 --> 00:03:38.770 five consecutive numbers? 00:03:38.770 --> 00:03:41.020 Well now, there's two ways of doing this. 00:03:41.020 --> 00:03:43.250 Now that we already know that x is 38, we know that the other 00:03:43.250 --> 00:03:52.130 numbers are going to be -- well this is 38, 39, 40, 41, 42. 00:03:52.130 --> 00:03:53.960 Well we could just average these four numbers. 00:03:53.960 --> 00:03:57.900 You could just say 38 plus 39 plus 40 plus 41 plus 42. 00:03:57.900 --> 00:03:59.360 And well we already know what those -- I don't 00:03:59.360 --> 00:04:00.060 even have to do the math. 00:04:00.060 --> 00:04:03.320 You already know that they average up, they sum up to 200 00:04:03.320 --> 00:04:06.860 and then we divide the sum by 5, because there are 5 numbers. 00:04:06.860 --> 00:04:09.310 So the average is 40. 00:04:14.380 --> 00:04:16.490 There are a couple ways you could think about that. 00:04:16.490 --> 00:04:19.420 One, you see 40's just a middle number so that makes sense. 00:04:19.420 --> 00:04:22.230 And the only time we can really say it's the middle number 00:04:22.230 --> 00:04:25.800 is when the numbers are distributed evenly around 40. 00:04:25.800 --> 00:04:27.620 If we had a number that was much smaller than 40 or 00:04:27.620 --> 00:04:29.270 something, you couldn't just necessarily pick 00:04:29.270 --> 00:04:29.740 the middle number. 00:04:29.740 --> 00:04:32.090 But in this case these are consecutive and makes sense. 00:04:32.090 --> 00:04:34.230 Another way we could have done this problem, if you were, say, 00:04:34.230 --> 00:04:37.840 taking the SAT and they were to ask you the sum of five 00:04:37.840 --> 00:04:41.500 numbers is 200, what's the average of the numbers? 00:04:41.500 --> 00:04:43.800 Well you say, well, all I have to do is divide that 200 00:04:43.800 --> 00:04:46.270 by 5 and I'll get 40. 00:04:46.270 --> 00:04:47.700 Let's do another problem and I'll make it a 00:04:47.700 --> 00:04:50.070 little bit harder. 00:04:50.070 --> 00:05:08.540 Let's say the sum of seven odd numbers, and let me make up a 00:05:08.540 --> 00:05:12.060 good -- I hope this one works, I'm going to try to do it in 00:05:12.060 --> 00:05:27.535 my head -- is 217, what is the largest number? 00:05:30.750 --> 00:05:35.640 I shouldn't say number -- seven odd integers. 00:05:35.640 --> 00:05:38.210 Actually it becomes a much harder problem if it was just 00:05:38.210 --> 00:05:40.840 seven odd -- well actually, the only thing that could be odd 00:05:40.840 --> 00:05:42.900 are integers anyway, so you could almost assume it. 00:05:42.900 --> 00:05:45.400 But the sum of seven odd integers is 217. 00:05:45.400 --> 00:05:50.160 What is the largest of the integers? 00:05:50.160 --> 00:05:53.200 As you can tell I'm doing this on the fly. 00:05:53.200 --> 00:05:55.750 Actually my wife just diagnosed me with, she thinks I 00:05:55.750 --> 00:05:57.030 have benign vertigo. 00:05:57.030 --> 00:05:59.490 I got very dizzy this morning when I went to work, so you 00:05:59.490 --> 00:06:00.850 have to forgive me for that as well. 00:06:00.850 --> 00:06:03.010 That's impairing me even more. 00:06:03.010 --> 00:06:05.080 So let's do this problem. 00:06:05.080 --> 00:06:08.385 Let's say that x is the largest. 00:06:12.080 --> 00:06:14.400 Then what would the number right below x be? 00:06:14.400 --> 00:06:16.550 Would it be x minus 1? 00:06:16.550 --> 00:06:19.990 Well, if x is an odd number, x minus 1 would 00:06:19.990 --> 00:06:21.750 be an even number. 00:06:21.750 --> 00:06:26.850 So in order to get the number right below it, we have to 00:06:26.850 --> 00:06:30.290 do x minus 2 to get another odd number. 00:06:30.290 --> 00:06:32.630 My apologies -- it should say the sum of seven 00:06:32.630 --> 00:06:34.170 consecutive odd. 00:06:34.170 --> 00:06:36.160 I don't know if you assumed that. 00:06:36.160 --> 00:06:39.130 I'm trying my best today to confuse you. 00:06:39.130 --> 00:06:43.840 The sum of seven consecutive odd integers is 217. 00:06:43.840 --> 00:06:46.530 What is the largest of the integers? 00:06:46.530 --> 00:06:49.420 So if x is the largest, then to next smallest one would be x 00:06:49.420 --> 00:06:53.030 minus 2, right, because it's consecutive odd numbers, 00:06:53.030 --> 00:06:54.250 not just consecutive. 00:06:54.250 --> 00:06:58.020 So consecutive odd numbers are like 1, 3, 5, 7 -- you're 00:06:58.020 --> 00:06:59.050 skipping the evens, right? 00:06:59.050 --> 00:07:01.470 So that's why you're going up or down by two, depending 00:07:01.470 --> 00:07:02.740 how you view it. 00:07:02.740 --> 00:07:05.345 So the next one down would be x minus 2, then we'll have x 00:07:05.345 --> 00:07:13.450 minus 4, x minus 6, x minus 8, x minus 10, x minus 12. 00:07:13.450 --> 00:07:14.060 I think that's it. 00:07:14.060 --> 00:07:16.700 One, two, three, four, five, six, seven, right. 00:07:16.700 --> 00:07:18.060 Those are seven numbers. 00:07:18.060 --> 00:07:19.300 They're separated by two. 00:07:19.300 --> 00:07:21.990 X is the largest of them, right? 00:07:21.990 --> 00:07:24.210 We can assume that they're odd because apparently 00:07:24.210 --> 00:07:26.310 the problem will work out so that they're odd. 00:07:26.310 --> 00:07:28.320 So what is the sum of these seven numbers? 00:07:28.320 --> 00:07:31.850 Well the seven x's just add up to 7x. 00:07:31.850 --> 00:07:38.630 And then let's see, 2 and 4 is 6, 6 and 6 is 12, 12 and 8 is 00:07:38.630 --> 00:07:43.330 20, 20 and 10 is 30, 30 and 12 is 32. 00:07:43.330 --> 00:07:51.840 So 7x minus 32 is equal to 217. 00:07:51.840 --> 00:07:53.520 We just solved for x. 00:07:53.520 --> 00:07:58.310 7x is equal to -- let's see, if we add 32 to both sides 00:07:58.310 --> 00:08:04.380 of this equation we get 249. 00:08:04.380 --> 00:08:10.070 Let's see, 7 goes into 249 -- is that right? 00:08:13.060 --> 00:08:13.750 Right. 00:08:13.750 --> 00:08:17.685 So 7 goes into 249 -- did I do this addition properly? 00:08:17.685 --> 00:08:19.070 I want to make sure. 00:08:19.070 --> 00:08:28.770 2 plus 4 is 6, 6 plus 6 is 12, 12 plus 8 is 20, 20 plus 10 00:08:28.770 --> 00:08:32.820 is 30, 30 plus 12 is 42. 00:08:32.820 --> 00:08:34.210 Oh, here you go. 00:08:34.210 --> 00:08:37.320 See, my mathematical spider sense could tell that something 00:08:37.320 --> 00:08:39.090 was fishy about this. 00:08:39.090 --> 00:08:41.470 So that's 7x minus 42. 00:08:41.470 --> 00:08:46.870 So if we add 42 to both sides it's 7x is equal to 259. 00:08:46.870 --> 00:08:49.900 See how brave I am, I do this thing in real time. 00:08:49.900 --> 00:08:50.680 259. 00:08:50.680 --> 00:08:55.530 So 7 goes into 259 -- let's see, 7 goes into 25 three 00:08:55.530 --> 00:09:01.900 times, 3 times 7 is 21, 49 -- it goes into it 37 times. 00:09:01.900 --> 00:09:06.530 So we get x is equal to 37 and we're done. 00:09:06.530 --> 00:09:09.630 So just to review because I think had a lot of errors in 00:09:09.630 --> 00:09:11.460 this problem when I presented it. 00:09:11.460 --> 00:09:13.830 The question was the sum of seven consecutive 00:09:13.830 --> 00:09:16.900 odd integers is 217. 00:09:16.900 --> 00:09:18.830 What is the largest of the integers? 00:09:18.830 --> 00:09:21.390 I said x is the largest, and then if x is the largest, the 00:09:21.390 --> 00:09:23.650 next smaller one will x minus 2. 00:09:23.650 --> 00:09:25.140 Because we're not saying just consecutive integers, 00:09:25.140 --> 00:09:28.010 we're saying consecutive odd integers, right? 00:09:28.010 --> 00:09:32.320 So if x is 37, which is what we solved for, then x minus 2 is 00:09:32.320 --> 00:09:38.490 35, this is 33, this is 31, this is 29, this is 00:09:38.490 --> 00:09:41.490 27, this is 25. 00:09:41.490 --> 00:09:46.050 And then we just added up all the x's and I'll add up all 00:09:46.050 --> 00:09:48.910 the constants and said, well they add up to 217. 00:09:48.910 --> 00:09:50.600 And then we just solved for x. 00:09:50.600 --> 00:09:53.460 I think you're now ready to try some of these problems. 00:09:53.460 --> 00:09:54.940 Have fun.

Basic addition

https://www.youtube.com/watch?v=AuX7nPBqDts

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https://www.youtube.com/api/timedtext?v=AuX7nPBqDts&ei=f2eUZb3_CtSchcIP5dKy6Ag&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=93FF1700B3C61D21BD05D8E580C790B34946FBF1.C864891DF8BF0F5B5DD6963CD8C90BC3805EB884&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:01.160 --> 00:00:04.730 Welcome to the presentation on basic addition. 00:00:04.730 --> 00:00:06.810 I know what you're thinking, Sal, addition doesn't 00:00:06.810 --> 00:00:08.260 seem so basic to me. 00:00:08.260 --> 00:00:10.940 Well, I apologize. 00:00:10.940 --> 00:00:13.170 Hopefully by the end of this presentation or in a couple 00:00:13.170 --> 00:00:14.840 of weeks it will seem basic. 00:00:14.840 --> 00:00:16.990 So let's get started with, I guess we could 00:00:16.990 --> 00:00:18.390 say, some problems. 00:00:18.390 --> 00:00:23.100 Well let's say I start with an old classic. 00:00:23.100 --> 00:00:25.950 1 plus 1. 00:00:25.950 --> 00:00:29.440 And I think you already know how to do this, but I'll kind 00:00:29.440 --> 00:00:31.960 of show you a way of doing this in case you don't have this 00:00:31.960 --> 00:00:35.050 memorized or you haven't already mastered this. 00:00:35.050 --> 00:00:40.930 You say, well, if I have 1, let's call that an avocado. 00:00:40.930 --> 00:00:45.030 If I have 1 avocado and then you were to give me 00:00:45.030 --> 00:00:48.870 another avocado, how many avocados do I now have? 00:00:48.870 --> 00:00:49.340 Well, let's see. 00:00:49.340 --> 00:00:52.300 I have 1, 2 avocados. 00:00:52.300 --> 00:00:54.740 So 1 plus 1 is equal to 2. 00:00:54.740 --> 00:00:55.910 Now, I know what you're thinking. 00:00:55.910 --> 00:00:58.390 That was too easy, so let me give you something a 00:00:58.390 --> 00:01:00.410 little bit more difficult. 00:01:00.410 --> 00:01:01.840 I like the avocados. 00:01:01.840 --> 00:01:04.130 I might stick with that theme. 00:01:04.130 --> 00:01:09.420 What is 3 plus 4? 00:01:09.420 --> 00:01:11.980 This is, I think, a more difficult problem. 00:01:11.980 --> 00:01:14.070 Well, let's stick with the avocados. 00:01:14.070 --> 00:01:17.710 And in case you don't know what an avocado is, it's actually 00:01:17.710 --> 00:01:18.890 a very delicious fruit. 00:01:18.890 --> 00:01:21.210 It's actually the fattiest of all the fruits. 00:01:21.210 --> 00:01:22.340 You probably didn't even think it was a fruit, 00:01:22.340 --> 00:01:25.470 even if you ate one. 00:01:25.470 --> 00:01:32.270 Let's say I have 3 avocados-- 1, 2, 3. 00:01:32.270 --> 00:01:35.120 And let's say you were to give me 4 more avocados. 00:01:35.120 --> 00:01:38.620 So let me put this 4 in yellow so you know that these are 00:01:38.620 --> 00:01:40.610 the ones you're giving me. 00:01:40.610 --> 00:01:45.780 1, 2, 3, 4. 00:01:45.780 --> 00:01:48.980 So how many total avocados do I have now? 00:01:48.980 --> 00:01:55.790 That's 1, 2, 3, 4, 5, 6, 7 avocados. 00:01:55.790 --> 00:01:59.000 So 3 plus 4 is equal to 7. 00:01:59.000 --> 00:02:00.290 And now I'm going to introduce you to another way of 00:02:00.290 --> 00:02:01.070 thinking about this. 00:02:01.070 --> 00:02:02.180 It's called the number line. 00:02:02.180 --> 00:02:05.490 And actually, I think this is how I do it in my head when I 00:02:05.490 --> 00:02:08.920 forget-- if I don't have it memorized. 00:02:08.920 --> 00:02:11.450 So number line, I just write all the numbers in order. 00:02:11.450 --> 00:02:13.850 And I go high enough just so all the numbers I'm 00:02:13.850 --> 00:02:15.470 using are kind of in it. 00:02:15.470 --> 00:02:18.000 So you know the first number is 0, which is nothing. 00:02:18.000 --> 00:02:20.020 Maybe you don't know, but now you know. 00:02:20.020 --> 00:02:36.250 And then you go to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 00:02:36.250 --> 00:02:37.600 Keeps going-- 11. 00:02:37.600 --> 00:02:39.360 So we're saying 3 plus 4. 00:02:39.360 --> 00:02:41.740 So let's start at 3. 00:02:41.740 --> 00:02:45.500 So I have 3 here and we're going to add 4 to that 3. 00:02:45.500 --> 00:02:48.430 So all we do is we go up the number line, or we go to the 00:02:48.430 --> 00:02:50.850 right on the number line, 4 more. 00:02:50.850 --> 00:02:59.130 So we go 1, 2, 3, 4. 00:02:59.130 --> 00:03:01.360 Notice all we did is we just increased it by one, by 00:03:01.360 --> 00:03:02.610 two, by three, by four. 00:03:02.610 --> 00:03:04.090 And then we ended up at 7. 00:03:04.090 --> 00:03:06.030 And that was our answer. 00:03:06.030 --> 00:03:07.180 We can do a couple of different ones. 00:03:09.690 --> 00:03:14.030 What if I asked you what 8 plus 1 is? 00:03:14.030 --> 00:03:14.960 Well, you might already know it. 00:03:14.960 --> 00:03:16.390 You know, plus 1 is just the next number. 00:03:16.390 --> 00:03:18.225 But if we look at the number line you start 00:03:18.225 --> 00:03:22.500 at 8 and you add 1. 00:03:22.500 --> 00:03:26.340 8 plus 1 is equal to 9. 00:03:26.340 --> 00:03:27.445 Let's do some harder problems. 00:03:30.370 --> 00:03:32.830 And just so you know, if you're a little daunted by this 00:03:32.830 --> 00:03:34.550 initially, you can always draw the circles. 00:03:34.550 --> 00:03:36.060 You can always do the number line. 00:03:36.060 --> 00:03:39.670 And eventually, over time, the more practice you do-- you'll 00:03:39.670 --> 00:03:41.330 hopefully memorize these and you'll do these problems 00:03:41.330 --> 00:03:42.440 in like half a second. 00:03:42.440 --> 00:03:43.240 I promise you. 00:03:43.240 --> 00:03:46.250 You just got to keep practicing. 00:03:46.250 --> 00:03:48.380 I want to draw the number line again, actually, I have a line 00:03:48.380 --> 00:03:50.740 tool, so I should give you all those ugly looking lines 00:03:50.740 --> 00:03:52.260 that I've been giving you. 00:03:52.260 --> 00:03:52.840 Look at that. 00:03:52.840 --> 00:03:54.660 That's amazing. 00:03:54.660 --> 00:03:56.810 Let me see. 00:03:56.810 --> 00:04:00.230 Look at that. 00:04:00.230 --> 00:04:01.490 That's a nice looking line. 00:04:01.490 --> 00:04:04.650 I'm going to feel bad to erase it later on. 00:04:04.650 --> 00:04:06.920 So let me draw a number line. 00:04:06.920 --> 00:04:28.590 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. 00:04:28.590 --> 00:04:30.060 So let's do a hard problem. 00:04:33.180 --> 00:04:35.290 I'm going to do it in different colors now. 00:04:35.290 --> 00:04:38.550 5 plus 6. 00:04:38.550 --> 00:04:40.690 So if you want, you could pause the video and try this. 00:04:40.690 --> 00:04:42.720 You might already know the answer. 00:04:42.720 --> 00:04:45.140 And the reason why I say this is a hard problem is because 00:04:45.140 --> 00:04:48.660 the answer has more numbers than figures, so you can't 00:04:48.660 --> 00:04:50.770 necessarily do it on your fingers. 00:04:50.770 --> 00:04:53.980 So let's get started with this problem. 00:04:53.980 --> 00:04:55.780 Actually, my phone is ringing, but I'm going to ignore the 00:04:55.780 --> 00:04:58.900 phone because you're more important. 00:04:58.900 --> 00:05:00.980 OK, let's start at the 5. 00:05:00.980 --> 00:05:05.400 So we start at the 5 and we're going to add 6 to it. 00:05:05.400 --> 00:05:13.980 So we go 1, 2, 3, 4, 5, 6. 00:05:13.980 --> 00:05:16.640 And we're at 11. 00:05:16.640 --> 00:05:19.670 So 5 plus 6 is equal to 11. 00:05:19.670 --> 00:05:22.475 Now I'm going to ask you a question, what is 6 plus 5? 00:05:26.690 --> 00:05:28.540 Well, we're now going to see that. 00:05:28.540 --> 00:05:30.890 Can you switch the two numbers and get the same answer? 00:05:30.890 --> 00:05:32.260 Well, let's try that. 00:05:32.260 --> 00:05:33.600 And I'm going to try it in a different color so we 00:05:33.600 --> 00:05:34.520 don't get all confused. 00:05:34.520 --> 00:05:36.930 So let's start at 6. 00:05:36.930 --> 00:05:39.880 Ignore the yellow for now and add 5 to it. 00:05:39.880 --> 00:05:45.040 1, 2, 3, 4, 5. 00:05:45.040 --> 00:05:47.040 We get to the same place. 00:05:47.040 --> 00:05:48.840 And I think you might want to try this on a bunch of 00:05:48.840 --> 00:05:50.600 problems and you'll see it always works out. 00:05:50.600 --> 00:05:53.900 That it doesn't matter what order-- 5 plus 6 is the 00:05:53.900 --> 00:05:55.910 same thing as 6 plus 5. 00:05:55.910 --> 00:05:56.670 And that makes sense. 00:05:56.670 --> 00:05:58.960 If I have 5 avocados and you give me 6, 00:05:58.960 --> 00:05:59.740 I'm going to have 11. 00:05:59.740 --> 00:06:01.845 If I have 6 avocados and you gave me 5, I'm going 00:06:01.845 --> 00:06:05.350 to have 11 either way. 00:06:05.350 --> 00:06:07.190 Since this number line is so nice, I want to do a few 00:06:07.190 --> 00:06:08.170 more problems using it. 00:06:08.170 --> 00:06:10.720 Although as I use it I'm sure I'll just continue to confuse 00:06:10.720 --> 00:06:12.460 you because I'll write so much on top of it. 00:06:12.460 --> 00:06:14.290 But let's see. 00:06:14.290 --> 00:06:17.250 I'll use white now. 00:06:17.250 --> 00:06:22.320 What is 8 plus 7? 00:06:22.320 --> 00:06:26.830 Well, if you can still read this, 8 is right here. 00:06:26.830 --> 00:06:28.820 We're going to add 7 to it. 00:06:28.820 --> 00:06:36.910 1, 2, 3, 4, 5, 6, 7. 00:06:36.910 --> 00:06:38.110 We go to 15. 00:06:38.110 --> 00:06:39.390 8 plus 7 is 15. 00:06:42.450 --> 00:06:45.550 So hopefully that gives you a sense of how to do 00:06:45.550 --> 00:06:48.200 these types of problems. 00:06:48.200 --> 00:06:50.570 I guess this and you're going to learn multiplication in a 00:06:50.570 --> 00:06:53.880 little bit, but these types of problems are-- when you're 00:06:53.880 --> 00:06:55.830 getting started off in mathematics, these kind of 00:06:55.830 --> 00:06:58.390 require the most practice and to some degree, you have to 00:06:58.390 --> 00:06:59.430 start memorizing them. 00:06:59.430 --> 00:07:03.150 But over time, when you look back, I want you to remember 00:07:03.150 --> 00:07:05.650 how you feel while you're watching this video right now. 00:07:05.650 --> 00:07:09.780 And then I want you to watch this video in like 3 years and 00:07:09.780 --> 00:07:11.930 remember how you felt when you're watching it now. 00:07:11.930 --> 00:07:13.340 And you're going to be, oh my God. 00:07:13.340 --> 00:07:16.080 This was so easy because you're going to learn it so fast. 00:07:16.080 --> 00:07:19.940 So anyway, I think you have an idea. 00:07:19.940 --> 00:07:22.500 If you don't know the answer to any of the additional problems 00:07:22.500 --> 00:07:25.735 that we give in the exercises you can press the hints and 00:07:25.735 --> 00:07:28.560 it'll draw circles and you can just count up the circles. 00:07:28.560 --> 00:07:30.020 Or if you want to do it on your own so you get 00:07:30.020 --> 00:07:32.260 the problem right, you could draw the circles. 00:07:32.260 --> 00:07:34.370 Or you could draw a number line like we did in 00:07:34.370 --> 00:07:36.710 this presentation. 00:07:36.710 --> 00:07:40.330 I think you might be ready to tackle the addition problems. 00:07:40.330 --> 00:07:41.870 Have fun.

Level 2 Addition

https://www.youtube.com/watch?v=27Kp7HJYj2c

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https://www.youtube.com/api/timedtext?v=27Kp7HJYj2c&ei=fmeUZb2uMfmyvdIPvIGVuAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249838&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A3C2F585239047A90EA261F0D7B73C4775B7ABDB.3B764CA697B9CAB0064E134DD45DEB80AF962A1C&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.040 --> 00:00:04.980 Welcome to the presentation on level two addition. 00:00:04.980 --> 00:00:07.590 Well I think we should get started with some problems, and 00:00:07.590 --> 00:00:10.100 hopefully as we work through them, you'll have an 00:00:10.100 --> 00:00:13.491 understanding of how to do these types of problems. 00:00:13.491 --> 00:00:18.650 Let me make sure the pen tool is correct. 00:00:18.650 --> 00:00:27.480 Let's say I had 11 plus 4. 00:00:27.480 --> 00:00:32.570 So first you'd say hey, Sal, 11 plus 4, I don't know how to 00:00:32.570 --> 00:00:34.250 add two-digit numbers yet. 00:00:34.250 --> 00:00:35.960 Well there's a couple of ways we can think about this. 00:00:35.960 --> 00:00:39.720 First I'll show you how all you have to know is how to add 00:00:39.720 --> 00:00:42.210 one-digit numbers and you can use something called carrying 00:00:42.210 --> 00:00:43.510 to solve the whole problem. 00:00:43.510 --> 00:00:45.610 Then we'll actually try to visually represent it to show 00:00:45.610 --> 00:00:47.100 you how you could actually do this type of problem 00:00:47.100 --> 00:00:48.640 in your head as well. 00:00:48.640 --> 00:00:51.860 So what you do with these types of problems is you first look 00:00:51.860 --> 00:00:56.040 at the right-most digit on the 11. 00:00:56.040 --> 00:00:57.510 We call this the ones place, right? 00:00:57.510 --> 00:01:00.410 Because this 1 is 1, and we call this the tens place. 00:01:00.410 --> 00:01:04.180 I know I'm going to confuse you a lot, but that's when 00:01:04.180 --> 00:01:06.570 everything looks easier later on. 00:01:06.570 --> 00:01:08.780 So you look at this ones place, you say there's a 1 there. 00:01:08.780 --> 00:01:11.820 You take that 1 and you add it to the number right below it. 00:01:11.820 --> 00:01:13.850 So 1 plus 4 is 5. 00:01:13.850 --> 00:01:16.670 You knew that, right? 00:01:16.670 --> 00:01:20.920 You know that 1 plus 4 is equal to 5. 00:01:20.920 --> 00:01:21.680 That's all I did here. 00:01:21.680 --> 00:01:25.730 I just said this 1 plus this 4 is equal to 5. 00:01:25.730 --> 00:01:27.300 Now I go to this 1. 00:01:27.300 --> 00:01:29.520 This 1 plus -- well there's nothing here other than a plus 00:01:29.520 --> 00:01:30.870 sign and that's not a number. 00:01:30.870 --> 00:01:33.030 So this 1 plus nothing is 1. 00:01:33.030 --> 00:01:35.560 So we put a 1 here. 00:01:35.560 --> 00:01:40.770 And we get 11 plus 4 is equal to 15. 00:01:40.770 --> 00:01:44.590 Just so you know that this system actually works, let's 00:01:44.590 --> 00:01:46.490 actually draw it out in a couple of different ways 00:01:46.490 --> 00:01:48.620 just to give you the intuition of 11 plus 4. 00:01:48.620 --> 00:01:55.260 So if I had 11 balls -- one, two, three, four, five, six, 00:01:55.260 --> 00:01:58.910 seven, eight, nine, ten, eleven. 00:01:58.910 --> 00:01:59.720 That's eleven, right? 00:02:15.760 --> 00:02:17.840 So that's 11 and we're going to add 4 to it. 00:02:17.840 --> 00:02:22.740 So one, two, three, four. 00:02:22.740 --> 00:02:26.430 So now all we have to do is count how many total circles 00:02:26.430 --> 00:02:28.060 or balls we have now. 00:02:28.060 --> 00:02:34.000 That's one, two, three, four, five, six, seven, eight, 00:02:34.000 --> 00:02:40.570 nine, ten, eleven, twelve, thirteen, fourteen, fifteen. 00:02:40.570 --> 00:02:42.130 I don't recommend that you do this every time you do 00:02:42.130 --> 00:02:44.000 a problem because it'll take you a long time. 00:02:44.000 --> 00:02:46.070 But hey, if you ever get confused, it's better to take a 00:02:46.070 --> 00:02:48.350 long time than to get it wrong. 00:02:48.350 --> 00:02:50.650 Let's think about another way of representing this, because 00:02:50.650 --> 00:02:53.300 I think different visual approaches appeal in different 00:02:53.300 --> 00:02:54.480 ways to different people. 00:02:54.480 --> 00:02:56.210 Let's draw a number line. 00:02:56.210 --> 00:02:58.010 I don't know if you've seen a number line before but 00:02:58.010 --> 00:03:00.930 you're going to see it now. 00:03:00.930 --> 00:03:03.690 And a number line, all I do is I draw out all 00:03:03.690 --> 00:03:04.795 the numbers in order. 00:03:04.795 --> 00:03:14.310 So 0, 1, 2, 3, 4, 5, 6 -- I'm doing them small so I know I 00:03:14.310 --> 00:03:35.910 can get to 15 -- 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18. 00:03:35.910 --> 00:03:36.450 And so on. 00:03:36.450 --> 00:03:39.610 And these arrows mean that the numbers keep going 00:03:39.610 --> 00:03:41.180 in both directions. 00:03:41.180 --> 00:03:42.923 I know this is a little early for you to learn this, but 00:03:42.923 --> 00:03:43.890 actually the numbers can actually keep going to the 00:03:43.890 --> 00:03:45.260 left below zero as well. 00:03:45.260 --> 00:03:46.810 I'll leave you to think about that. 00:03:46.810 --> 00:03:48.990 But anyway, so let's go back to this problem. 00:03:48.990 --> 00:03:52.180 So we have 11, so let me circle 11 -- let me see where 11 00:03:52.180 --> 00:03:52.980 is on the number line. 00:03:52.980 --> 00:03:55.210 11 is here, right? 00:03:55.210 --> 00:03:56.820 This is 11. 00:03:56.820 --> 00:03:58.520 And we're adding 4. 00:03:58.520 --> 00:04:03.020 So when you add, that means we're increase 11 by 4. 00:04:03.020 --> 00:04:05.120 So that when you increase we're going to go up the number line, 00:04:05.120 --> 00:04:07.010 right, or we're going to go the right on the number line 00:04:07.010 --> 00:04:08.410 because the numbers are getting bigger. 00:04:08.410 --> 00:04:15.300 So we'd go 1, 2, 3, 4 -- bam. 00:04:15.300 --> 00:04:16.910 We're at 15. 00:04:16.910 --> 00:04:19.930 Once again, this takes a long time, but if you ever get 00:04:19.930 --> 00:04:23.450 confused or you forget what 1 plus 4 is, although I don't 00:04:23.450 --> 00:04:26.620 think you should, then you could just do it this way. 00:04:26.620 --> 00:04:29.500 Let's do some maybe harder problems now. 00:04:32.740 --> 00:04:41.300 Let's do 28 plus 7. 00:04:43.840 --> 00:04:46.660 8 plus 7 -- I'll tell you, frankly, even to this 00:04:46.660 --> 00:04:50.560 day, I sometimes get confused with 8 plus 7. 00:04:50.560 --> 00:04:53.700 If you know the answer then you already know how to do this 00:04:53.700 --> 00:04:55.450 problem, you can just write whatever the answer 00:04:55.450 --> 00:04:55.770 is down here. 00:04:55.770 --> 00:04:58.570 But let draw it on the number line, because I think a little 00:04:58.570 --> 00:05:01.620 bit of more basic addition practice isn't unwarranted 00:05:01.620 --> 00:05:02.750 at this point. 00:05:02.750 --> 00:05:04.920 So we could do out with the number line again. 00:05:04.920 --> 00:05:05.690 8 plus 7. 00:05:09.030 --> 00:05:11.570 I'm not going to start at 0, I'll start at like 5, because 00:05:11.570 --> 00:05:13.440 if you keep going you'll get to 0 eventually. 00:05:13.440 --> 00:05:29.600 So let's see 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 00:05:29.600 --> 00:05:33.276 16, 17, 18 and so on. 00:05:33.276 --> 00:05:35.500 It'll keep going all the way to 100 and 1,000 and a 00:05:35.500 --> 00:05:37.480 million billion trillion. 00:05:37.480 --> 00:05:38.600 So what are we doing? 00:05:38.600 --> 00:05:40.450 We start at 8 because this is 8 plus 7. 00:05:40.450 --> 00:05:44.405 We want to figure out what 8 plus 7 is. 00:05:44.405 --> 00:05:46.780 So we start at 8. 00:05:46.780 --> 00:05:47.790 We're going to add 7 to it. 00:05:47.790 --> 00:05:49.540 Let me change colors. 00:05:49.540 --> 00:05:58.030 So you go 1, 2, 3, 4, 5, 6, 7. 00:05:58.030 --> 00:05:59.580 Oh, that's 15 shows up again. 00:05:59.580 --> 00:06:02.590 So 8 plus 7 is equal to 15. 00:06:02.590 --> 00:06:05.500 Over time as you do practice, I think you'll memorize these 00:06:05.500 --> 00:06:09.240 that 8 plus 7 is 15 or whatever, 6 plus 7 is 00:06:09.240 --> 00:06:10.710 13 or any of these. 00:06:10.710 --> 00:06:13.420 But in the interim it actually doesn't hurt to do this number 00:06:13.420 --> 00:06:16.370 line because you actually are visualizing what's happening. 00:06:16.370 --> 00:06:18.260 And you can do it with the circles as well. 00:06:18.260 --> 00:06:19.590 So we know 8 plus 7 is 15. 00:06:19.590 --> 00:06:22.280 So this is a new thing you're going to learn right now. 00:06:22.280 --> 00:06:24.260 You don't write the whole 15 down here. 00:06:24.260 --> 00:06:29.010 You write the 5 -- you write this five right here. 00:06:29.010 --> 00:06:33.300 And then that 1, you carry the 1. 00:06:33.300 --> 00:06:35.170 You put it up there. 00:06:35.170 --> 00:06:38.270 I think in a future presentation I'll explain why 00:06:38.270 --> 00:06:41.840 this works and maybe you might even kind of have an intuition 00:06:41.840 --> 00:06:45.862 because the 1 is in the tens place, and this is 00:06:45.862 --> 00:06:47.320 the tens place. 00:06:47.320 --> 00:06:48.810 I don't want to confuse you. 00:06:48.810 --> 00:06:50.870 So you have that 1 and now you add it to the 00:06:50.870 --> 00:06:54.670 2, and you get 35. 00:06:54.670 --> 00:06:59.060 Because 1 plus 2 is equal to 3, right? 00:06:59.060 --> 00:07:00.050 So you're done. 00:07:00.050 --> 00:07:02.090 35. 00:07:02.090 --> 00:07:03.760 You might ask, well, does that make sense that 00:07:03.760 --> 00:07:06.720 28 plus 7 is 35? 00:07:06.720 --> 00:07:09.000 And there's a couple of ways I'd like to think about this. 00:07:09.000 --> 00:07:12.800 Well, 8 plus 7 we know is 15. 00:07:12.800 --> 00:07:14.240 And I don't know how comfortable we are 00:07:14.240 --> 00:07:15.000 with higher numbers. 00:07:15.000 --> 00:07:18.500 But 18 plus 7 -- so look at this pattern. 00:07:18.500 --> 00:07:22.640 8 plus 7 is equal to 15. 00:07:22.640 --> 00:07:25.360 18 plus 7 -- you're probably saying, Sal, where are you 00:07:25.360 --> 00:07:27.720 getting the 18 from, but take my word for it. 00:07:27.720 --> 00:07:30.380 18 plus 7 is 25. 00:07:30.380 --> 00:07:36.930 28 plus 7 is equal to 35, which is the one that we just did. 00:07:36.930 --> 00:07:38.710 This is a check mark. 00:07:38.710 --> 00:07:41.890 If you kept going, you said 38 plus 7, that 00:07:41.890 --> 00:07:43.380 actually equals 45. 00:07:43.380 --> 00:07:45.860 So you might see a little pattern here, and then you can 00:07:45.860 --> 00:07:47.910 just sit and think about this for a little bit if you like. 00:07:47.910 --> 00:07:49.740 Maybe you pause the video. 00:07:49.740 --> 00:07:52.350 Another way you could think about this if you still don't 00:07:52.350 --> 00:07:57.870 believe me is you say OK, if I have 28, if I add 1 I get 29, 00:07:57.870 --> 00:08:01.970 if I add 2 I get 30, if I add 3 I get 31. 00:08:01.970 --> 00:08:05.410 If I add 4 I get 32. 00:08:05.410 --> 00:08:08.425 If I add 5 I get 33. 00:08:08.425 --> 00:08:11.550 If I add 6 I get 34. 00:08:11.550 --> 00:08:14.110 And if I add 7 I get 35 again. 00:08:14.110 --> 00:08:16.150 Right, all I did is I kept saying oh, if I had one more 00:08:16.150 --> 00:08:19.800 I'll get a larger little bit -- the number a little bit larger. 00:08:19.800 --> 00:08:21.230 Let's do some more problems, and I think we'll 00:08:21.230 --> 00:08:21.940 do a couple more. 00:08:21.940 --> 00:08:24.680 Let's do it a little faster because you might get what 00:08:24.680 --> 00:08:26.110 we're doing here now. 00:08:26.110 --> 00:08:27.580 Let's do a hard one. 00:08:27.580 --> 00:08:33.300 Let's do 99 plus 9. 00:08:33.300 --> 00:08:35.400 So what's 9 plus 9? 00:08:35.400 --> 00:08:37.670 So if you don't know what it is, you can work it out 00:08:37.670 --> 00:08:40.720 either using the number line or drawing the circles. 00:08:40.720 --> 00:08:41.990 That's a fair way to do it, although you should 00:08:41.990 --> 00:08:44.070 eventually kind of know it. 00:08:44.070 --> 00:08:46.110 9 plus 9 it turns out is 18. 00:08:50.770 --> 00:08:55.320 You put the 8 down here and you carry the 1. 00:08:55.320 --> 00:08:56.810 And now you just say 1 plus 9. 00:08:56.810 --> 00:08:58.150 Well you know what 1 plus 9 is. 00:08:58.150 --> 00:09:01.480 1 plus 9 is equal to 10. 00:09:01.480 --> 00:09:04.640 So there's nowhere to carry this 1, so you write the 00:09:04.640 --> 00:09:07.460 whole thing down here. 00:09:07.460 --> 00:09:12.280 So 99 plus 9 is equal to 108. 00:09:12.280 --> 00:09:15.820 Let's do one more problem. 00:09:15.820 --> 00:09:22.450 Let's say 56 plus 7. 00:09:22.450 --> 00:09:23.780 Well what's 6 plus 7. 00:09:23.780 --> 00:09:30.070 Well 6 plus 7 is 13, right? 00:09:30.070 --> 00:09:32.650 If you get confused, draw out everything again. 00:09:32.650 --> 00:09:33.980 And then you get 1 plus 5. 00:09:33.980 --> 00:09:35.570 1 plus 5 is 6. 00:09:35.570 --> 00:09:36.960 63. 00:09:36.960 --> 00:09:38.770 And you might want to give yourself a bunch of problems 00:09:38.770 --> 00:09:41.580 and I think you're also now, if you understand what we did, 00:09:41.580 --> 00:09:45.060 ready to try the level two addition problems. 00:09:45.060 --> 00:09:46.580 Have fun.

Quadratic inequalities

https://www.youtube.com/watch?v=ZNtzWpU80-0

vtt

https://www.youtube.com/api/timedtext?v=ZNtzWpU80-0&ei=gmeUZZCwIKT6vdIPvKuywAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249842&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7CA63F0ACDC957643127E9D005CCE3FCE1C46483.C153D96B312138FE8A7B57ED7313749FCF021A4C&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.960 --> 00:00:03.530 Welcome to the presentation on quadratic inequalities. 00:00:03.530 --> 00:00:06.450 I know that sounds very complicated, but hopefully 00:00:06.450 --> 00:00:09.140 you'll see it's actually not that difficult. 00:00:09.140 --> 00:00:10.830 Or at least, maybe the problems we're going to work on 00:00:10.830 --> 00:00:12.170 aren't that difficult. 00:00:12.170 --> 00:00:15.890 Well, let's get started with some problems and hopefully 00:00:15.890 --> 00:00:17.730 you'll see where this is kind of slightly different 00:00:17.730 --> 00:00:20.050 than solving regular quadratic equations. 00:00:20.050 --> 00:00:27.730 So let's say I had the inequality x squared plus 00:00:27.730 --> 00:00:33.480 3x is greater than 10. 00:00:33.480 --> 00:00:36.970 And remember, whenever you solve a quadratic or I guess 00:00:36.970 --> 00:00:38.640 you would call it a second degree equation-- I guess 00:00:38.640 --> 00:00:39.500 this is an inequality. 00:00:39.500 --> 00:00:41.460 I shouldn't use the word equation. 00:00:41.460 --> 00:00:43.680 It's tempting to sometimes do it the same way you'd do a 00:00:43.680 --> 00:00:45.470 linear equation, kind of getting all the x terms 00:00:45.470 --> 00:00:47.070 on one side and all the constants on the other. 00:00:47.070 --> 00:00:49.950 But it never works because you actually have an x term and 00:00:49.950 --> 00:00:51.570 then you have an x squared term. 00:00:51.570 --> 00:00:53.950 So you actually want to get it in kind of what I would call 00:00:53.950 --> 00:00:55.750 the-- I don't know if it's actually called this-- the 00:00:55.750 --> 00:00:58.730 standard form where you actually have all of the terms 00:00:58.730 --> 00:01:00.710 on one side and then a 0 on the other side. 00:01:00.710 --> 00:01:03.030 And then you can either factor it or use the 00:01:03.030 --> 00:01:03.910 quadratic equation. 00:01:03.910 --> 00:01:05.200 So let's do that. 00:01:05.200 --> 00:01:06.410 Well this is pretty easy. 00:01:06.410 --> 00:01:10.890 We just have to subtract 10 from both sides and we get x 00:01:10.890 --> 00:01:22.040 squared plus 3x minus 10 is greater than 0. 00:01:22.040 --> 00:01:23.920 Now let's see if we can factor it. 00:01:23.920 --> 00:01:27.050 Are there two numbers that when you multiply it become negative 00:01:27.050 --> 00:01:31.190 10 and when you add it become positive 3? 00:01:31.190 --> 00:01:31.840 Well, yeah. 00:01:31.840 --> 00:01:33.670 Positive 5 and negative 2. 00:01:33.670 --> 00:01:36.510 And once again, at this point I think you already know how to 00:01:36.510 --> 00:01:41.450 do factoring, so this should be hopefully, obvious to you. 00:01:41.450 --> 00:01:53.060 So it's x plus 5 times x minus 2 is greater than 0. 00:01:53.060 --> 00:01:55.840 Now this is the part where it's going to become a little bit 00:01:55.840 --> 00:02:00.810 more difficult than just your traditional factoring problem. 00:02:00.810 --> 00:02:02.650 We have two numbers, I guess you could view it. 00:02:02.650 --> 00:02:03.670 We have x plus 5. 00:02:03.670 --> 00:02:04.850 I view that as one number. 00:02:04.850 --> 00:02:06.490 Or I guess we have two expressions. 00:02:06.490 --> 00:02:10.300 We have x plus 5 and we have x minus 2. 00:02:10.300 --> 00:02:12.320 And when we're multiplying them we're getting 00:02:12.320 --> 00:02:15.020 something greater than 0. 00:02:15.020 --> 00:02:16.910 Now let's think about what happens when you 00:02:16.910 --> 00:02:19.460 multiply numbers. 00:02:19.460 --> 00:02:21.980 If they're both positive and you multiply them, then 00:02:21.980 --> 00:02:23.320 you get a positive number. 00:02:23.320 --> 00:02:26.710 And if they're both negative and you multiply them, then you 00:02:26.710 --> 00:02:28.610 also get a positive number. 00:02:28.610 --> 00:02:32.590 So we know that either both of these expressions are the same 00:02:32.590 --> 00:02:35.870 sign, that they're both greater than 0, they're both positive. 00:02:35.870 --> 00:02:38.650 Or we know that they're both negative. 00:02:38.650 --> 00:02:40.240 And I know this might be a little confusing, but just 00:02:40.240 --> 00:02:43.840 think of it as-- if I told you that-- I'll do something 00:02:43.840 --> 00:02:44.530 slightly separate out here. 00:02:44.530 --> 00:02:50.330 If I told you that a times b is greater than 0 we know that 00:02:50.330 --> 00:02:57.470 either a is greater than 0 and b is greater than 0. 00:02:57.470 --> 00:02:59.420 Which just means that they're both positive. 00:02:59.420 --> 00:03:07.230 Or a is less than 0 and b is less than 0. 00:03:07.230 --> 00:03:08.700 Which means that they're both negative. 00:03:08.700 --> 00:03:11.200 All we know is that they both have to be the same sign in 00:03:11.200 --> 00:03:14.400 order for their product to be greater than 0. 00:03:14.400 --> 00:03:16.270 Now we just do the same thing here. 00:03:16.270 --> 00:03:21.140 So we know that either both of these are positive, so x plus 00:03:21.140 --> 00:03:30.890 5 is greater than 0 and x minus 2 is greater than 0. 00:03:30.890 --> 00:03:37.240 Or-- now this is a little confusing, but if you work 00:03:37.240 --> 00:03:39.190 through these problems it actually makes a lot of sense. 00:03:39.190 --> 00:03:41.750 Or they're both negative. 00:03:41.750 --> 00:03:50.170 Or x plus 5 is less than 0 and x minus 2 is less than 0. 00:03:50.170 --> 00:03:53.350 I know that's confusing, but just think of it in terms of we 00:03:53.350 --> 00:03:55.350 have two expressions: they're either both positive or 00:03:55.350 --> 00:03:56.700 they're either both negative. 00:03:56.700 --> 00:03:58.110 Because when you multiple them you get something 00:03:58.110 --> 00:03:59.450 larger than 0. 00:03:59.450 --> 00:04:00.440 Well, let's solve this side. 00:04:00.440 --> 00:04:04.880 So this says that x is greater than negative 5 00:04:04.880 --> 00:04:11.640 and x is greater than 2. 00:04:11.640 --> 00:04:13.560 We just 2 both sides of this equation. 00:04:16.100 --> 00:04:22.540 Or, and if we solve this side-- x is less than negative 00:04:22.540 --> 00:04:27.820 5 and x is less than 2. 00:04:27.820 --> 00:04:30.520 I just solved both of these inequalities right here. 00:04:30.520 --> 00:04:33.530 Now we can actually simplify this because here we say that 00:04:33.530 --> 00:04:37.860 x is greater than negative 5 and x is greater than 2. 00:04:37.860 --> 00:04:42.600 So in order for x ti be greater than negative 5 and for x to be 00:04:42.600 --> 00:04:45.450 greater than 2, this just simplifies as saying, well, 00:04:45.450 --> 00:04:47.000 x is just greater than 2. 00:04:47.000 --> 00:04:48.870 Because if x is greater than 2, it's definitely 00:04:48.870 --> 00:04:50.230 greater than negative 5. 00:04:50.230 --> 00:04:52.710 So it just simplifies to this. 00:04:52.710 --> 00:04:57.370 And we'd say or-- and here we said x is less than negative 00:04:57.370 --> 00:05:00.530 5 or x is less than 2. 00:05:00.530 --> 00:05:03.330 Well, we know if x is less than negative 5, then x is 00:05:03.330 --> 00:05:04.920 definitely less than 2. 00:05:04.920 --> 00:05:10.850 So we could just simplify it to or x is less than negative 5. 00:05:10.850 --> 00:05:13.710 So the solutions to this problem is x could be greater 00:05:13.710 --> 00:05:17.760 than 2 or x could be less than negative 5. 00:05:17.760 --> 00:05:19.500 And so let's just think about how that looks 00:05:19.500 --> 00:05:20.510 on the number line. 00:05:24.454 --> 00:05:28.080 So if 2 is here, x could be greater than 2. 00:05:28.080 --> 00:05:29.190 So it's all of these numbers. 00:05:32.470 --> 00:05:34.585 And if this is negative 5-- I shouldn't have done it 00:05:34.585 --> 00:05:37.250 so close to the bottom. 00:05:37.250 --> 00:05:38.610 x is less the negative 5. 00:05:38.610 --> 00:05:42.250 So these are the numbers that satisfy this equation. 00:05:42.250 --> 00:05:44.460 And I'll leave it up to you to try out to see that 00:05:44.460 --> 00:05:45.920 they actually work. 00:05:45.920 --> 00:05:48.370 Let's try another one and hopefully, I can 00:05:48.370 --> 00:05:49.520 confuse you even more. 00:05:52.830 --> 00:06:07.070 Let's say I have minus x times 2x minus 14 is greater 00:06:07.070 --> 00:06:09.410 than or equal to 24. 00:06:09.410 --> 00:06:11.580 Well, the first thing we want to do is just manipulate this 00:06:11.580 --> 00:06:13.430 so it looks in the standard form. 00:06:13.430 --> 00:06:21.980 So we get negative 2x squared plus 14x-- I'm just 00:06:21.980 --> 00:06:27.050 distributing the minus x-- is greater than or equal to 24. 00:06:27.050 --> 00:06:29.895 I don't like any coefficient it front of my x squared term, 00:06:29.895 --> 00:06:32.910 so let's divide both sides of this equation by negative 2. 00:06:32.910 --> 00:06:36.220 So we get x squared-- we divided by negative 00:06:36.220 --> 00:06:39.440 2-- minus 7x. 00:06:39.440 --> 00:06:42.630 And remember, when you divide by a negative number you switch 00:06:42.630 --> 00:06:45.130 the sign on the inequality, or you switch the direction 00:06:45.130 --> 00:06:46.170 of the inequality. 00:06:46.170 --> 00:06:48.800 So we're dividing by negative 2, so we switched it. 00:06:48.800 --> 00:06:50.245 We went from greater than or equal to, to less 00:06:50.245 --> 00:06:51.120 than or equal to. 00:06:51.120 --> 00:06:55.780 And then 24 divided by negative 2 is minus 12. 00:06:55.780 --> 00:06:58.250 And now we can just bring this minus 12 onto the left-hand 00:06:58.250 --> 00:06:58.850 side of the equation. 00:06:58.850 --> 00:06:59.920 Add 12 to both sides. 00:06:59.920 --> 00:07:09.560 We get x squared minus 7x plus 12 is less than or equal to 0. 00:07:09.560 --> 00:07:12.670 And then we can just factor that and we get, what is that? 00:07:12.670 --> 00:07:21.680 It's x minus 3 times x minus 4 is less than or equal to 0. 00:07:21.680 --> 00:07:24.230 So now we know that when we multiply these two terms 00:07:24.230 --> 00:07:26.150 we get a negative number. 00:07:26.150 --> 00:07:29.350 So that means that these expressions have to be 00:07:29.350 --> 00:07:31.780 of different signs. 00:07:31.780 --> 00:07:32.370 Does that make sense? 00:07:32.370 --> 00:07:34.030 If I tell you I have two number and I multiply them, 00:07:34.030 --> 00:07:34.830 I get a negative number. 00:07:34.830 --> 00:07:37.030 You know that they have to be of different signs. 00:07:37.030 --> 00:07:45.040 So we know that either x minus 3 is less than or equal to 0 00:07:45.040 --> 00:07:50.380 and x minus 4 is greater than or equal to 0. 00:07:50.380 --> 00:07:51.750 So that's one case. 00:07:51.750 --> 00:07:57.780 And the other case is x minus 3 is greater than or equal to 0, 00:07:57.780 --> 00:08:00.040 which means x minus 3 is positive. 00:08:00.040 --> 00:08:04.690 And x minus 4 is less than or equal to 0-- 00:08:04.690 --> 00:08:05.740 oh, I went to the edge. 00:08:05.740 --> 00:08:07.900 So let's solve this and hopefully it'll simplify more. 00:08:07.900 --> 00:08:12.040 So this just says that x is less than or equal to 3. 00:08:12.040 --> 00:08:18.340 And this says x is greater than or equal to 4. 00:08:18.340 --> 00:08:20.915 So both of these things have to be true. x has to be less than 00:08:20.915 --> 00:08:26.030 or equal to 3 and x has to be greater than or equal to 4. 00:08:26.030 --> 00:08:27.350 Well, let me ask you a question. 00:08:27.350 --> 00:08:32.390 Can something be both less than or equal to 3 and greater 00:08:32.390 --> 00:08:33.490 than or equal to 4? 00:08:33.490 --> 00:08:34.110 Well, no. 00:08:34.110 --> 00:08:37.520 So we know that this situation can't happen. 00:08:37.520 --> 00:08:40.150 There's no numbers that's less than or equal to 3 and 00:08:40.150 --> 00:08:41.560 greater than or equal to 4. 00:08:41.560 --> 00:08:42.700 So let's look at this situation. 00:08:42.700 --> 00:08:48.580 This says x is greater than or equal to 3 and x is 00:08:48.580 --> 00:08:50.770 less than or equal to 4. 00:08:50.770 --> 00:08:51.460 Can this happen? 00:08:51.460 --> 00:08:51.870 Sure. 00:08:51.870 --> 00:08:54.930 That just means that x is some number between 3 and 4. 00:08:54.930 --> 00:08:59.380 If we were to draw this on the number line, we would get-- 00:08:59.380 --> 00:09:03.690 if this is 3, this is 4. 00:09:03.690 --> 00:09:06.750 And it's greater than or equal to so we fill it in. 00:09:06.750 --> 00:09:09.250 And less than or equal to so we'd fill it in. 00:09:09.250 --> 00:09:11.580 And it would be any number between 3 and 4 would 00:09:11.580 --> 00:09:12.485 satisfy this equation. 00:09:12.485 --> 00:09:14.730 And I'll leave it up to you to try it out. 00:09:14.730 --> 00:09:17.240 I know this is confusing at first, and this is actually 00:09:17.240 --> 00:09:19.160 something that they normally don't teach really well, I 00:09:19.160 --> 00:09:21.420 think, in most high schools until 10th or 11th grade. 00:09:21.420 --> 00:09:25.510 But just think about you're multiplying two expressions. 00:09:25.510 --> 00:09:27.630 If the answer is negative then they must be 00:09:27.630 --> 00:09:28.390 of different signs. 00:09:28.390 --> 00:09:31.360 If the answer is positive they must be the same sign. 00:09:31.360 --> 00:09:33.020 And then you just work through the logic. 00:09:33.020 --> 00:09:35.700 And you say, well, no number can be less than 3 and greater 00:09:35.700 --> 00:09:37.900 than 4, so this doesn't apply. 00:09:37.900 --> 00:09:39.490 And then you do this side and you're like, oh, this 00:09:39.490 --> 00:09:40.410 situation does work. 00:09:40.410 --> 00:09:42.580 It's any number between 3 and 4. 00:09:42.580 --> 00:09:44.090 Hopefully that gives you a sense of how to do 00:09:44.090 --> 00:09:45.110 these type of problems. 00:09:45.110 --> 00:09:47.000 I'll let you do the exercises now. 00:09:47.000 --> 00:09:48.540 Have fun.

Algebra: Solving Inequalities

https://www.youtube.com/watch?v=VgDe_D8ojxw

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https://www.youtube.com/api/timedtext?v=VgDe_D8ojxw&ei=f2eUZejwIP2qmLAP4fiLyAo&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=589755C6A9CE48ADBB413DD95E44BD6318B9DFD0.492511AF11AF817215D13B8D0C94B8BD23B9BB1F&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.100 --> 00:00:03.775 Welcome to the presentation on solving inequalities, 00:00:03.775 --> 00:00:06.890 or I guess you call them algebra inequalities. 00:00:06.890 --> 00:00:09.120 So let's get started. 00:00:09.120 --> 00:00:14.500 If I were to tell you that, well, let's just say x is 00:00:14.500 --> 00:00:17.840 greater than 5, right? 00:00:17.840 --> 00:00:22.700 So x could be 5.01, it could be 5.5, it could be a million. 00:00:22.700 --> 00:00:26.680 It just can't be 4, or 3, or 0, or negative 8, and actually, 00:00:26.680 --> 00:00:28.500 just for convenience, let's actually draw that 00:00:28.500 --> 00:00:30.970 on the number line. 00:00:30.970 --> 00:00:33.260 That's the number line. 00:00:33.260 --> 00:00:38.270 And if this is 5, x can't be equal to 5, so we draw a big 00:00:38.270 --> 00:00:41.400 circle here, and then we would color in all the values 00:00:41.400 --> 00:00:42.060 that x could be. 00:00:42.060 --> 00:00:46.510 So x could be just the small-- it could be 5.0000001. 00:00:46.510 --> 00:00:49.110 It just has to be a little bit bigger than 5, and any of those 00:00:49.110 --> 00:00:51.310 would satisfy it, right? 00:00:51.310 --> 00:00:53.660 So let's just write some numbers that satisfy. 00:00:53.660 --> 00:00:56.120 6 would satisfy it, 10 would satisfy it, 100 00:00:56.120 --> 00:00:58.060 would satisfy it. 00:00:58.060 --> 00:01:01.640 Now, if I were to multiply, or I guess divide, both sides of 00:01:01.640 --> 00:01:05.700 this, I guess we could say, equation or this inequality 00:01:05.700 --> 00:01:10.000 by negative 1, I want to understand what happens. 00:01:10.000 --> 00:01:18.360 So what's the relation between negative x and negative 5? 00:01:18.360 --> 00:01:21.500 When I say what's the relation, is it greater than or is 00:01:21.500 --> 00:01:24.410 it less than negative 5? 00:01:24.410 --> 00:01:27.530 Well, 6 is a value that works for x. 00:01:27.530 --> 00:01:33.170 So negative 6, is that greater than or less than negative 5? 00:01:33.170 --> 00:01:36.810 Well, negative 6 is less than negative 5, right? 00:01:36.810 --> 00:01:40.665 So let me draw the number line here. 00:01:40.665 --> 00:01:44.622 If we have negative 5 here, and let's just draw a circle around 00:01:44.622 --> 00:01:47.450 it because we know it's not going to be equal to negative 5 00:01:47.450 --> 00:01:49.770 because we're deciding between greater than or less than. 00:01:49.770 --> 00:01:51.500 So we're saying 6 works for x. 00:01:51.500 --> 00:01:55.580 So negative 6 is here, right? 00:01:55.580 --> 00:01:58.880 So negative 6 is less than negative 5, so is negative 10, 00:01:58.880 --> 00:02:02.330 so is negative 100, so is negative a million, right? 00:02:02.330 --> 00:02:07.920 So it turns out that negative x is less than negative 5. 00:02:07.920 --> 00:02:11.540 And this is really all you have to remember when you are 00:02:11.540 --> 00:02:13.940 working with inequalities in algebra. 00:02:13.940 --> 00:02:17.260 Inequalities, you can treat them just the way-- a greater 00:02:17.260 --> 00:02:19.050 than or less than sign, you could treat them exactly the 00:02:19.050 --> 00:02:20.930 way you would treat an equal sign. 00:02:20.930 --> 00:02:26.320 The only difference is, if you multiply or divide both sides 00:02:26.320 --> 00:02:30.290 of the equation by a negative number, you swap it. 00:02:30.290 --> 00:02:31.850 That's all you have to remember. 00:02:31.850 --> 00:02:33.290 Let's do some problems, and hopefully, that'll 00:02:33.290 --> 00:02:34.370 hit the point home. 00:02:34.370 --> 00:02:37.230 And if you ever forget, you just have to try-- you just 00:02:37.230 --> 00:02:39.770 remember this: if x is greater than 5, well, then negative 00:02:39.770 --> 00:02:40.840 x is less than negative 5. 00:02:40.840 --> 00:02:42.480 And keep trying out numbers. 00:02:42.480 --> 00:02:45.090 That's what's going to give you the best intuition. 00:02:45.090 --> 00:02:47.650 Let's do some problems. 00:02:47.650 --> 00:02:55.120 So if I said that 3x plus 2 is, let's say, less than or equal 00:02:55.120 --> 00:02:57.820 to 1-- well, this is a pretty easy equation to solve. 00:02:57.820 --> 00:03:01.540 We just say 3x-- let's subtract 2 from both sides, and when you 00:03:01.540 --> 00:03:04.490 add or subtract, you don't do anything to the inequality. 00:03:04.490 --> 00:03:09.180 So if you subtract 2 from both sides, you get 3x is less than 00:03:09.180 --> 00:03:13.150 or equal to negative 1, right? 00:03:13.150 --> 00:03:16.730 And then, now we're going to divide both sides by 3. 00:03:16.730 --> 00:03:21.690 We get x is less than or equal to negative 1/3, right? 00:03:21.690 --> 00:03:23.020 And notice, we didn't change anything because 00:03:23.020 --> 00:03:28.470 we divided both sides by a positive 3, right? 00:03:28.470 --> 00:03:30.270 We could have actually done this equation in a 00:03:30.270 --> 00:03:31.250 slightly different way. 00:03:31.250 --> 00:03:34.240 What if we subtracted 1 from both sides? 00:03:34.240 --> 00:03:36.270 So this is another way of solving it. 00:03:36.270 --> 00:03:41.780 What if we said 3x plus 1 is less than or equal to 0, right? 00:03:41.780 --> 00:03:44.820 I just subtracted 1 from both sides, and now let's subtract 00:03:44.820 --> 00:03:45.920 3x from both sides. 00:03:45.920 --> 00:03:51.010 And we get 1 is less than or equal to minus 3x, right? 00:03:51.010 --> 00:03:54.180 I subtracted 3x from here, so I have to subtract 3x from here. 00:03:54.180 --> 00:03:56.480 Now, I would have to divide both sides by a 00:03:56.480 --> 00:03:58.360 negative number, right? 00:03:58.360 --> 00:04:01.780 Because I'm going to divide both sides by negative 3. 00:04:01.780 --> 00:04:06.170 So I get negative 1/3 on this side, and based on what we had 00:04:06.170 --> 00:04:08.060 just learned, since we're dividing by a negative number, 00:04:08.060 --> 00:04:10.350 we want to swap the inequality, right? 00:04:10.350 --> 00:04:11.580 It was less than or equal, now it's going to be 00:04:11.580 --> 00:04:14.840 greater than or equal to x. 00:04:14.840 --> 00:04:16.480 Now, did we get the same answer when we did it 00:04:16.480 --> 00:04:18.520 both-- two different ways? 00:04:18.520 --> 00:04:22.310 Here, we got x is less than or equal to negative 1/3, and 00:04:22.310 --> 00:04:25.440 here we got negative 1/3 is greater than or equal to x. 00:04:25.440 --> 00:04:27.110 Well, that's the same answer, right? x is less than or 00:04:27.110 --> 00:04:28.760 equal to negative 1/3. 00:04:28.760 --> 00:04:30.530 And that's-- I always find that to be the cool 00:04:30.530 --> 00:04:31.210 thing about algebra. 00:04:31.210 --> 00:04:33.500 You can tackle the problem in a bunch of different ways, and 00:04:33.500 --> 00:04:35.170 you should always get to the right answer as long as 00:04:35.170 --> 00:04:37.940 you, I guess, do it right. 00:04:37.940 --> 00:04:40.840 Let's do a couple more problems. 00:04:40.840 --> 00:04:42.680 Oh, let's erase this thing. 00:04:42.680 --> 00:04:43.710 There you go. 00:04:43.710 --> 00:04:45.450 I'll do a slightly harder one. 00:04:45.450 --> 00:04:56.760 Let's say negative 8x plus 7 is greater than 5x plus 2. 00:04:56.760 --> 00:05:00.030 Let's subtract 5x from both sides. 00:05:00.030 --> 00:05:05.540 Negative 13x plus 7 is greater than 2. 00:05:05.540 --> 00:05:07.300 Now, we could subtract 7 from both sides. 00:05:07.300 --> 00:05:12.780 Negative 13x is greater than minus 5. 00:05:12.780 --> 00:05:14.180 Now, we're going to divide both sides of this 00:05:14.180 --> 00:05:17.050 equation by negative 13. 00:05:17.050 --> 00:05:18.720 Well, very easy. 00:05:18.720 --> 00:05:23.120 It's just x, and on this side, negative 5 divided by 00:05:23.120 --> 00:05:24.990 negative 13 is 5/13, right? 00:05:24.990 --> 00:05:26.200 The negatives cancel out. 00:05:26.200 --> 00:05:31.382 And since we divided by a negative, we switch the sign. 00:05:31.382 --> 00:05:33.750 x is less than 5/13. 00:05:33.750 --> 00:05:35.620 And once again, just like the beginning, if you don't believe 00:05:35.620 --> 00:05:36.930 me, try out some numbers. 00:05:36.930 --> 00:05:39.460 And I remember when I first learned this, I didn't believe 00:05:39.460 --> 00:05:41.780 the teacher, so I did try out numbers, and that's how I got 00:05:41.780 --> 00:05:43.690 convinced that it actually works. 00:05:43.690 --> 00:05:47.020 When you multiply or divide both sides of this equation 00:05:47.020 --> 00:05:50.600 by a negative sign, you swap the inequality. 00:05:50.600 --> 00:05:53.510 And remember, that's only when you multiply or divide, not 00:05:53.510 --> 00:05:55.700 when you add or subtract. 00:05:55.700 --> 00:05:58.630 I think that should give you a good idea of how 00:05:58.630 --> 00:05:59.350 to do these problems. 00:05:59.350 --> 00:06:01.160 There's really not much new here. 00:06:01.160 --> 00:06:05.360 You do an inequality-- or I guess you could call this an 00:06:05.360 --> 00:06:08.450 inequality equation-- you do it exactly the same way you do a 00:06:08.450 --> 00:06:12.090 normal linear equation, the only difference being is if you 00:06:12.090 --> 00:06:15.270 multiply or you divide both sides of the equation by a 00:06:15.270 --> 00:06:19.250 negative number, then you swap the inequality. 00:06:19.250 --> 00:06:22.810 I think you're ready now to try some practice problems. 00:06:22.810 --> 00:06:24.300 Have fun.

Why borrowing works

https://www.youtube.com/watch?v=fWan_T0enj4

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https://www.youtube.com/api/timedtext?v=fWan_T0enj4&ei=f2eUZd2UHcbUhcIPl72LqAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4ADD11D12495639E55D5C733F86847EB852794D9.D8D5911562995FF8C64527422CFDDC6DC7BCF987&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.850 --> 00:00:05.080 Welcome to the presentation on why, not how, borrowing works. 00:00:05.080 --> 00:00:06.750 And I think this is very important because a lot of 00:00:06.750 --> 00:00:11.500 people who even know math fairly well or have an advanced 00:00:11.500 --> 00:00:15.630 degree still aren't completely sure on why borrowing works. 00:00:15.630 --> 00:00:17.710 That's the focus of this presentation. 00:00:17.710 --> 00:00:20.500 Let's say I have the subtraction problem 00:00:20.500 --> 00:00:23.450 1,000-- that's a 0. 00:00:23.450 --> 00:00:31.840 1,005 minus 616. 00:00:31.840 --> 00:00:34.050 What I'm going to do is I'm going to write the same problem 00:00:34.050 --> 00:00:35.300 in a slightly different way. 00:00:35.300 --> 00:00:37.640 We could call this the expanded form. 00:00:37.640 --> 00:00:40.080 1,005-- what I'm going to do is I'm going to separate 00:00:40.080 --> 00:00:42.330 the digits out into their respective places. 00:00:42.330 --> 00:00:49.720 So that is equal to 1,000 plus let's say zero 100's 00:00:49.720 --> 00:00:53.550 plus zero 10's plus 5. 00:00:53.550 --> 00:00:56.830 1,005 is just 1,000 plus 0 plus 0 plus 5. 00:00:56.830 --> 00:01:00.340 And then that's minus 616. 00:01:00.340 --> 00:01:08.910 So that's minus 600 minus 10 minus 6. 00:01:08.910 --> 00:01:13.440 616 could be rewritten as 600 plus 10 plus 6. 00:01:13.440 --> 00:01:15.150 And I put a minus there because we're subtracting 00:01:15.150 --> 00:01:16.130 the whole thing. 00:01:16.130 --> 00:01:18.940 So let's do this problem. 00:01:18.940 --> 00:01:23.220 Well, if you're familiar with how you borrow is, this 5 is 00:01:23.220 --> 00:01:25.940 less than this 6, so we have to somehow make this 5 a bigger 00:01:25.940 --> 00:01:28.220 number so that we could subtract the 6 from it. 00:01:28.220 --> 00:01:31.060 Well, we know from traditional borrowing that we have to 00:01:31.060 --> 00:01:33.890 borrow 1 from someplace and make this it into a 15. 00:01:33.890 --> 00:01:36.240 But what I want to see actually, is understand where 00:01:36.240 --> 00:01:38.530 that 1 or actually where that 10 comes from. 00:01:38.530 --> 00:01:41.090 Because if you're turning this 5 into a 15 you actually 00:01:41.090 --> 00:01:43.090 have to add 10 to it. 00:01:43.090 --> 00:01:45.550 Well, if we look at this top number, the only place that 00:01:45.550 --> 00:01:49.970 a 10 could come from is here, is from this 1,000. 00:01:49.970 --> 00:01:52.950 But what we're going to do since this is the 1,000's 00:01:52.950 --> 00:01:57.040 place, instead of borrowing 10 from here, which would make it 00:01:57.040 --> 00:01:59.480 kind of a very messy problem, I'm going to borrow 00:01:59.480 --> 00:02:02.430 1,000 from here. 00:02:02.430 --> 00:02:03.960 I'm going to get rid of this 1,000. 00:02:03.960 --> 00:02:08.110 And I have a 1,000 that I took from this 1,000. 00:02:08.110 --> 00:02:12.740 I have 1,000 now that I can distribute into 00:02:12.740 --> 00:02:14.790 these 3 buckets. 00:02:14.790 --> 00:02:17.360 Into the 100's, 10's and 1's buckets. 00:02:17.360 --> 00:02:21.270 Well, we need 10 here, so let's put 10 here. 00:02:21.270 --> 00:02:24.320 So it's 10 plus 5 is equal to 15. 00:02:24.320 --> 00:02:25.040 We got our 15. 00:02:27.820 --> 00:02:32.030 If we took 10 from the 1,000 then we have 990 left. 00:02:32.030 --> 00:02:37.960 So we could put 900 here and 90 here. 00:02:37.960 --> 00:02:41.250 Notice, we just said-- so we had 1,000 and we just rewrote 00:02:41.250 --> 00:02:44.040 it as 900 plus 90 plus 10. 00:02:44.040 --> 00:02:45.670 And we added this 10 to this 5. 00:02:45.670 --> 00:02:47.670 And now we could do this subtraction just how we 00:02:47.670 --> 00:02:49.110 would do a normal problem. 00:02:49.110 --> 00:02:52.710 15 minus 6 is 9. 00:02:52.710 --> 00:02:56.300 90 minus 10 is 80. 00:02:56.300 --> 00:03:00.730 900 minus 600 is 300. 00:03:00.730 --> 00:03:06.550 So 300 plus 80 plus 9 is 389. 00:03:06.550 --> 00:03:08.940 And let's see how we would have done it traditionally and make 00:03:08.940 --> 00:03:12.790 sure that it would have kind of translated into the same way. 00:03:12.790 --> 00:03:15.210 Well, the way I teach it and I don't know if this is actually 00:03:15.210 --> 00:03:20.440 the traditional way of teaching borrowing, is I say, OK, I need 00:03:20.440 --> 00:03:23.480 to turn this 5 into a 15. 00:03:23.480 --> 00:03:25.220 So I have to borrow a 1 from someplace. 00:03:25.220 --> 00:03:26.840 Well, we know from this side of the problem that we actually 00:03:26.840 --> 00:03:28.630 borrowed a 10 because that's why it turned to 15. 00:03:28.630 --> 00:03:30.540 If we're going to borrow 1, I'd say, well, can I 00:03:30.540 --> 00:03:31.660 borrow the 1 from the 0? 00:03:31.660 --> 00:03:32.100 No. 00:03:32.100 --> 00:03:33.730 Can I borrow the 1 from this 0? 00:03:33.730 --> 00:03:34.440 No. 00:03:34.440 --> 00:03:36.630 I could borrow it from here, but I'm borrowing 00:03:36.630 --> 00:03:38.980 it from 100, right? 00:03:38.980 --> 00:03:43.110 So 100 minus 1 is 99. 00:03:43.110 --> 00:03:44.500 So that's the how I do it. 00:03:44.500 --> 00:03:47.560 And I say 15 minus 6 is 9. 00:03:47.560 --> 00:03:49.370 9 minus 1 is 8. 00:03:49.370 --> 00:03:51.570 And 9 minus 6 is 300. 00:03:51.570 --> 00:03:55.600 So this way that I just did it is clearly faster and, I guess 00:03:55.600 --> 00:03:57.780 you could say it's easier, but a lot of people might say, well 00:03:57.780 --> 00:03:59.200 Sal, that looks like a little bit of magic. 00:03:59.200 --> 00:04:02.160 You just took that 5, put a 1 on it, and then you borrowed 00:04:02.160 --> 00:04:04.600 a 1 from this 100 here. 00:04:04.600 --> 00:04:07.080 But really, what I did is right here. 00:04:07.080 --> 00:04:12.600 I took 1,000 from this 1 and I redistributed that 00:04:12.600 --> 00:04:17.050 1,000 amongst the 100's, 10's, and 1's place. 00:04:17.050 --> 00:04:18.010 Let me do another example. 00:04:18.010 --> 00:04:19.810 I think it might make it a little bit more clearer 00:04:19.810 --> 00:04:22.050 of why borrowing works. 00:04:25.130 --> 00:04:26.820 Let me do a simpler problem. 00:04:26.820 --> 00:04:29.210 I actually started off with a problem that tends to confuse 00:04:29.210 --> 00:04:30.670 the most number of people. 00:04:30.670 --> 00:04:45.290 Let's say I had 732 minus-- Let me do a fairly simple one. 00:04:45.290 --> 00:04:46.573 Minus 23. 00:04:49.150 --> 00:04:50.820 Sometimes those 3's just come out weird. 00:04:50.820 --> 00:04:55.120 Well, we just learned that's the same thing as 700 plus 00:04:55.120 --> 00:05:03.240 30 plus 2 minus 20 minus 3. 00:05:03.240 --> 00:05:07.150 Well, we see this 2, 2 is less than 3, so we can't subtract. 00:05:07.150 --> 00:05:09.050 Wouldn't it be great if we could get a 10 from someplace? 00:05:09.050 --> 00:05:10.870 We could get a 10 from here. 00:05:10.870 --> 00:05:16.960 We make this into 20 and add the 10 to the 2 and we get 12. 00:05:16.960 --> 00:05:21.940 And notice, 700 plus 20 plus 12 is still 732. 00:05:21.940 --> 00:05:24.420 So we really didn't change the number up top at all. 00:05:24.420 --> 00:05:28.520 We just redistributed its quantity amongst the 00:05:28.520 --> 00:05:29.200 different places. 00:05:29.200 --> 00:05:30.260 And now we're ready to subtract. 00:05:30.260 --> 00:05:32.340 12 minus 3 is 9. 00:05:32.340 --> 00:05:36.610 20 minus 20 is 0 and then you just bring down the 700. 00:05:36.610 --> 00:05:42.190 You get 700 plus 0 plus 9, which is the same thing as 709. 00:05:42.190 --> 00:05:45.260 And that's the reason why this borrowing will work. 00:05:45.260 --> 00:05:47.380 Well, we say, oh, let's borrow 1 from the 3. 00:05:47.380 --> 00:05:48.290 Makes it a 2. 00:05:48.290 --> 00:05:49.750 This becomes a 12. 00:05:49.750 --> 00:05:52.200 And then we subtract. 00:05:52.200 --> 00:05:54.520 9 0 7. 00:05:54.520 --> 00:05:57.370 Let's do another problem, one last one. 00:05:57.370 --> 00:05:59.160 And once again, you don't have to do it this way. 00:05:59.160 --> 00:06:00.850 You don't have to every time you do a subtraction 00:06:00.850 --> 00:06:01.480 problem do it this way. 00:06:01.480 --> 00:06:03.520 Although if you ever get confused, you can do it this 00:06:03.520 --> 00:06:05.700 way and you won't make a mistake, and you'll actually 00:06:05.700 --> 00:06:06.990 understand what you're doing. 00:06:06.990 --> 00:06:08.930 But if you're on a test and you have to do things really fast 00:06:08.930 --> 00:06:10.860 you should do it the conventional way. 00:06:10.860 --> 00:06:13.570 But it takes a lot of practice to make sure you never are 00:06:13.570 --> 00:06:15.710 doing something improper. 00:06:15.710 --> 00:06:16.390 And that's the problem. 00:06:16.390 --> 00:06:18.280 People learn just the rules, and then they forget the 00:06:18.280 --> 00:06:19.980 rules, and then they forgot how to do it. 00:06:19.980 --> 00:06:22.620 If you learn what you're doing, you'll never really forget it 00:06:22.620 --> 00:06:26.230 because it should make some sense to you. 00:06:26.230 --> 00:06:28.000 Let's do another problem. 00:06:28.000 --> 00:06:35.970 If I had 512 minus 38. 00:06:35.970 --> 00:06:38.350 Well, let's keep doing it that way I just showed you. 00:06:38.350 --> 00:06:45.020 That's the same thing as 500 plus 10 plus 00:06:45.020 --> 00:06:51.080 2 minus 30 minus 8. 00:06:51.080 --> 00:06:52.130 Well, 2 is less than 8. 00:06:52.130 --> 00:06:53.290 I need a 10 from someplace. 00:06:53.290 --> 00:06:55.290 Well, one option we can do is we can just get 00:06:55.290 --> 00:06:56.600 the 10 from here. 00:06:56.600 --> 00:06:58.720 So then that becomes 0. 00:06:58.720 --> 00:07:00.220 And then this will become a 12. 00:07:00.220 --> 00:07:05.370 Notice that 500 plus 0 plus 12, same thing as 512 still. 00:07:05.370 --> 00:07:06.270 So we could subtract. 00:07:06.270 --> 00:07:09.770 12 minus 8 is 4. 00:07:09.770 --> 00:07:14.860 But here we see this 0 is less than 30, so we can't subtract. 00:07:14.860 --> 00:07:17.290 But we can borrow from the 500. 00:07:17.290 --> 00:07:22.930 Well, all we need is 100, so if we turn this into 100 then we 00:07:22.930 --> 00:07:25.260 took the 100 from the 500. 00:07:25.260 --> 00:07:28.070 This becomes 400. 00:07:28.070 --> 00:07:31.430 I just rewrote 500 as 400 plus 100. 00:07:31.430 --> 00:07:32.340 Now I can subtract. 00:07:32.340 --> 00:07:35.510 100 minus 30 is 70. 00:07:35.510 --> 00:07:38.790 Bring down the 400. 00:07:38.790 --> 00:07:42.560 And this is the same thing as 474. 00:07:42.560 --> 00:07:44.490 And the way you learn how to do it in school is you say, oh, 00:07:44.490 --> 00:07:47.815 well, 2 is less than 8, so let me borrow the 1. 00:07:47.815 --> 00:07:48.930 It becomes 12. 00:07:48.930 --> 00:07:50.890 This becomes a 0. 00:07:50.890 --> 00:07:56.120 0 is less than 3, so let me borrow 1 from this 5. 00:07:56.120 --> 00:07:57.140 Make this 4. 00:07:57.140 --> 00:07:58.710 This becomes 10. 00:07:58.710 --> 00:08:01.270 So then you say 12 minus 8 is 4. 00:08:01.270 --> 00:08:05.780 10 minus 3 is 7 and you bring down the 4. 00:08:05.780 --> 00:08:09.400 Hopefully what I've done here will give you an intuition 00:08:09.400 --> 00:08:10.510 of why borrowing works. 00:08:10.510 --> 00:08:12.760 And this is something that actually I didn't quite 00:08:12.760 --> 00:08:16.690 understand until a while after I learned how to borrow. 00:08:16.690 --> 00:08:19.650 And if you learned this, you'll realize that what you're doing 00:08:19.650 --> 00:08:21.260 here isn't really magic. 00:08:21.260 --> 00:08:23.780 And hopefully, you'll never forget what you're actually 00:08:23.780 --> 00:08:25.400 doing and you can always kind of think about what's 00:08:25.400 --> 00:08:28.900 fundamentally happening to the numbers when you borrow. 00:08:28.900 --> 00:08:31.580 I hope you found that useful. 00:08:31.580 --> 00:08:32.400 Talk to later. 00:08:32.400 --> 00:08:33.700 Bye.

Advanced ratio problems

https://www.youtube.com/watch?v=PASSD2OcU0c

vtt

https://www.youtube.com/api/timedtext?v=PASSD2OcU0c&ei=gGeUZc3dKoKfxN8P6JyQwAY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249840&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=28B9865A12E39B749A4963082729D303DE224173.413C9419C9E11B8BA226282C04DCE0151B8BFDB8&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.920 --> 00:00:04.020 Welcome to the presentation on more advanced ratio problems. 00:00:04.020 --> 00:00:06.240 Let's get started with some problems. 00:00:06.240 --> 00:00:11.870 So let's say that I have a class, and then-- oh the pen is 00:00:11.870 --> 00:00:17.180 messed up --OK, so in the class the total number of 00:00:17.180 --> 00:00:21.840 students is 57. 00:00:21.840 --> 00:00:26.850 And I would also tell you that the ratio of girls 00:00:26.850 --> 00:00:34.450 to boys is equal to 4:15. 00:00:34.450 --> 00:00:36.425 So now this the interesting part, so far it 00:00:36.425 --> 00:00:37.530 doesn't look to tough. 00:00:37.530 --> 00:00:42.450 My question is how many boys need to leave the room, so how 00:00:42.450 --> 00:01:02.760 many boys need to leave for the ratio of girls to 00:01:02.760 --> 00:01:10.960 boys to be 4:11. 00:01:10.960 --> 00:01:12.590 This is fascinating. 00:01:12.590 --> 00:01:15.100 So, a good place to start is just to figure out how many 00:01:15.100 --> 00:01:17.460 girls and how many boys there are in this classroom. 00:01:17.460 --> 00:01:19.650 And we already learned how to do that in the introduction 00:01:19.650 --> 00:01:21.130 to ratio problems. 00:01:21.130 --> 00:01:27.140 We know that the girls plus the boys is equal to 57, right, 00:01:27.140 --> 00:01:29.550 because there are 57 kids in the room. 00:01:29.550 --> 00:01:31.810 And we also know, just multiplying-- taking this 00:01:31.810 --> 00:01:36.400 equation --and multiplying both sides by b, we also know that 00:01:36.400 --> 00:01:44.030 the girls are equal to 4/15 times the boys, right? 00:01:44.030 --> 00:01:47.040 And then we can just substitute that back into this equation, 00:01:47.040 --> 00:01:58.180 and then we get 4/15b plus b is equal to 57, is the same thing 00:01:58.180 --> 00:02:04.110 as 19/15b is equal to 57. 00:02:06.910 --> 00:02:10.210 Let me clean this up a little bit. 00:02:10.210 --> 00:02:14.630 That's separate, and then let me go here. 00:02:14.630 --> 00:02:23.820 And we say b is equal to 57-- oh, woops --it's actually 57 00:02:23.820 --> 00:02:27.930 times 15 all of that over 19, right? 00:02:27.930 --> 00:02:31.430 I just multiply both sides by 15/19. 00:02:31.430 --> 00:02:36.260 So 57 divided by 19 is 3. 00:02:36.260 --> 00:02:40.520 So b is equal to 45. 00:02:40.520 --> 00:02:44.240 And we know there're a total of 57 kids in the class-- g plus 00:02:44.240 --> 00:02:48.220 b is 57 --so we know that there are 12 girls, right? 00:02:48.220 --> 00:02:50.370 57 minus 45. 00:02:50.370 --> 00:02:51.110 Good. 00:02:51.110 --> 00:02:54.240 So now we know that the current boys and girls 00:02:54.240 --> 00:02:58.850 are 45 boys and 12 girls. 00:02:58.850 --> 00:02:59.870 So let's write that down. 00:02:59.870 --> 00:03:07.520 So there's 12 girls and 45 boys. 00:03:07.520 --> 00:03:11.010 Now, the question says, how many boys need to leave for 00:03:11.010 --> 00:03:13.550 the ratio of girls to boys equals 4 4:11? 00:03:13.550 --> 00:03:15.690 So this is the number of girls right now, 12, 00:03:15.690 --> 00:03:17.170 this is the number boys. 00:03:17.170 --> 00:03:21.770 Let's say x is a number of boys that need to leave the room. 00:03:21.770 --> 00:03:26.950 So if x boys leave the room the new ratio will be 12 girls to 00:03:26.950 --> 00:03:31.720 the 45 boys minus the x boys that leave, right? 00:03:31.720 --> 00:03:34.490 If that confuses you, sit and look at that for a second. 00:03:34.490 --> 00:03:38.010 We start off with 12 girls and 45 boys in the room. 00:03:38.010 --> 00:03:40.260 And we're saying x boys are going to leave, so the new 00:03:40.260 --> 00:03:43.820 ratio is going to be 12:45 minus x. 00:03:43.820 --> 00:03:47.090 And we know from this part of the problem that that new 00:03:47.090 --> 00:03:51.420 ratio is going to equal 4:11. 00:03:51.420 --> 00:03:54.120 There, we just set up a equation with one unknown 00:03:54.120 --> 00:03:56.050 and we can solve for x. 00:03:56.050 --> 00:03:57.130 I hope that doesn't confuse you much. 00:03:57.130 --> 00:03:59.090 All we did is we figured out how many boys, how many 00:03:59.090 --> 00:04:00.380 girls are in the room now. 00:04:00.380 --> 00:04:02.900 We said x is the number boys that need to leave. 00:04:02.900 --> 00:04:06.770 And we said the new ratio is going to be girls to the new 00:04:06.770 --> 00:04:09.280 number of boys, which is 45 minus x, and that's going to 00:04:09.280 --> 00:04:11.400 be equal to the new ratio. 00:04:11.400 --> 00:04:13.070 So let's solve for x. 00:04:13.070 --> 00:04:19.230 Well, 12 times 11 is what that's 132. 00:04:19.230 --> 00:04:30.500 132 is equal to 4 times 45, 160, 180, minus 4x. 00:04:30.500 --> 00:04:33.320 And then if you solve for x, I think you know how to do this 00:04:33.320 --> 00:04:42.270 right now, and we can say minus 4x is equal to minus 48. 00:04:42.270 --> 00:04:44.890 x is equal to 12. 00:04:44.890 --> 00:04:45.810 There we solved it. 00:04:45.810 --> 00:04:50.960 So we say that if 12 boys left the room, the new ratio of 00:04:50.960 --> 00:04:53.430 girls to boys would be 4:11. 00:04:53.430 --> 00:04:54.620 And does that make sense? 00:04:54.620 --> 00:04:57.810 Well if 12 boys left the room, then the new ratio of girls to 00:04:57.810 --> 00:05:03.440 boys would be 12:33, right? 00:05:03.440 --> 00:05:06.630 Because 45 minus 12 is 33. 00:05:06.630 --> 00:05:08.870 And that's the same thing as if you divide the 00:05:08.870 --> 00:05:10.020 top and bottom by 3. 00:05:10.020 --> 00:05:11.930 That's 4:11. 00:05:11.930 --> 00:05:13.150 So there, we got it right. 00:05:13.150 --> 00:05:16.200 So what looked like a very hard problem actually wasn't so bad 00:05:16.200 --> 00:05:21.940 when you just sit down and work through the algebra. 00:05:21.940 --> 00:05:22.930 Let's do another problem. 00:05:27.930 --> 00:05:35.470 Let's say --this thing sometimes malfunctions --OK, 00:05:35.470 --> 00:05:41.630 let's say that the ratio of apples to bananas in a 00:05:41.630 --> 00:05:46.550 basket is equal to 5:19. 00:05:46.550 --> 00:06:03.160 And when we add 23 bananas the ratio of apples to bananas-- 00:06:03.160 --> 00:06:06.330 and actually let's write it right now, we now have 23 00:06:06.330 --> 00:06:17.370 bananas more --is equal to 10:61. 00:06:17.370 --> 00:06:20.820 So the question is, what is the total amount of 00:06:20.820 --> 00:06:21.730 fruit in the basket? 00:06:26.760 --> 00:06:33.020 Amount of fruit-- --ah, that's so messy --after 00:06:33.020 --> 00:06:33.815 adding the bananas. 00:06:40.790 --> 00:06:42.780 So I actually gave you a hint just when I wrote 00:06:42.780 --> 00:06:43.700 down initial problem. 00:06:43.700 --> 00:06:48.580 We're saying the ratio of a to b-- so let a equal the number 00:06:48.580 --> 00:06:52.710 of apples, and b equal the number bananas --so the ratio 00:06:52.710 --> 00:06:55.450 of apples to bananas equals 5:19. 00:06:55.450 --> 00:07:01.080 When I add 23 bananas, now the new ratio's going to be the 00:07:01.080 --> 00:07:04.460 number of apples to b plus 23. 00:07:04.460 --> 00:07:06.600 The new ratio is 10:61. 00:07:06.600 --> 00:07:07.540 So how do we solve this? 00:07:07.540 --> 00:07:10.080 Well, once again we have two equations and two unknowns. 00:07:10.080 --> 00:07:12.630 We know that-- I guess let's take this equation first, 00:07:12.630 --> 00:07:18.200 because it's a little more complicated --we know if we 00:07:18.200 --> 00:07:32.890 cross multiply that 61a is equal to 10b plus 230 and if we 00:07:32.890 --> 00:07:36.980 divide both sides by 61, we know that a is equal to 00:07:36.980 --> 00:07:45.830 10/61b plus 230/61. 00:07:45.830 --> 00:07:46.670 Right? 00:07:46.670 --> 00:07:50.070 And we could take this equation and multiply both sides by b 00:07:50.070 --> 00:07:54.850 and we could say that a is equal to 5/19b. 00:07:54.850 --> 00:07:55.810 Right? 00:07:55.810 --> 00:07:58.440 Well both of these are equal to a so we could set them 00:07:58.440 --> 00:07:59.800 equal to each other. 00:07:59.800 --> 00:08:24.550 And you get 5/19b is equal to 10/61b plus 230 times 61. 00:08:24.550 --> 00:08:26.070 And we solve for b. 00:08:26.070 --> 00:08:28.640 While this might seem complicated to you at first, 00:08:28.640 --> 00:08:31.110 but it's just a basic linear equation. 00:08:31.110 --> 00:08:33.100 And for the sake of time, because I only have 2 minutes 00:08:33.100 --> 00:08:33.960 left in this youtube. 00:08:33.960 --> 00:08:39.500 I'm just going to solve for b, and you get b is equal to 38. 00:08:39.500 --> 00:08:44.700 if b is equal to 38, we know that the initial ratio is 5:19. 00:08:44.700 --> 00:08:45.620 So that's pretty easy. 00:08:45.620 --> 00:08:53.250 We just say a is equal to 5/19 times 38 is equal to 10. 00:08:53.250 --> 00:08:56.540 So the initial number of apples was 10, the initial number 00:08:56.540 --> 00:08:58.840 bananas is 38, right? 00:08:58.840 --> 00:09:03.420 So initially we started off with 48 pieces of fruit, and 00:09:03.420 --> 00:09:09.180 then we're going to add 23 more pieces of fruit, right? 00:09:09.180 --> 00:09:14.660 And 43 plus 28, that's what, 71 pieces of fruit. 00:09:14.660 --> 00:09:16.250 So, let me review real quick what we said. 00:09:16.250 --> 00:09:20.430 We said the ratio of apples to bananas is 5:19. 00:09:20.430 --> 00:09:21.990 That's a is the number of apples, b is the 00:09:21.990 --> 00:09:23.380 number of bananas. 00:09:23.380 --> 00:09:27.470 When I add 23 bananas, I now have b plus 23 bananas, the 00:09:27.470 --> 00:09:31.520 new ratio of apples to total number bananas is 10:61. 00:09:31.520 --> 00:09:33.700 And I just used both of these equations. 00:09:33.700 --> 00:09:35.610 Two equations and two unknowns. 00:09:35.610 --> 00:09:38.830 Solved for a, and then substituted, and 00:09:38.830 --> 00:09:39.920 I solved for b. 00:09:39.920 --> 00:09:40.740 Nothing fancy here. 00:09:40.740 --> 00:09:42.120 I know there's a lot of fractions here, but if you 00:09:42.120 --> 00:09:44.770 just work through this, the fractions actually work out. 00:09:44.770 --> 00:09:46.750 And I was able to solve for a and b. 00:09:46.750 --> 00:09:51.120 Add the 23 pieces and I got 71 total pieces of fruit. 00:09:51.120 --> 00:09:53.190 I think you're now ready to try some of the more 00:09:53.190 --> 00:09:54.980 difficult ratio problems. 00:09:54.980 --> 00:09:56.490 Have fun!

Introduction to Ratios

https://www.youtube.com/watch?v=UsPmg_Ne1po

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https://www.youtube.com/api/timedtext?v=UsPmg_Ne1po&ei=gGeUZZzPKs_oxN8P8ZKE6Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249840&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=98B22D71EEB15414C14BEAB236BF55126379996C.42EE595145F3266E78558B13FE7E5EB17B9702EC&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:03.022 Welcome to the presentation on ratios. 00:00:03.022 --> 00:00:05.580 Now, I'm going to start just giving you a definition 00:00:05.580 --> 00:00:07.770 of ratios, and this I got from Wikipedia. 00:00:07.770 --> 00:00:11.420 "A ratio is a quantity that denotes the proportional amount 00:00:11.420 --> 00:00:14.980 of magnitude of one quantity relative to another." So I'm 00:00:14.980 --> 00:00:16.600 going to tell you from the beginning that I think a ratio 00:00:16.600 --> 00:00:18.870 is something that's easier to understand than to give a 00:00:18.870 --> 00:00:20.580 definition for because I don't think that Wikipedia 00:00:20.580 --> 00:00:22.090 definition is that useful. 00:00:22.090 --> 00:00:24.490 Let me give you some examples. 00:00:24.490 --> 00:00:29.210 If there are-- let's say there are-- let me make 00:00:29.210 --> 00:00:30.640 this pen size is right. 00:00:30.640 --> 00:00:39.480 Let's say there 10 boys and 2 girls in a class. 00:00:39.480 --> 00:00:47.270 So the ratio of boys to girls would be 10:2 or 10/2. 00:00:47.270 --> 00:00:49.070 Those are two different ways of writing it. 00:00:49.070 --> 00:00:51.030 And we know from fractions that that's also the same 00:00:51.030 --> 00:00:57.720 thing as 5:1 or 5/1. 00:00:57.720 --> 00:00:59.320 We want to keep the 1 there because we know that it's a 00:00:59.320 --> 00:01:01.520 ratio of one thing to another thing. 00:01:01.520 --> 00:01:02.560 So what does that mean? 00:01:02.560 --> 00:01:06.240 All that means is that for every 5 boys there's 1 girl. 00:01:06.240 --> 00:01:10.240 And so if we told you that the ratio of boys to girls in a 00:01:10.240 --> 00:01:16.290 room is 5:1, and we told you that there are-- let's say that 00:01:16.290 --> 00:01:23.300 we told you that there are 100 girls, then we'd know that, 00:01:23.300 --> 00:01:26.290 well, for every 1 of those girls, there's 5 boys, so 00:01:26.290 --> 00:01:28.860 that means that there'd be 500 boys, right? 00:01:31.440 --> 00:01:34.795 Or you could also look at that as the ratio of boys to girls 00:01:34.795 --> 00:01:39.320 is 500/100, which equals 5/1. 00:01:41.820 --> 00:01:43.730 And this is the typical way that a ratio is written: 00:01:43.730 --> 00:01:47.690 500:100 of boys girls. 00:01:47.690 --> 00:01:50.620 Now, let me ask you a couple of questions based on that. 00:01:50.620 --> 00:01:53.340 I think you get the general idea. 00:01:53.340 --> 00:02:04.380 If I told you that the ratio of, let's say, red balls to 00:02:04.380 --> 00:02:15.190 green balls in a bag is 2:3. 00:02:15.190 --> 00:02:33.330 And then if I also told you that there are 40 red balls, 00:02:33.330 --> 00:02:36.750 how many green balls are there? 00:02:36.750 --> 00:02:40.730 Well, what we say is the ratio of red balls to green balls-- 00:02:40.730 --> 00:02:45.240 so we know that there are 40 red balls, and then we want to 00:02:45.240 --> 00:02:47.970 solve for the number of green balls, that that 00:02:47.970 --> 00:02:49.770 is equal to 2/3. 00:02:52.360 --> 00:02:53.970 And then we could just solve this. 00:02:53.970 --> 00:02:55.440 We just cross multiply. 00:02:55.440 --> 00:03:04.325 40 times 3 is 120 is equal to 2g, and then we just solve. 00:03:04.325 --> 00:03:08.190 We just say g equals 60. 00:03:08.190 --> 00:03:10.560 And there's an easier way of doing this kind of in your 00:03:10.560 --> 00:03:13.270 head, and this is the algebraic way that'll always work. 00:03:13.270 --> 00:03:15.440 But you could also just say-- let me write this a little bit. 00:03:15.440 --> 00:03:17.690 This is a 3 down here. 00:03:17.690 --> 00:03:20.570 You can also say, well, to get from 2 to 40, you have to 00:03:20.570 --> 00:03:26.005 multiply by 20, so to get from 3 to g, I'm also going 00:03:26.005 --> 00:03:28.170 to multiply by 20. 00:03:28.170 --> 00:03:30.090 And so 3 times 20 is 60. 00:03:30.090 --> 00:03:31.270 That's another way to do it. 00:03:31.270 --> 00:03:33.110 A lot of you might actually find it more intuitive 00:03:33.110 --> 00:03:33.900 just to think about it. 00:03:33.900 --> 00:03:38.510 Well, if for every 2 red balls, there are 3 green balls. 00:03:38.510 --> 00:03:43.310 Then if there are 40 red balls, then it makes sense that there 00:03:43.310 --> 00:03:47.640 would be 60 green balls because for every 20, there'd be 30, 00:03:47.640 --> 00:03:49.020 for every 40, there'd be 60. 00:03:49.020 --> 00:03:52.940 I hope I'm not completely confusing you. 00:03:52.940 --> 00:03:54.070 Let me give you another example. 00:03:54.070 --> 00:04:11.790 Let's say the ratio of boys to girls is equal to 2 to 7. 00:04:11.790 --> 00:04:20.950 And if I were tell you that the total class has 180 kids in it, 00:04:20.950 --> 00:04:23.050 can we figure out how many boys and girls there 00:04:23.050 --> 00:04:24.620 are in the class? 00:04:24.620 --> 00:04:26.110 Well, let's think about it. 00:04:26.110 --> 00:04:36.000 Well, we know that the boys to girls is equal to 2/7, and we 00:04:36.000 --> 00:04:42.750 also know that the boys plus girls is equal to 180. 00:04:42.750 --> 00:04:45.330 So here, we have a system of two equations and two unknowns. 00:04:45.330 --> 00:04:47.380 And you could actually, if you really think about it, 00:04:47.380 --> 00:04:49.210 you could actually solve this without algebra. 00:04:49.210 --> 00:04:51.920 But I'll show you the algebraic way, because when problems get 00:04:51.920 --> 00:04:54.590 complicated, this'll always work. 00:04:54.590 --> 00:04:56.730 So what we can do is we can do substitution. 00:04:56.730 --> 00:05:03.220 We know that b is equal to 2/7g, right? 00:05:03.220 --> 00:05:07.650 I just multiplied both sides of this equation by g. 00:05:07.650 --> 00:05:11.570 It cancels out there, and then times g, and you get this. 00:05:11.570 --> 00:05:14.940 And then we can just substitute that back in for b. 00:05:14.940 --> 00:05:24.620 So then we have 2/7g plus g is equal to 180. 00:05:24.620 --> 00:05:31.290 And what's 2/7g plus-- we could 1g or 7/7g? 00:05:31.290 --> 00:05:33.900 Well, you could do the fraction, but it's 2/7 plus 1 00:05:33.900 --> 00:05:41.420 is the same thing as-- that's equal to 2/7 plus 7/7, right, 00:05:41.420 --> 00:05:45.430 because that's just 1g is equal to 180. 00:05:45.430 --> 00:05:47.670 And I'm jumping around on the chalkboard on purpose to 00:05:47.670 --> 00:05:51.170 intentionally confuse you. 00:05:51.170 --> 00:05:52.550 OK, this is where I am. 00:05:52.550 --> 00:05:55.400 So 2/7 plus 7/7g equals 180. 00:05:55.400 --> 00:06:00.580 So we have 9/7g is equal to 180. 00:06:00.580 --> 00:06:02.640 And then we just multiply both sides times the 00:06:02.640 --> 00:06:07.560 reciprocal of 7/9. 00:06:07.560 --> 00:06:07.930 Oops! 00:06:07.930 --> 00:06:08.490 That's not a g. 00:06:08.490 --> 00:06:09.570 That's a 9. 00:06:09.570 --> 00:06:12.965 Once, again, an intentional device to confuse you. 00:06:12.965 --> 00:06:23.450 These cancel out, and you get g equals 180 times 7/9. 00:06:23.450 --> 00:06:27.420 Well, 180 divided by 9, this is just 20, right? 00:06:27.420 --> 00:06:30.650 So g is equal to 140. 00:06:30.650 --> 00:06:34.250 So if there's 140 girls in the room, how many boys 00:06:34.250 --> 00:06:34.850 are there going to be? 00:06:34.850 --> 00:06:39.060 Well, we know that the whole class has 180 people, and we 00:06:39.060 --> 00:06:41.960 know b plus g is 180, so there's going to be-- the 00:06:41.960 --> 00:06:45.610 boys are equal to 40. 00:06:45.610 --> 00:06:50.420 And this is really about as difficult as I guess we could 00:06:50.420 --> 00:06:52.650 say basic ratio problems get. 00:06:52.650 --> 00:06:54.280 There's nothing really difficult about ratios. 00:06:54.280 --> 00:06:57.710 They're just representing, for every amount of one thing, how 00:06:57.710 --> 00:06:59.350 much do you have of the other thing? 00:06:59.350 --> 00:07:04.170 And then you can use that ratio if you have some other 00:07:04.170 --> 00:07:06.840 information in terms of how many total people there are, or 00:07:06.840 --> 00:07:09.000 how many total objects there are, or how much of one object 00:07:09.000 --> 00:07:11.810 there is, you can use that to figure out how much of the 00:07:11.810 --> 00:07:16.110 other object there is or how many total objects there are. 00:07:16.110 --> 00:07:19.840 I think you're now ready to try some of the ratio problems, and 00:07:19.840 --> 00:07:22.230 I'm going to do another presentation on what I would 00:07:22.230 --> 00:07:25.260 consider slightly more advanced ratio problems. 00:07:25.260 --> 00:07:27.330 So have fun.

Averages

https://www.youtube.com/watch?v=9VZsMY15xeU

vtt

https://www.youtube.com/api/timedtext?v=9VZsMY15xeU&ei=f2eUZY_LKu6Mp-oPkdCksAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=DDE3E0D2C6094345125BB77FE3D618EBD13E2482.582D17345A8418BC6858C2B79385CF1998E6069C&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.840 --> 00:00:03.510 Welcome to the presentation on averages. 00:00:03.510 --> 00:00:05.980 Averages is probably a concept that you've already used 00:00:05.980 --> 00:00:08.020 before, maybe not in a mathematical way. 00:00:08.020 --> 00:00:11.790 But people will talk in terms of, the average voter wants a 00:00:11.790 --> 00:00:14.750 politician to do this, or the average student in a class 00:00:14.750 --> 00:00:15.760 wants to get out early. 00:00:15.760 --> 00:00:17.760 So you're probably already familiar with the 00:00:17.760 --> 00:00:18.760 concept of an average. 00:00:18.760 --> 00:00:21.230 And you probably already intuitively knew that an 00:00:21.230 --> 00:00:27.710 average is just a number that represents the different values 00:00:27.710 --> 00:00:29.290 that a group could have. 00:00:29.290 --> 00:00:32.970 But it can represent that as one number as opposed to giving 00:00:32.970 --> 00:00:34.350 all the different values. 00:00:34.350 --> 00:00:36.010 And let's give a couple of examples of how to compute 00:00:36.010 --> 00:00:38.070 an average, and you might already know how to do this. 00:00:38.070 --> 00:00:47.150 So let's say I had the numbers 1, 3, 5, and 20. 00:00:47.150 --> 00:00:51.980 And I asked you, what is the average of these four numbers? 00:00:51.980 --> 00:00:54.380 Well, what we do is, we literally just 00:00:54.380 --> 00:00:55.280 add up the numbers. 00:00:55.280 --> 00:00:57.410 And then divide by the number of numbers we have. 00:00:57.410 --> 00:01:01.220 So we say 1 plus 3 is 4. 00:01:01.220 --> 00:01:01.960 So let me write that. 00:01:01.960 --> 00:01:10.630 1 plus 3 plus 5 plus 20 equals, let's see, 1 plus 3 is 4. 00:01:10.630 --> 00:01:12.190 4 plus 5 is 9. 00:01:12.190 --> 00:01:15.310 9 plus 20 is 29. 00:01:15.310 --> 00:01:18.430 And we had 4 numbers; one, two, three, four. 00:01:18.430 --> 00:01:22.210 So 4 goes into 29. 00:01:22.210 --> 00:01:26.300 And it goes, 7, 7, 28. 00:01:26.300 --> 00:01:29.900 And then we have 10, I didn't have to do that decimal 00:01:29.900 --> 00:01:31.260 there, oh well. 00:01:31.260 --> 00:01:33.845 2, 8, 25. 00:01:36.910 --> 00:01:40.880 So 4 goes into 29 7.25 times. 00:01:40.880 --> 00:01:49.590 So the average of these four numbers is equal to 7.25. 00:01:49.590 --> 00:01:54.000 And that might make sense to you because 7.25 is someplace 00:01:54.000 --> 00:01:55.700 in between these numbers. 00:01:55.700 --> 00:02:00.450 And we can kind of view this, 7.25, as one way to represent 00:02:00.450 --> 00:02:03.090 these four numbers without having to list these 00:02:03.090 --> 00:02:03.990 four numbers. 00:02:03.990 --> 00:02:06.480 There are other representations you'll learn later on. 00:02:06.480 --> 00:02:07.960 Like the mode. 00:02:07.960 --> 00:02:12.700 You'll also the mean, which we'll talk about later, 00:02:12.700 --> 00:02:14.610 is actually the same thing as the average. 00:02:14.610 --> 00:02:16.680 But the average is just one number that you can use to 00:02:16.680 --> 00:02:19.400 represent a set of numbers. 00:02:19.400 --> 00:02:22.730 So let's do some problems which I think are going to 00:02:22.730 --> 00:02:26.180 be close to your heart. 00:02:26.180 --> 00:02:31.830 Let's say on the first four tests of an exam, I got a-- 00:02:31.830 --> 00:02:39.690 let's see, I got an 80, an 81. 00:02:39.690 --> 00:02:45.250 An 87, and an 88. 00:02:45.250 --> 00:02:47.580 What's my average in the class so far? 00:02:47.580 --> 00:02:50.890 Well, all I have to do is add up these four numbers. 00:02:50.890 --> 00:03:00.790 So I say, 80 plus 81 plus 87 plus 88. 00:03:00.790 --> 00:03:02.370 Well, zero plus 1 is 1. 00:03:02.370 --> 00:03:06.190 1 plus 7 is 8. 00:03:06.190 --> 00:03:07.490 8 plus 8 is 16. 00:03:07.490 --> 00:03:10.770 I just ran eight miles, so I'm a bit tired. 00:03:10.770 --> 00:03:13.310 And, 4/8, so that's 32. 00:03:13.310 --> 00:03:14.210 Plus 1 is 33. 00:03:16.950 --> 00:03:20.750 And now we divide this number by 4. 00:03:20.750 --> 00:03:25.300 4 goes into 336. 00:03:25.300 --> 00:03:27.000 Goes into 33, 8 times. 00:03:27.000 --> 00:03:28.900 8 times 4 is 32. 00:03:31.850 --> 00:03:34.000 33 minus 32 is 1, 16. 00:03:34.000 --> 00:03:34.460 4. 00:03:34.460 --> 00:03:39.750 So the average is equal to 84. 00:03:39.750 --> 00:03:43.220 So depending on what school you go to that's either a B or a C. 00:03:43.220 --> 00:03:49.250 So, so far my average after the first four exams is an 84. 00:03:49.250 --> 00:03:51.740 Now let's make this a little bit more difficult. 00:03:51.740 --> 00:04:00.150 We know that the average after four exams, at four 00:04:00.150 --> 00:04:04.980 exams, is equal to 84. 00:04:04.980 --> 00:04:09.200 If I were to ask you what do I have to get on the next test to 00:04:09.200 --> 00:04:16.010 average an 88, to average an 88 in the class. 00:04:20.310 --> 00:04:23.490 So let's say that x is what I get on the next test. 00:04:28.180 --> 00:04:31.990 So now what we can say is, is that the first four exams, I 00:04:31.990 --> 00:04:36.910 could either list out the first four exams that I took. 00:04:36.910 --> 00:04:38.950 Or I already know what the average is. 00:04:38.950 --> 00:04:42.210 So I know the sum of the first four exams is 00:04:42.210 --> 00:04:45.470 going to 4 times 84. 00:04:45.470 --> 00:04:51.420 And now I want to add the, what I get on the 5th exam, x. 00:04:51.420 --> 00:04:55.640 And I'm going to divide that by all five exams. 00:04:55.640 --> 00:04:58.770 So in other words, this number is the average 00:04:58.770 --> 00:05:00.220 of my first five exams. 00:05:00.220 --> 00:05:02.510 We just figured out the average of the first four exams. 00:05:02.510 --> 00:05:06.620 But now, we sum up the first four exams here. 00:05:06.620 --> 00:05:09.220 We add what I got on the fifth exam, and then we divide it 00:05:09.220 --> 00:05:11.940 by 5, because now we're averaging five exams. 00:05:11.940 --> 00:05:16.030 And I said that I need to get in an 88 in the class. 00:05:16.030 --> 00:05:18.230 And now we solve for x. 00:05:18.230 --> 00:05:20.080 Let me make some space here. 00:05:22.710 --> 00:05:25.080 So, 5 times 88 is, let's see. 00:05:25.080 --> 00:05:31.110 5 times 80 is 400, so it's 440. 00:05:31.110 --> 00:05:36.030 440 equals 4 times 84, we just saw that, is 00:05:36.030 --> 00:05:39.910 320 plus 16 is 336. 00:05:39.910 --> 00:05:44.210 336 plus x is equal to 440. 00:05:44.210 --> 00:05:47.200 Well, it turns out if you subtract 336 from both sides, 00:05:47.200 --> 00:05:51.550 you get x is equal to 104. 00:05:51.550 --> 00:05:55.560 So unless you have a exam that has some bonus problems on it, 00:05:55.560 --> 00:06:00.760 it's probably impossible for you to get ah an 88 average in 00:06:00.760 --> 00:06:03.180 the class after just the next exam. 00:06:03.180 --> 00:06:05.380 You'd have to get 104 on that next exam. 00:06:05.380 --> 00:06:06.810 And let's just look at what we just did. 00:06:06.810 --> 00:06:10.380 We said, after 4 exams we had an 84. 00:06:10.380 --> 00:06:14.530 What do I have to get on that next exam to average an 88 00:06:14.530 --> 00:06:17.090 in the class after 5 exams? 00:06:17.090 --> 00:06:19.920 And that's what we solved for when we got x. 00:06:19.920 --> 00:06:23.670 Now, let's ask another question. 00:06:23.670 --> 00:06:29.830 I said after four exams, after four exams, I 00:06:29.830 --> 00:06:32.080 had an 84 average. 00:06:34.670 --> 00:06:39.050 If I said that there are 6 exams in the class, and the 00:06:39.050 --> 00:06:42.170 highest score I could get on an exam is 100, what is the 00:06:42.170 --> 00:06:44.730 highest average I can finish in the class if I were to really 00:06:44.730 --> 00:06:48.030 study hard and get 100 on the next 2 exams? 00:06:48.030 --> 00:06:51.250 Well, once again, what we'll want to do is assume we get 00:06:51.250 --> 00:06:54.520 100 on the next 2 exams and then take the average. 00:06:54.520 --> 00:06:56.890 So we'll have to solve all 6 exams. 00:06:56.890 --> 00:06:58.820 So we're going to have the average of 6, so in the 00:06:58.820 --> 00:07:01.000 denominator we're going to have 6. 00:07:01.000 --> 00:07:05.370 The first four exams, the sum, as we already learned, is 4 00:07:05.370 --> 00:07:07.710 exams times the 84 average. 00:07:07.710 --> 00:07:09.590 And this dot is just times. 00:07:09.590 --> 00:07:12.140 Plus, and there's going to be 2 more exams, right? 00:07:12.140 --> 00:07:13.740 Because there's 6 exams in the class. 00:07:13.740 --> 00:07:15.550 And I'm going to get 100 in each. 00:07:15.550 --> 00:07:18.260 So that's 200. 00:07:18.260 --> 00:07:19.230 And what's this average? 00:07:19.230 --> 00:07:23.050 Well, 4 times 84, we already said, is 336. 00:07:23.050 --> 00:07:27.250 Plus 200 over 6. 00:07:27.250 --> 00:07:31.090 So that's 536 over 6. 00:07:31.090 --> 00:07:33.840 6 goes into 5 36. 00:07:33.840 --> 00:07:36.580 I don't know if if I gave myself enough space. 00:07:36.580 --> 00:07:41.190 But 6 goes into 53, 8 times. 00:07:41.190 --> 00:07:42.990 48. 00:07:42.990 --> 00:07:44.940 56. 00:07:44.940 --> 00:07:46.640 9 times. 00:07:46.640 --> 00:07:49.850 9 times 6 is 54. 00:07:49.850 --> 00:07:54.730 6 minus is 20 6 goes into-- so we'll see it's actually 00:07:54.730 --> 00:07:57.630 89.333333, goes on forever. 00:07:57.630 --> 00:07:59.770 So 89.3 repeating. 00:07:59.770 --> 00:08:03.950 So no matter how hard I try in this class, the best I can do. 00:08:03.950 --> 00:08:05.970 Because I only have two exams left, even if I were to get 00:08:05.970 --> 00:08:07.740 100 on the next two exams. 00:08:07.740 --> 00:08:13.090 I can finish the class with an 89.333 average. 00:08:13.090 --> 00:08:16.180 Hopefully, I think some of this might have been a little 00:08:16.180 --> 00:08:16.990 bit of a review for you. 00:08:16.990 --> 00:08:19.170 You already had kind of a sense of what an average is. 00:08:19.170 --> 00:08:22.340 And hopefully these last two problems not only taught you 00:08:22.340 --> 00:08:25.100 how to do some algebra problems involving average, but they'll 00:08:25.100 --> 00:08:27.830 also help you figure out how well you have to do on your 00:08:27.830 --> 00:08:30.350 exams to get an A in your math class. 00:08:30.350 --> 00:08:33.310 I think you're now ready for the average module. 00:08:33.310 --> 00:08:34.860 Have fun.

Introduction to the quadratic equation

https://www.youtube.com/watch?v=IWigvJcCAJ0

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https://www.youtube.com/api/timedtext?v=IWigvJcCAJ0&ei=f2eUZeKwKqeNp-oPycK80AI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=82C8C43ADB5615541096A6BF9193AC052FACD769.2E80705AFE1EB12EE4E206AFA2CE5A672127A31D&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.010 --> 00:00:04.520 Welcome to the presentation on using the quadratic equation. 00:00:04.520 --> 00:00:06.730 So the quadratic equation, it sounds like something 00:00:06.730 --> 00:00:07.810 very complicated. 00:00:07.810 --> 00:00:09.930 And when you actually first see the quadratic equation, you'll 00:00:09.930 --> 00:00:11.590 say, well, not only does it sound like something 00:00:11.590 --> 00:00:13.110 complicated, but it is something complicated. 00:00:13.110 --> 00:00:14.930 But hopefully you'll see, over the course of this 00:00:14.930 --> 00:00:16.580 presentation, that it's actually not hard to use. 00:00:16.580 --> 00:00:19.040 And in a future presentation I'll actually show you 00:00:19.040 --> 00:00:21.300 how it was derived. 00:00:21.300 --> 00:00:24.810 So, in general, you've already learned how to factor a 00:00:24.810 --> 00:00:25.810 second degree equation. 00:00:25.810 --> 00:00:30.910 You've learned that if I had, say, x squared minus 00:00:30.910 --> 00:00:40.340 x, minus 6, equals 0. 00:00:40.340 --> 00:00:42.970 If I had this equation. x squared minus x minus x equals 00:00:42.970 --> 00:00:48.720 zero, that you could factor that as x minus 3 and 00:00:48.720 --> 00:00:52.210 x plus 2 equals 0. 00:00:52.210 --> 00:00:54.955 Which either means that x minus 3 equals 0 or 00:00:54.955 --> 00:00:57.073 x plus 2 equals 0. 00:00:57.073 --> 00:01:03.512 So x minus 3 equals 0 or x plus 2 equals 0. 00:01:03.512 --> 00:01:08.500 So, x equals 3 or negative 2. 00:01:08.500 --> 00:01:17.980 And, a graphical representation of this would be, if I had the 00:01:17.980 --> 00:01:26.150 function f of x is equal to x squared minus x minus 6. 00:01:26.150 --> 00:01:28.760 So this axis is the f of x axis. 00:01:28.760 --> 00:01:32.670 You might be more familiar with the y axis, and for the purpose 00:01:32.670 --> 00:01:34.780 of this type of problem, it doesn't matter. 00:01:34.780 --> 00:01:36.270 And this is the x axis. 00:01:36.270 --> 00:01:40.430 And if I were to graph this equation, x squared minus x, 00:01:40.430 --> 00:01:42.380 minus 6, it would look something like this. 00:01:42.380 --> 00:01:50.130 A bit like -- this is f of x equals minus 6. 00:01:50.130 --> 00:01:52.900 And the graph will kind of do something like this. 00:01:57.150 --> 00:02:00.030 Go up, it will keep going up in that direction. 00:02:00.030 --> 00:02:03.150 And know it goes through minus 6, because when x equals 0, 00:02:03.150 --> 00:02:05.110 f of x is equal to minus 6. 00:02:05.110 --> 00:02:07.800 So I know it goes through this point. 00:02:07.800 --> 00:02:11.520 And I know that when f of x is equal to 0, so f of x is equal 00:02:11.520 --> 00:02:14.960 to 0 along the x axis, right? 00:02:14.960 --> 00:02:16.600 Because this is 1. 00:02:16.600 --> 00:02:17.870 This is 0. 00:02:17.870 --> 00:02:19.160 This is negative 1. 00:02:19.160 --> 00:02:21.510 So this is where f of x is equal to 0, along 00:02:21.510 --> 00:02:23.420 this x axis, right? 00:02:23.420 --> 00:02:29.210 And we know it equals 0 at the points x is equal to 3 and 00:02:29.210 --> 00:02:32.330 x is equal to minus 2. 00:02:32.330 --> 00:02:34.360 That's actually what we solved here. 00:02:34.360 --> 00:02:36.440 Maybe when we were doing the factoring problems we didn't 00:02:36.440 --> 00:02:38.940 realize graphically what we were doing. 00:02:38.940 --> 00:02:42.070 But if we said that f of x is equal to this function, we're 00:02:42.070 --> 00:02:43.270 setting that equal to 0. 00:02:43.270 --> 00:02:44.820 So we're saying this function, when does 00:02:44.820 --> 00:02:48.220 this function equal 0? 00:02:48.220 --> 00:02:49.390 When is it equal to 0? 00:02:49.390 --> 00:02:51.720 Well, it's equal to 0 at these points, right? 00:02:51.720 --> 00:02:55.360 Because this is where f of x is equal to 0. 00:02:55.360 --> 00:02:57.490 And then what we were doing when we solved this by 00:02:57.490 --> 00:03:01.970 factoring is, we figured out, the x values that made f of x 00:03:01.970 --> 00:03:04.160 equal to 0, which is these two points. 00:03:04.160 --> 00:03:06.740 And, just a little terminology, these are also called 00:03:06.740 --> 00:03:09.860 the zeroes, or the roots, of f of x. 00:03:12.470 --> 00:03:14.810 Let's review that a little bit. 00:03:14.810 --> 00:03:23.700 So, if I had something like f of x is equal to x squared plus 00:03:23.700 --> 00:03:29.550 4x plus 4, and I asked you, where are the zeroes, or 00:03:29.550 --> 00:03:31.770 the roots, of f of x. 00:03:31.770 --> 00:03:33.970 That's the same thing as saying, where does f of x 00:03:33.970 --> 00:03:36.300 interject intersect the x axis? 00:03:36.300 --> 00:03:38.210 And it intersects the x axis when f of x is 00:03:38.210 --> 00:03:39.440 equal to 0, right? 00:03:39.440 --> 00:03:42.120 If you think about the graph I had just drawn. 00:03:42.120 --> 00:03:45.720 So, let's say if f of x is equal to 0, then we could 00:03:45.720 --> 00:03:51.860 just say, 0 is equal to x squared plus 4x plus 4. 00:03:51.860 --> 00:03:53.940 And we know, we could just factor that, that's x 00:03:53.940 --> 00:03:57.080 plus 2 times x plus 2. 00:03:57.080 --> 00:04:07.090 And we know that it's equal to 0 at x equals minus 2. 00:04:10.170 --> 00:04:13.940 x equals minus 2. 00:04:13.940 --> 00:04:18.270 Well, that's a little -- x equals minus 2. 00:04:18.270 --> 00:04:22.380 So now, we know how to find the 0's when the the actual 00:04:22.380 --> 00:04:24.560 equation is easy to factor. 00:04:24.560 --> 00:04:27.500 But let's do a situation where the equation is actually 00:04:27.500 --> 00:04:28.850 not so easy to factor. 00:04:32.120 --> 00:04:39.750 Let's say we had f of x is equal to minus 10x 00:04:39.750 --> 00:04:45.380 squared minus 9x plus 1. 00:04:45.380 --> 00:04:47.580 Well, when I look at this, even if I were to divide it by 10 I 00:04:47.580 --> 00:04:48.650 would get some fractions here. 00:04:48.650 --> 00:04:53.130 And it's very hard to imagine factoring this quadratic. 00:04:53.130 --> 00:04:54.860 And that's what's actually called a quadratic equation, or 00:04:54.860 --> 00:04:57.580 this second degree polynomial. 00:04:57.580 --> 00:04:59.600 But let's set it -- So we're trying to solve this. 00:04:59.600 --> 00:05:02.420 Because we want to find out when it equals 0. 00:05:02.420 --> 00:05:07.130 Minus 10x squared minus 9x plus 1. 00:05:07.130 --> 00:05:09.090 We want to find out what x values make this 00:05:09.090 --> 00:05:11.260 equation equal to zero. 00:05:11.260 --> 00:05:13.730 And here we can use a tool called a quadratic equation. 00:05:13.730 --> 00:05:15.625 And now I'm going to give you one of the few things in math 00:05:15.625 --> 00:05:18.030 that's probably a good idea to memorize. 00:05:18.030 --> 00:05:21.330 The quadratic equation says that the roots of a quadratic 00:05:21.330 --> 00:05:24.810 are equal to -- and let's say that the quadratic equation is 00:05:24.810 --> 00:05:31.900 a x squared plus b x plus c equals 0. 00:05:31.900 --> 00:05:35.790 So, in this example, a is minus 10. 00:05:35.790 --> 00:05:39.940 b is minus 9, and c is 1. 00:05:39.940 --> 00:05:48.040 The formula is the roots x equals negative b plus or minus 00:05:48.040 --> 00:05:58.060 the square root of b squared minus 4 times a times c, 00:05:58.060 --> 00:06:00.230 all of that over 2a. 00:06:00.230 --> 00:06:02.843 I know that looks complicated, but the more you use it, you'll 00:06:02.843 --> 00:06:04.400 see it's actually not that bad. 00:06:04.400 --> 00:06:07.720 And this is a good idea to memorize. 00:06:07.720 --> 00:06:10.730 So let's apply the quadratic equation to this equation 00:06:10.730 --> 00:06:12.670 that we just wrote down. 00:06:12.670 --> 00:06:15.260 So, I just said -- and look, the a is just the coefficient 00:06:15.260 --> 00:06:18.610 on the x term, right? 00:06:18.610 --> 00:06:20.300 a is the coefficient on the x squared term. 00:06:20.300 --> 00:06:23.570 b is the coefficient on the x term, and c is the constant. 00:06:23.570 --> 00:06:25.100 So let's apply it tot this equation. 00:06:25.100 --> 00:06:26.250 What's b? 00:06:26.250 --> 00:06:28.700 Well, b is negative 9. 00:06:28.700 --> 00:06:29.970 We could see here. 00:06:29.970 --> 00:06:33.980 b is negative 9, a is negative 10. 00:06:33.980 --> 00:06:34.970 c is 1. 00:06:34.970 --> 00:06:36.090 Right? 00:06:36.090 --> 00:06:42.350 So if b is negative 9 -- so let's say, that's negative 9. 00:06:42.350 --> 00:06:49.260 Plus or minus the square root of negative 9 squared. 00:06:49.260 --> 00:06:49.810 Well, that's 81. 00:06:53.140 --> 00:06:56.940 Minus 4 times a. 00:06:56.940 --> 00:06:59.760 a is minus 10. 00:06:59.760 --> 00:07:03.240 Minus 10 times c, which is 1. 00:07:03.240 --> 00:07:05.110 I know this is messy, but hopefully you're 00:07:05.110 --> 00:07:06.470 understanding it. 00:07:06.470 --> 00:07:09.560 And all of that over 2 times a. 00:07:09.560 --> 00:07:14.050 Well, a is minus 10, so 2 times a is minus 20. 00:07:14.050 --> 00:07:14.990 So let's simplify that. 00:07:14.990 --> 00:07:19.410 Negative times negative 9, that's positive 9. 00:07:19.410 --> 00:07:26.460 Plus or minus the square root of 81. 00:07:26.460 --> 00:07:30.660 We have a negative 4 times a negative 10. 00:07:30.660 --> 00:07:31.870 This is a minus 10. 00:07:31.870 --> 00:07:33.280 I know it's very messy, I really apologize 00:07:33.280 --> 00:07:34.380 for that, times 1. 00:07:34.380 --> 00:07:39.410 So negative 4 times negative 10 is 40, positive 40. 00:07:39.410 --> 00:07:41.040 Positive 40. 00:07:41.040 --> 00:07:46.070 And then we have all of that over negative 20. 00:07:46.070 --> 00:07:48.300 Well, 81 plus 40 is 121. 00:07:48.300 --> 00:07:52.330 So this is 9 plus or minus the square root 00:07:52.330 --> 00:07:58.290 of 121 over minus 20. 00:07:58.290 --> 00:08:01.620 Square root of 121 is 11. 00:08:01.620 --> 00:08:03.170 So I'll go here. 00:08:03.170 --> 00:08:06.184 Hopefully you won't lose track of what I'm doing. 00:08:06.184 --> 00:08:13.720 So this is 9 plus or minus 11, over minus 20. 00:08:13.720 --> 00:08:19.090 And so if we said 9 plus 11 over minus 20, that is 9 00:08:19.090 --> 00:08:22.540 plus 11 is 20, so this is 20 over minus 20. 00:08:22.540 --> 00:08:23.730 Which equals negative 1. 00:08:23.730 --> 00:08:24.900 So that's one root. 00:08:24.900 --> 00:08:28.260 That's 9 plus -- because this is plus or minus. 00:08:28.260 --> 00:08:33.790 And the other root would be 9 minus 11 over negative 20. 00:08:33.790 --> 00:08:37.720 Which equals minus 2 over minus 20. 00:08:37.720 --> 00:08:40.700 Which equals 1 over 10. 00:08:40.700 --> 00:08:42.690 So that's the other root. 00:08:42.690 --> 00:08:48.950 So if we were to graph this equation, we would see that it 00:08:48.950 --> 00:08:52.640 actually intersects the x axis. 00:08:52.640 --> 00:08:57.770 Or f of x equals 0 at the point x equals negative 00:08:57.770 --> 00:09:01.690 1 and x equals 1/10. 00:09:01.690 --> 00:09:04.080 I'm going to do a lot more examples in part 2, because I 00:09:04.080 --> 00:09:06.100 think, if anything, I might have just confused 00:09:06.100 --> 00:09:08.120 you with this one. 00:09:08.120 --> 00:09:11.680 So, I'll see you in the part 2 of using the 00:09:11.680 --> 00:09:12.150 quadratic equation.

Quadratic equation part 2

https://www.youtube.com/watch?v=y19jYxzY8Y8

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en

WEBVTT Kind: captions Language: en 00:00:00.890 --> 00:00:03.590 Welcome to part two of the presentation on 00:00:03.590 --> 00:00:05.660 quadratic equations. 00:00:05.660 --> 00:00:08.470 Well, I think I thoroughly confused you the last time 00:00:08.470 --> 00:00:11.170 around, so let me see if I can fix that a bit by doing 00:00:11.170 --> 00:00:12.770 several more examples. 00:00:12.770 --> 00:00:15.430 So let's just start with a review of what the 00:00:15.430 --> 00:00:16.380 quadratic equation is. 00:00:16.380 --> 00:00:19.650 The quadratic equation says, if I'm trying to solve the 00:00:19.650 --> 00:00:31.590 equation Ax squared plus Bx plus C equals 0, then the 00:00:31.590 --> 00:00:35.440 solution or the solutions because there's usually two 00:00:35.440 --> 00:00:38.970 times that it intersects the x-axis, or two solutions for 00:00:38.970 --> 00:00:47.610 this equation is x equals minus B plus or minus the square root 00:00:47.610 --> 00:00:56.390 of B squared minus 4 times A times C. 00:00:56.390 --> 00:01:00.270 And all of that over 2A. 00:01:00.270 --> 00:01:02.040 So let's do a problem and hopefully this should make 00:01:02.040 --> 00:01:02.690 a little more sense. 00:01:02.690 --> 00:01:04.620 That's a 2 on the bottom. 00:01:04.620 --> 00:01:13.890 So let's say I had the equation minus 9x squared minus 00:01:13.890 --> 00:01:19.950 9x plus 6 equals 0. 00:01:19.950 --> 00:01:22.230 So in this example what's A? 00:01:22.230 --> 00:01:25.410 Well, A is the coefficient on the x squared term. 00:01:25.410 --> 00:01:29.820 The x squared term is here, the coefficient is minus 9. 00:01:29.820 --> 00:01:30.620 So let's write that. 00:01:30.620 --> 00:01:34.120 A equals minus 9. 00:01:34.120 --> 00:01:35.400 What does B equal? 00:01:35.400 --> 00:01:39.180 B is the coefficient on the x term, so that's this term here. 00:01:39.180 --> 00:01:43.220 So B is also equal to minus 9. 00:01:43.220 --> 00:01:47.140 And C is the constant term, which in this example is 6. 00:01:47.140 --> 00:01:49.550 So C is equal to 6. 00:01:49.550 --> 00:01:52.070 Now we just substitute these values into the actual 00:01:52.070 --> 00:01:53.260 quadratic equation. 00:01:53.260 --> 00:01:59.600 So negative B, so it's negative times negative 9. 00:01:59.600 --> 00:02:00.780 That's B. 00:02:00.780 --> 00:02:08.110 Plus or minus the square root of B squared, so that's 81. 00:02:08.110 --> 00:02:08.390 Right? 00:02:08.390 --> 00:02:10.030 Negative 9 squared. 00:02:10.030 --> 00:02:14.720 Minus 4 times negative 9. 00:02:14.720 --> 00:02:16.140 That's A. 00:02:16.140 --> 00:02:19.480 Times C, which is 6. 00:02:19.480 --> 00:02:23.950 And all of that over 2 times negative 9, which 00:02:23.950 --> 00:02:25.630 is minus 18, right? 00:02:25.630 --> 00:02:26.720 2 times negative 9-- 2A. 00:02:29.230 --> 00:02:33.760 Let's try to simplify this up here. 00:02:33.760 --> 00:02:37.930 Well, negative negative 9, that's positive 9. 00:02:37.930 --> 00:02:46.480 Plus or minus the square root of 81. 00:02:46.480 --> 00:02:47.900 Let's see. 00:02:47.900 --> 00:02:50.270 This is negative 4 times negative 9. 00:02:50.270 --> 00:02:53.470 Negative 4 times negative 9 is positive 36. 00:02:53.470 --> 00:02:58.310 And then positive 36 times 6 is-- let's see. 00:02:58.310 --> 00:03:01.330 30 times 6 is 180. 00:03:01.330 --> 00:03:07.890 And then 180 plus another 36 is 216. 00:03:07.890 --> 00:03:10.980 Plus 216, is that right? 00:03:10.980 --> 00:03:14.490 180 plus 36 is 216. 00:03:14.490 --> 00:03:16.840 All of that over 2A. 00:03:16.840 --> 00:03:19.570 2A we already said is minus 19. 00:03:19.570 --> 00:03:20.740 So we simplify that more. 00:03:20.740 --> 00:03:28.090 That's 9 plus or minus the square root 81 plus 216. 00:03:28.090 --> 00:03:30.400 That's 80 plus 217. 00:03:30.400 --> 00:03:38.040 That's 297. 00:03:38.040 --> 00:03:41.900 And all of that over minus 18. 00:03:41.900 --> 00:03:45.020 Now, this is actually-- the hardest part with the quadratic 00:03:45.020 --> 00:03:47.720 equation is oftentimes just simplifying this expression. 00:03:47.720 --> 00:03:50.860 We have to figure out if we can simplify this radical. 00:03:50.860 --> 00:03:53.090 Well, let's see. 00:03:53.090 --> 00:03:56.490 One way to figure out if a number is divisible by 9 is to 00:03:56.490 --> 00:03:58.320 actually add up the digits and see if the digits 00:03:58.320 --> 00:03:59.260 are divisible by 9. 00:03:59.260 --> 00:03:59.950 In this case, it is. 00:03:59.950 --> 00:04:02.510 2 plus 9 plus 7 is equal to 18. 00:04:02.510 --> 00:04:04.600 So let's see how many times 9 goes into that. 00:04:04.600 --> 00:04:07.150 I'll do it on the side here; I don't want to be too messy. 00:04:07.150 --> 00:04:09.450 9 goes into 2 97. 00:04:13.630 --> 00:04:16.190 3 times 27. 00:04:16.190 --> 00:04:19.040 27-- it goes 33 times, right? 00:04:19.040 --> 00:04:24.290 So this is the same thing as 9 plus or minus the square root 00:04:24.290 --> 00:04:31.110 of 9 times 33 over minus 18. 00:04:31.110 --> 00:04:32.470 And 9 is a perfect square. 00:04:32.470 --> 00:04:34.650 That's why I actually wanted to see if 9 would work because 00:04:34.650 --> 00:04:36.390 that's the only way I could get it out of the radical, if 00:04:36.390 --> 00:04:37.390 it's a perfect square. 00:04:37.390 --> 00:04:40.410 As you learned in that exponent rules number one module. 00:04:40.410 --> 00:04:46.140 So this is equal to 9 plus or minus 3 times the square 00:04:46.140 --> 00:04:53.230 root of 33, and all of that over minus 18. 00:04:53.230 --> 00:04:54.570 We're almost done. 00:04:54.570 --> 00:04:57.840 We can actually simplify it because 9, 3, and minus 18 00:04:57.840 --> 00:05:00.650 are all divisible by 3. 00:05:00.650 --> 00:05:02.270 Let's divide everything by 3. 00:05:02.270 --> 00:05:14.370 3 plus or minus the square root of 33 over minus 6. 00:05:14.370 --> 00:05:15.610 And we are done. 00:05:15.610 --> 00:05:17.010 So as you see, the hardest thing with the quadratic 00:05:17.010 --> 00:05:20.110 equation is often just simplifying the expression. 00:05:20.110 --> 00:05:22.750 But what we've said, I know you might have lost track-- we did 00:05:22.750 --> 00:05:27.120 all this math-- is we said, this equation: minus 9x 00:05:27.120 --> 00:05:30.550 squared minus 9x plus 6. 00:05:30.550 --> 00:05:34.200 Now we found two x values that would satisfy this equation 00:05:34.200 --> 00:05:35.970 and make it equal to 0. 00:05:35.970 --> 00:05:39.830 One x value is x equals 3 plus the square root 00:05:39.830 --> 00:05:42.100 of 33 over minus 6. 00:05:42.100 --> 00:05:45.860 And the second value is 3 minus the square root 00:05:45.860 --> 00:05:50.160 of 33 over minus 6. 00:05:50.160 --> 00:05:52.250 And you might want to think about why we have 00:05:52.250 --> 00:05:53.370 that plus or minus. 00:05:53.370 --> 00:05:55.490 We have that plus or minus because a square root could 00:05:55.490 --> 00:05:59.550 actually be a positive or a negative number. 00:05:59.550 --> 00:06:02.180 Let's do another problem. 00:06:02.180 --> 00:06:05.890 Hopefully this one will be a little bit simpler. 00:06:09.210 --> 00:06:16.780 Let's say I wanted to solve minus 8x squared 00:06:16.780 --> 00:06:21.000 plus 5x plus 9. 00:06:21.000 --> 00:06:23.150 Now I'm going to assume that you've memorized the quadratic 00:06:23.150 --> 00:06:25.310 equation because that's something you should do. 00:06:25.310 --> 00:06:26.630 Or you should write it down on a piece of paper. 00:06:26.630 --> 00:06:31.630 But the quadratic equation is negative B-- So b is 5, right? 00:06:31.630 --> 00:06:34.160 We're trying to solve that equal to 0, so negative B. 00:06:34.160 --> 00:06:39.790 So negative 5, plus or minus the square root of B squared- 00:06:39.790 --> 00:06:44.030 that's 5 squared, 25. 00:06:44.030 --> 00:06:50.470 Minus 4 times A, which is minus 8. 00:06:50.470 --> 00:06:53.820 Times C, which is 9. 00:06:53.820 --> 00:06:56.400 And all of that over 2 times A. 00:06:56.400 --> 00:07:00.320 Well, A is minus 8, so all of that is over minus 16. 00:07:00.320 --> 00:07:04.090 So let's simplify this expression up here. 00:07:04.090 --> 00:07:09.440 Well, that's equal to minus 5 plus or minus 00:07:09.440 --> 00:07:13.630 the square root of 25. 00:07:13.630 --> 00:07:14.620 Let's see. 00:07:14.620 --> 00:07:18.220 4 times 8 is 32 and the negatives cancel out, so 00:07:18.220 --> 00:07:21.520 that's positive 32 times 9. 00:07:21.520 --> 00:07:24.480 Positive 32 times 9, let's see. 00:07:24.480 --> 00:07:26.720 30 times 9 is 270. 00:07:26.720 --> 00:07:31.110 It's 288. 00:07:31.110 --> 00:07:31.570 I think. 00:07:31.570 --> 00:07:31.800 Right? 00:07:36.130 --> 00:07:37.490 288. 00:07:37.490 --> 00:07:40.590 We have all of that over minus 16. 00:07:40.590 --> 00:07:42.560 Now simplify it more. 00:07:42.560 --> 00:07:47.760 Minus 5 plus or minus the square root-- 25 plus 00:07:47.760 --> 00:07:51.340 288 is 313 I believe. 00:07:56.950 --> 00:08:00.230 And all of that over minus 16. 00:08:00.230 --> 00:08:03.430 And I think, I'm not 100% sure, although I'm pretty sure. 00:08:03.430 --> 00:08:04.570 I haven't checked it. 00:08:04.570 --> 00:08:10.370 That 313 can't be factored into a product of a perfect 00:08:10.370 --> 00:08:11.690 square and another number. 00:08:11.690 --> 00:08:13.670 In fact, it actually might be a prime number. 00:08:13.670 --> 00:08:15.600 That's something that you might want to check out. 00:08:15.600 --> 00:08:18.200 So if that is the case and we've got it in completely 00:08:18.200 --> 00:08:21.840 simplified form, and we say there are two solutions, two 00:08:21.840 --> 00:08:24.940 x values that will make this equation true. 00:08:24.940 --> 00:08:30.750 One of them is x is equal to minus 5 plus the square 00:08:30.750 --> 00:08:35.830 root of 313 over minus 16. 00:08:35.830 --> 00:08:44.110 And the other one is x is equal to minus 5 minus the square 00:08:44.110 --> 00:08:49.660 root of 313 over minus 16. 00:08:49.660 --> 00:08:51.760 Hopefully those two examples will give you a good 00:08:51.760 --> 00:08:53.940 sense of how to use the quadratic equation. 00:08:53.940 --> 00:08:55.860 I might add some more modules. 00:08:55.860 --> 00:08:58.230 And then, once you master this, I'll actually teach you how to 00:08:58.230 --> 00:09:00.370 solve quadratic equations when you actually get a negative 00:09:00.370 --> 00:09:01.910 number under the radical. 00:09:01.910 --> 00:09:03.140 Very interesting. 00:09:03.140 --> 00:09:06.760 Anyway, I hope you can do the module now and maybe I'll add a 00:09:06.760 --> 00:09:10.370 few more presentations because this isn't the easiest module. 00:09:10.370 --> 00:09:11.840 But I hope you have fun. 00:09:11.840 --> 00:09:13.140 Bye.

i and Imaginary numbers

https://www.youtube.com/watch?v=rDLDGQMKT3M

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https://www.youtube.com/api/timedtext?v=rDLDGQMKT3M&ei=f2eUZZ6FKtqMmLAPpOaA6A0&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249839&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=074B6B327C744323EE4A5C12D1AF7A83D4E9B2F9.107D5FF9E126DA36C0C80816A22C87F2160569A9&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.970 --> 00:00:04.540 Welcome to the presentation on i and imaginary numbers. 00:00:04.540 --> 00:00:08.200 So let me just start out with a definition. 00:00:08.200 --> 00:00:14.340 i is equal to the square root of negative 1. 00:00:14.340 --> 00:00:18.790 Or to view it another way, you could say that i squared 00:00:18.790 --> 00:00:22.030 is equal to negative 1. 00:00:22.030 --> 00:00:23.260 Now why is this special? 00:00:23.260 --> 00:00:27.440 Well, we knew or we've learned that any number when you 00:00:27.440 --> 00:00:29.680 square it is equal to a positive number, right? 00:00:29.680 --> 00:00:32.820 If I had negative 1 times negative 1, that 00:00:32.820 --> 00:00:34.460 equals positive 1. 00:00:34.460 --> 00:00:36.150 You don't have to write the positive every time but that 00:00:36.150 --> 00:00:38.780 equal positive 1 and so does 1 times 1, that 00:00:38.780 --> 00:00:40.570 equals positive 1. 00:00:40.570 --> 00:00:46.260 So if we think about the square root of a number, so far all 00:00:46.260 --> 00:00:48.080 we've learned is taking the square roots of positive 00:00:48.080 --> 00:00:52.880 numbers, and that makes sense to us because the notion of a 00:00:52.880 --> 00:00:54.790 square root of a negative number didn't really 00:00:54.790 --> 00:00:56.860 exist until now. 00:00:56.860 --> 00:01:00.960 So what we've done is we've set this definition that the number 00:01:00.960 --> 00:01:04.760 i, and i isn't a variable, it's an actual number. 00:01:04.760 --> 00:01:09.440 It's value is equal to the square root of negative 1. 00:01:09.440 --> 00:01:14.160 Now I won't go into all of the philosophical musings about 00:01:14.160 --> 00:01:18.570 whether i, as a number i, or any imaginary numbers 00:01:18.570 --> 00:01:20.780 actually exist. 00:01:20.780 --> 00:01:22.500 Maybe I'll make another presentation on that. 00:01:22.500 --> 00:01:27.390 But they exist enough to be very useful to many 00:01:27.390 --> 00:01:28.530 engineers and physicists. 00:01:28.530 --> 00:01:34.240 So I'll leave you with that, and I'll also just hint at -- 00:01:34.240 --> 00:01:36.400 well, I won't go into the whole e to the i pi 00:01:36.400 --> 00:01:37.170 equals negative 1. 00:01:37.170 --> 00:01:40.360 But that blows my mind, but I won't go into that. 00:01:40.360 --> 00:01:44.030 And when you think about whether i really exists, you 00:01:44.030 --> 00:01:47.900 should also think about whether anything really exists. 00:01:47.900 --> 00:01:51.820 So, I've diversed for too long, so let's get back to what I was 00:01:51.820 --> 00:01:53.750 saying before. i is equal to the square root of 00:01:53.750 --> 00:01:57.030 negative 1, and i squared equals negative 1. 00:01:57.030 --> 00:01:59.800 So let's think about the implications of this. 00:01:59.800 --> 00:02:04.730 If I were to say -- well, i to the first power, just like 00:02:04.730 --> 00:02:07.580 anything else, is equal to itself, right? 00:02:07.580 --> 00:02:13.920 i squared, I've already said using this definition, i 00:02:13.920 --> 00:02:16.910 squared is equal to negative 1. 00:02:16.910 --> 00:02:19.750 i to the third power, well that would just equal i 00:02:19.750 --> 00:02:22.890 squared times i, right? 00:02:22.890 --> 00:02:26.560 And i squared is negative 1, so it would be negative 1 times i. 00:02:26.560 --> 00:02:30.380 And that just equals negative i. 00:02:30.380 --> 00:02:39.330 And i to the fourth would equal i to the third times i. 00:02:39.330 --> 00:02:41.580 I'm just using my exponent rules here. 00:02:41.580 --> 00:02:48.430 Well i to the third is negative i times i, right? i to the 00:02:48.430 --> 00:02:52.460 third is negative i, and then we just kept that i. 00:02:52.460 --> 00:02:54.910 Well what's negative i times i? 00:02:54.910 --> 00:02:57.780 Well that's the same thing as negative 1 times 00:02:57.780 --> 00:03:00.660 i times i, right? 00:03:00.660 --> 00:03:02.690 And what's i times i? 00:03:02.690 --> 00:03:04.020 Well, the definition. 00:03:04.020 --> 00:03:07.330 i times i, i squared, is equal to negative 1. 00:03:07.330 --> 00:03:12.710 So that equals negative 1 times negative 1, which equals 1. 00:03:12.710 --> 00:03:13.880 Interesting. 00:03:13.880 --> 00:03:15.260 Let me clean this up a little bit. 00:03:18.370 --> 00:03:20.620 Actually let me start with i to the zero power. 00:03:20.620 --> 00:03:22.750 i to the zero power, well we know anything to the zero 00:03:22.750 --> 00:03:23.690 power is equal to 1. 00:03:23.690 --> 00:03:27.250 So we'll keep that -- that still equals 1. 00:03:27.250 --> 00:03:31.490 i to the first power is equal to i. 00:03:31.490 --> 00:03:36.470 i squared by definition is equal to negative 1. 00:03:36.470 --> 00:03:40.710 i to the third power, I just showed you, is equal to 00:03:40.710 --> 00:03:43.780 negative i, and that makes sense because that's 00:03:43.780 --> 00:03:46.160 just i squared times i. 00:03:46.160 --> 00:03:51.930 And i to the fourth power equaled 1 again. 00:03:51.930 --> 00:03:56.500 If I did i to the fifth power, well that's just equal to i to 00:03:56.500 --> 00:04:01.110 the fourth power times i, right? 00:04:01.110 --> 00:04:04.320 I'll write that down, i to the fourth times i. 00:04:07.640 --> 00:04:11.260 i to the fourth is equal to 1, right? 00:04:11.260 --> 00:04:14.770 This is equal to 1, so 1 times i is equal to i. 00:04:14.770 --> 00:04:16.760 Do you see a pattern here? 00:04:16.760 --> 00:04:19.010 i to the zero power equals 1. 00:04:19.010 --> 00:04:21.240 i to the negative 1 power equals i. 00:04:21.240 --> 00:04:24.340 i squared is equal to negative 1. 00:04:24.340 --> 00:04:27.350 i to the third power equals negative i. 00:04:27.350 --> 00:04:29.510 i the fourth equals 1 again. 00:04:33.130 --> 00:04:38.740 So i to the fourth equals i to zero, and i to the one power 00:04:38.740 --> 00:04:41.280 is equal to i to the fifth. 00:04:41.280 --> 00:04:45.640 I think you'll find out that i to the zero, and you could try 00:04:45.640 --> 00:04:49.410 this out if you don't believe me, i to the zero is equal to i 00:04:49.410 --> 00:04:55.645 to the fourth, which equals i to the eighth, which equals i 00:04:55.645 --> 00:04:57.250 to the twelfth, I think you see the pattern. 00:04:57.250 --> 00:05:03.950 Any multiple of 4 equals 1. 00:05:03.950 --> 00:05:11.610 And i to the first equals i to the fifth, equals i to the 00:05:11.610 --> 00:05:16.900 ninth equals i to the thirteenth, equals i. 00:05:16.900 --> 00:05:20.915 So that's i to any power that is a multiple 00:05:20.915 --> 00:05:23.110 of 4 plus 1, right? 00:05:23.110 --> 00:05:27.740 Because 5 is equal to 4 plus 1, nine is equal to 8 plus 1. 00:05:27.740 --> 00:05:30.790 And we could do a similar pattern. i squared is equal to 00:05:30.790 --> 00:05:36.500 i to the sixth, equals i to the tenth, and so on, and 00:05:36.500 --> 00:05:39.190 that equals negative 1. 00:05:39.190 --> 00:05:43.792 And finally i to the third is equal to i to the seventh, 00:05:43.792 --> 00:05:50.990 which equals i to the eleventh, and so on, equals negative i. 00:05:50.990 --> 00:05:52.060 So why is this useful? 00:05:52.060 --> 00:05:53.190 We see a pattern. 00:05:53.190 --> 00:05:55.300 It's a cycle of four. 00:05:55.300 --> 00:05:57.940 In this pattern, if we look at this, we can use 00:05:57.940 --> 00:06:00.420 this to determine what i to any power is. 00:06:00.420 --> 00:06:06.460 So if I were to ask you what i to the hundredth power is, 00:06:06.460 --> 00:06:08.860 well, you could just work it out, you could say, well that's 00:06:08.860 --> 00:06:10.810 just equal to i times i to the ninety-ninth and 00:06:10.810 --> 00:06:11.500 so far so down. 00:06:11.500 --> 00:06:14.390 But if we use the cycle, we see that hundred is 00:06:14.390 --> 00:06:17.490 a multiple of 4, right? 00:06:17.490 --> 00:06:19.680 4 times 25 is 100. 00:06:19.680 --> 00:06:22.640 So i to the hundredth will fall into this category, 00:06:22.640 --> 00:06:23.310 this first one. 00:06:23.310 --> 00:06:24.570 It's a multiple of 4. 00:06:24.570 --> 00:06:28.000 So we know that i to the hundredth power is equal to 1. 00:06:28.000 --> 00:06:32.230 Similarly, if I said i to the hundred and one power, that's 00:06:32.230 --> 00:06:33.820 going to equal i, right? 00:06:33.820 --> 00:06:37.650 Because that equals 100 plus 1. 00:06:37.650 --> 00:06:39.500 So it puts you into this category. 00:06:39.500 --> 00:06:43.920 Equal to a multiple of 4, 100 is a multiple of 4, and 101 00:06:43.920 --> 00:06:47.440 is a multiple of 4 plus 1. 00:06:47.440 --> 00:06:51.330 i to the hundred and second power, similarly would 00:06:51.330 --> 00:06:54.120 equal negative 1. 00:06:54.120 --> 00:06:59.640 i to the hundred and third power would equal negative i. 00:06:59.640 --> 00:07:03.100 I hope you understand what I'm doing here, and all I did is I 00:07:03.100 --> 00:07:05.670 defined i as the square root of negative 1, and then I 00:07:05.670 --> 00:07:08.560 kept multiplying i to figure out a pattern. 00:07:08.560 --> 00:07:12.270 I said i to the zero equals 1, i to the first equals i, i 00:07:12.270 --> 00:07:16.170 squared equals negative 1, i to the third equals negative i, 00:07:16.170 --> 00:07:18.080 and i to the fourth equals 1 again. 00:07:18.080 --> 00:07:19.980 And the pattern repeated itself. 00:07:19.980 --> 00:07:23.570 And then I used that pattern to be able to figure out i to any 00:07:23.570 --> 00:07:26.150 power, even it's a very high number. 00:07:26.150 --> 00:07:30.600 So a very simple way to think about it is if I had i to the 00:07:30.600 --> 00:07:33.670 three hundred and twenty three. 00:07:33.670 --> 00:07:38.300 What I do is I say if I were to divide 4 into 323, 00:07:38.300 --> 00:07:39.930 what's the remainder? 00:07:39.930 --> 00:07:42.770 Well, I know 4 goes into 320, right? 00:07:42.770 --> 00:07:45.320 4 times 80 is 320. 00:07:45.320 --> 00:07:56.180 So I know that when I divide 4 into 323, so 4 goes into 323 80 00:07:56.180 --> 00:07:58.940 times with a remainder of 3, right? 00:07:58.940 --> 00:08:00.560 And the remainder is what we care about. 00:08:00.560 --> 00:08:03.370 And this number is actually called a modulus. 00:08:03.370 --> 00:08:05.440 Maybe I'll do another module on modulus -- it's very important 00:08:05.440 --> 00:08:08.040 actually in computer programming. 00:08:08.040 --> 00:08:10.580 But since we know that when you divide this exponent by 4 the 00:08:10.580 --> 00:08:13.630 remainder is 3, we can say that this is the same thing as i to 00:08:13.630 --> 00:08:17.490 the third, which we've learned is negative i. 00:08:17.490 --> 00:08:22.410 Similarly, if I said i to the five hundred and second 00:08:22.410 --> 00:08:27.220 power, well I know 500 is divisible by 4, right? 00:08:27.220 --> 00:08:29.520 4 times 125 is 500. 00:08:29.520 --> 00:08:32.910 So the remainder is 2 if I were divide it by 4. 00:08:32.910 --> 00:08:36.410 So I could say that this is the same thing as i squared. 00:08:36.410 --> 00:08:41.350 And i squared we learned by definition is negative 1. 00:08:41.350 --> 00:08:46.150 If I were to ask you i to the thirty-seven? 00:08:46.150 --> 00:08:51.510 We know 36 was divisible by 4, so the remainder is 1. 00:08:51.510 --> 00:08:55.670 So it would be i to the 1, which equals i. 00:08:55.670 --> 00:08:58.270 Hopefully that gives you an indication of what i is. 00:08:58.270 --> 00:08:59.970 It might have been confusing the first time because we're 00:08:59.970 --> 00:09:03.940 dealing with a number that's "imaginary," and I'm teaching 00:09:03.940 --> 00:09:05.960 the cycle property of it. 00:09:05.960 --> 00:09:08.710 What you might want to do is review the video again, but 00:09:08.710 --> 00:09:10.990 then after that you could just try the module on i, which 00:09:10.990 --> 00:09:12.930 essentially just keeps working you through this 00:09:12.930 --> 00:09:14.070 type of problem. 00:09:14.070 --> 00:09:15.150 I hope you have fun. 00:09:15.150 --> 00:09:16.450 Bye.

Simplifying radicals

https://www.youtube.com/watch?v=6QJtWfIiyZo

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en

WEBVTT Kind: captions Language: en 00:00:01.290 --> 00:00:04.270 Welcome to the presentation on simplifying radicals. 00:00:04.270 --> 00:00:05.890 So let's get started with a little terminology 00:00:05.890 --> 00:00:06.490 out of the way. 00:00:06.490 --> 00:00:08.310 You're probably just wondering what a radical is and 00:00:08.310 --> 00:00:09.880 I'll just let you know. 00:00:13.190 --> 00:00:15.280 A radical is just that. 00:00:15.280 --> 00:00:17.470 Or you're probably more familiar calling that 00:00:17.470 --> 00:00:18.830 the square root symbol. 00:00:18.830 --> 00:00:21.680 So with the terminology out of the way, let's actually talk 00:00:21.680 --> 00:00:23.800 about what it means to simplify a radical. 00:00:23.800 --> 00:00:25.020 And some people would argue that what we're going to 00:00:25.020 --> 00:00:26.890 actually be doing is actually making it more complicated. 00:00:26.890 --> 00:00:28.140 But let's see. 00:00:32.390 --> 00:00:36.900 So if I were to give you the square root of 36. 00:00:36.900 --> 00:00:37.610 Hey, that's easy. 00:00:37.610 --> 00:00:41.790 That's just equal to 6 times 6 or you'd say the square 00:00:41.790 --> 00:00:43.850 root of 36 is just 6. 00:00:43.850 --> 00:00:50.390 Now, what if I asked you what the square root of 72 is? 00:00:50.390 --> 00:00:54.590 Well we know that 72 is 36 times 2. 00:00:54.590 --> 00:00:55.680 Let's write that. 00:00:55.680 --> 00:01:00.850 Square root of 72 is the same thing as the square 00:01:00.850 --> 00:01:03.310 root of 36 times 2. 00:01:07.920 --> 00:01:10.690 And the square root, if you remember from level 3 00:01:10.690 --> 00:01:13.520 exponents, square root is the same thing as something 00:01:13.520 --> 00:01:14.920 to the 1/2 power. 00:01:14.920 --> 00:01:15.860 So let's write it that way. 00:01:15.860 --> 00:01:18.710 And I'm just writing it this way just to show you how this 00:01:18.710 --> 00:01:20.580 radical simplification works and that it's really 00:01:20.580 --> 00:01:22.970 not a new concept. 00:01:22.970 --> 00:01:29.750 So this is the same thing as 36 times 2 to the 1/2 power. 00:01:29.750 --> 00:01:33.210 Because it's just a square root is the same thing as 1/2 power. 00:01:33.210 --> 00:01:35.860 And we learned from the exponent rules that when you 00:01:35.860 --> 00:01:38.890 multiply two numbers and then you raise that to the 1/2 00:01:38.890 --> 00:01:43.010 power, that that's the same thing as raising each of the 00:01:43.010 --> 00:01:47.500 numbers to the 1/2 power and then multiplying. 00:01:50.420 --> 00:01:53.330 Well that right there, that's the same thing as saying the 00:01:53.330 --> 00:01:58.480 square root is 36 times the square root of 2. 00:01:58.480 --> 00:02:00.780 And we already figured out what the square root of 36 is. 00:02:00.780 --> 00:02:01.810 It's 6. 00:02:01.810 --> 00:02:07.430 So that just equals 6 times the square root of 2. 00:02:07.430 --> 00:02:10.060 And you're probably wondering why I went through this step 00:02:10.060 --> 00:02:12.340 of changing the radical, the square root symbol, 00:02:12.340 --> 00:02:13.530 into the 1/2 power. 00:02:13.530 --> 00:02:16.160 And I did that just to show you that this is just an extension 00:02:16.160 --> 00:02:17.030 of the exponent rules. 00:02:17.030 --> 00:02:22.450 It isn't really a new concept, although, I guess sometimes 00:02:22.450 --> 00:02:24.690 it's not so obvious that they are the same concepts. 00:02:24.690 --> 00:02:26.480 I just wanted to point that out. 00:02:26.480 --> 00:02:28.470 So let's do another problem. 00:02:28.470 --> 00:02:30.820 I think as we do more and more problems, these will 00:02:30.820 --> 00:02:33.260 become more obvious. 00:02:33.260 --> 00:02:37.820 The square root of 50. 00:02:37.820 --> 00:02:40.920 Well, the square root of 50 -- 50 is the same 00:02:40.920 --> 00:02:47.150 thing as 25 times 2. 00:02:47.150 --> 00:02:49.790 And we know, based on what we just did and this is really 00:02:49.790 --> 00:02:53.670 just an exponent rule, square root of 25 times 2 is the same 00:02:53.670 --> 00:03:01.070 thing as the square root of 25 times the square root of 2. 00:03:01.070 --> 00:03:02.580 Well we know what the square root of 25 is. 00:03:02.580 --> 00:03:03.170 That's 5. 00:03:03.170 --> 00:03:09.700 So that just equals 5 times the square root of 2. 00:03:09.700 --> 00:03:14.610 Now, you might be saying, "Hey, Sal, you make it look easy, but 00:03:14.610 --> 00:03:19.210 how did you know to split 50 into 25 and 2?" Why didn't I 00:03:19.210 --> 00:03:24.010 say that 50 is equal to the square root of 5 and 10 or that 00:03:24.010 --> 00:03:28.800 50 is equal to the square root -- actually, I think 1 and 50? 00:03:28.800 --> 00:03:30.160 I don't know what other factors is 50. 00:03:30.160 --> 00:03:32.570 Well, anyway, I won't go into that right now. 00:03:32.570 --> 00:03:35.590 The reason why I picked 25 and 2 is because I wanted a factor 00:03:35.590 --> 00:03:39.640 of 50 -- I actually wanted the largest factor of 50 that 00:03:39.640 --> 00:03:40.880 is a perfect square. 00:03:40.880 --> 00:03:42.860 And that's 25. 00:03:42.860 --> 00:03:44.970 If I had done 5 and 10, there's really nothing I could have 00:03:44.970 --> 00:03:47.900 done with it because neither 5 nor 10 are perfect squares and 00:03:47.900 --> 00:03:50.610 same thing's with 1 and 50. 00:03:50.610 --> 00:03:52.270 So the way you should think about it, think about the 00:03:52.270 --> 00:03:55.960 factors of the original number and figure out if any of those 00:03:55.960 --> 00:03:57.890 factors are perfect squares. 00:03:57.890 --> 00:03:59.370 And there's no real mechanical way. 00:03:59.370 --> 00:04:02.280 You really just have to learn to recognize perfect squares. 00:04:02.280 --> 00:04:03.940 And you'll get familiar with them, of course. 00:04:03.940 --> 00:04:17.150 They're 1, 4, 9, 25, 16, 25, 36, 49, 64, et cetera. 00:04:17.150 --> 00:04:19.980 And maybe by doing this module, you'll actually learn to 00:04:19.980 --> 00:04:21.290 recognize them more readily. 00:04:21.290 --> 00:04:25.930 But if any of these numbers are a factor of the number under 00:04:25.930 --> 00:04:27.360 the radical sign, then you'll probably want to 00:04:27.360 --> 00:04:28.020 factor them out. 00:04:28.020 --> 00:04:30.130 And then you can take them out of the radical sign like 00:04:30.130 --> 00:04:32.620 we did up in this problem. 00:04:32.620 --> 00:04:33.730 Let's do a couple more. 00:04:37.640 --> 00:04:43.470 What is 7 times the square root of 27? 00:04:43.470 --> 00:04:45.420 And when I write the 7 right next to it, that just means 00:04:45.420 --> 00:04:47.510 times the square root of 27. 00:04:47.510 --> 00:04:50.665 Well, let's think about what other factors of 27 and 00:04:50.665 --> 00:04:52.050 whether any of them are a perfect square. 00:04:52.050 --> 00:04:56.710 Well, 3 is a factor of 27, but that's not a perfect square. 00:04:56.710 --> 00:04:58.260 9 is. 00:04:58.260 --> 00:05:02.800 So, we could say 7 -- that's equal to 7 times the 00:05:02.800 --> 00:05:08.490 square root of 9 times 3. 00:05:08.490 --> 00:05:11.860 And now, based on the rules we just learned, that's the same 00:05:11.860 --> 00:05:18.580 thing as 7 times the square root of 9 times the 00:05:18.580 --> 00:05:21.080 square root of 3. 00:05:21.080 --> 00:05:25.230 Well that just equals 7 times 3 because the square root of 9 is 00:05:25.230 --> 00:05:29.270 3 times the square root of 3. 00:05:29.270 --> 00:05:34.670 That equals 21 times the square root of 3. 00:05:34.670 --> 00:05:35.830 Done. 00:05:35.830 --> 00:05:37.710 Let's do another one. 00:05:37.710 --> 00:05:46.000 What is 9 times the square root of 18? 00:05:46.000 --> 00:05:48.210 Well once again, what are the factors of 18? 00:05:48.210 --> 00:05:50.010 Well do we have 6 and 3? 00:05:50.010 --> 00:05:52.280 1 and 18? 00:05:52.280 --> 00:05:54.550 None of the numbers I mentioned so far are perfect squares. 00:05:54.550 --> 00:05:56.540 But we also have 2 and 9. 00:05:56.540 --> 00:05:59.010 And 9 is a perfect square. 00:05:59.010 --> 00:05:59.770 Let's write that. 00:05:59.770 --> 00:06:07.020 That's equal to 9 times the square root of 2 times 9. 00:06:07.020 --> 00:06:11.560 Which is equal to 9 times the square root of 2 -- that's a 2 00:06:11.560 --> 00:06:15.580 -- times the square root of 9. 00:06:15.580 --> 00:06:20.110 Which equals 9 times the square root of 2 times 3, right? 00:06:20.110 --> 00:06:24.520 That's the square root of 9 which equals 27 times 00:06:24.520 --> 00:06:27.250 the square root of 2. 00:06:27.250 --> 00:06:28.130 There we go. 00:06:28.130 --> 00:06:30.160 Hopefully, you're starting to get the hang of these problems. 00:06:30.160 --> 00:06:33.070 Let's do another one. 00:06:33.070 --> 00:06:39.830 What is 4 times the square root of 25? 00:06:39.830 --> 00:06:43.000 Well 25 itself is a perfect square. 00:06:43.000 --> 00:06:45.620 This problem is so easy, it's a bit of a trick problem. 00:06:45.620 --> 00:06:47.130 25 itself is a perfect square. 00:06:47.130 --> 00:06:50.150 The square root is 5, so this is just equal to 4 times 00:06:50.150 --> 00:06:52.910 5, which is equal to 20. 00:06:52.910 --> 00:06:57.020 Square root of 25 is 5. 00:06:57.020 --> 00:06:58.220 Let's do another one. 00:06:58.220 --> 00:07:01.550 What's 3 times the square root of 29? 00:07:04.330 --> 00:07:06.190 Well 29 only has two factors. 00:07:06.190 --> 00:07:06.870 It's a prime number. 00:07:06.870 --> 00:07:09.450 It only has the factors 1 and 29. 00:07:09.450 --> 00:07:11.750 And neither of those numbers are perfect squares. 00:07:11.750 --> 00:07:14.220 So we really can't simplify this one anymore. 00:07:14.220 --> 00:07:19.340 So this is already in completely simplified form. 00:07:19.340 --> 00:07:20.480 Let's do a couple more. 00:07:23.780 --> 00:07:32.140 What about 7 times the square root of 320? 00:07:32.140 --> 00:07:35.700 Let's think about 320. 00:07:35.700 --> 00:07:38.320 Well we could actually do it in steps when we have 00:07:38.320 --> 00:07:39.810 larger numbers like this. 00:07:39.810 --> 00:07:43.290 I can look at it and say, well it does look like 4 -- actually 00:07:43.290 --> 00:07:47.200 it looks like 16 would go into this because 16 goes into 32. 00:07:47.200 --> 00:07:48.380 So let's try that. 00:07:48.380 --> 00:07:55.280 So that equals 7 times the square root of 16 times 20. 00:07:58.540 --> 00:08:04.310 That just equals 7 times the square root of 16 times 00:08:04.310 --> 00:08:06.960 the square root of 20. 00:08:06.960 --> 00:08:08.590 7 times the square root of 16. 00:08:08.590 --> 00:08:10.380 The square root of 16 is 4. 00:08:10.380 --> 00:08:11.630 So 7 times 4 is 28. 00:08:11.630 --> 00:08:17.110 So that's 28 times the square root of 20. 00:08:17.110 --> 00:08:19.100 Now are we done? 00:08:19.100 --> 00:08:21.800 Well actually, I think I can factor 20 even more because 00:08:21.800 --> 00:08:24.680 20 is equal to 4 times 5. 00:08:24.680 --> 00:08:28.650 So I can say this is equal to 28 times the square 00:08:28.650 --> 00:08:33.570 root of 4 times 5. 00:08:33.570 --> 00:08:38.270 The square root of 4 is 2 so that could just take the 2 out 00:08:38.270 --> 00:08:43.170 and that becomes 56 times the square root of 5. 00:08:43.170 --> 00:08:44.450 I hope that made sense to you. 00:08:44.450 --> 00:08:45.980 And this is actually a pretty important technique 00:08:45.980 --> 00:08:46.890 I just did here. 00:08:46.890 --> 00:08:49.060 Immediately when I look at 320. 00:08:49.060 --> 00:08:52.160 I don't know what the largest number is that goes into 320. 00:08:52.160 --> 00:08:54.150 It actually turns out that it's 64. 00:08:54.150 --> 00:08:56.250 But just looking at the number, I said, well I 00:08:56.250 --> 00:08:57.610 know that 4 goes into it. 00:08:57.610 --> 00:09:00.520 So I could have just pulled out 4 and then said, "Oh, that's 00:09:00.520 --> 00:09:02.090 equal to 4 times 80." And then I would have had 00:09:02.090 --> 00:09:03.210 to work with 80. 00:09:03.210 --> 00:09:05.760 In this case, I saw 32 and I was like, it looks like 16 goes 00:09:05.760 --> 00:09:09.260 into it and I factored out 16 first and when I took out the 00:09:09.260 --> 00:09:11.890 square root of 16, I multiplied the outside by 4 and 00:09:11.890 --> 00:09:13.160 that's how I got the 28. 00:09:13.160 --> 00:09:15.670 But then I reduced the number on the inside said, "Oh, well 00:09:15.670 --> 00:09:17.430 that still is divisible by a perfect square. 00:09:17.430 --> 00:09:20.270 It's still divisible by 4." And then I kept doing it until I 00:09:20.270 --> 00:09:25.250 was left with essentially, a prime number or a number that 00:09:25.250 --> 00:09:27.710 couldn't be reduced anymore under the radical. 00:09:27.710 --> 00:09:29.950 And it actually doesn't have to be prime. 00:09:29.950 --> 00:09:33.140 So hopefully, that gives you a good sense of how to do 00:09:33.140 --> 00:09:34.270 radical simplification. 00:09:34.270 --> 00:09:37.020 It's really just an extension of the exponent rules that 00:09:37.020 --> 00:09:40.180 you've already learned and hopefully as you do the module, 00:09:40.180 --> 00:09:41.890 you'll get good at it. 00:09:41.890 --> 00:09:43.420 Have fun.

Exponent rules part 2

https://www.youtube.com/watch?v=rEtuPhl6930

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en

WEBVTT Kind: captions Language: en 00:00:00.970 --> 00:00:04.640 Welcome to Part 2 on the presentation on Level 00:00:04.640 --> 00:00:06.110 1 exponent rules. 00:00:06.110 --> 00:00:07.960 So let's start off by reviewing the rules 00:00:07.960 --> 00:00:09.340 we've learned already. 00:00:09.340 --> 00:00:21.050 If I had 2 to the tenth times 2 to the fifth, we learned that 00:00:21.050 --> 00:00:25.130 since we're multiplying exponents with the same base, 00:00:25.130 --> 00:00:28.410 we can add the exponent, so this equals 2 to the fifteenth. 00:00:31.320 --> 00:00:37.510 We also learned that if it was 2 to the tenth over 2 to the 00:00:37.510 --> 00:00:41.690 fifth, we would actually subtract the exponents. 00:00:41.690 --> 00:00:46.150 So this would be 2 to the 10 minus 5, which 00:00:46.150 --> 00:00:48.680 equals 2 to the fifth. 00:00:48.680 --> 00:00:50.460 At the end of the last presentation, and I probably 00:00:50.460 --> 00:00:52.970 shouldn't have introduced it so fast, I introduced 00:00:52.970 --> 00:00:54.070 a new concept. 00:00:54.070 --> 00:01:02.740 What happens if I have 2 to the tenth to the fifth power? 00:01:02.740 --> 00:01:04.380 Well, let's think about what that means. 00:01:04.380 --> 00:01:07.340 When I raise something to the fifth power, that's just like 00:01:07.340 --> 00:01:14.240 saying 2 to the tenth times 2 to the tenth times 2 to the 00:01:14.240 --> 00:01:19.955 tenth times 2 to the tenth times 2 to the tenth, right? 00:01:19.955 --> 00:01:22.465 All I did is I took 2 to the tenth and I multiplied 00:01:22.465 --> 00:01:24.800 it by itself five times. 00:01:24.800 --> 00:01:26.370 That's the fifth power. 00:01:26.370 --> 00:01:30.550 Well, we know from this rule up here that we can add these 00:01:30.550 --> 00:01:33.280 exponents because they're all the same base. 00:01:33.280 --> 00:01:36.360 So if we add 10 plus 10 plus 10 plus 10 plus 00:01:36.360 --> 00:01:39.050 10, what do we get? 00:01:39.050 --> 00:01:42.480 Right, we get 2 to the fiftieth power. 00:01:42.480 --> 00:01:44.090 So essentially, what did we do here? 00:01:44.090 --> 00:01:50.420 All we did is we multiplied 10 times 5 to get 50. 00:01:50.420 --> 00:01:53.480 So that's our third exponent rule, that when I raise an 00:01:53.480 --> 00:01:56.620 exponent to a power and then I raise that whole expression to 00:01:56.620 --> 00:02:00.940 another power, I can multiply those two exponents. 00:02:00.940 --> 00:02:02.400 So let me give you another example. 00:02:02.400 --> 00:02:12.340 If I said 3 to the 7, and all of that to the negative 9, once 00:02:12.340 --> 00:02:17.460 again, all I do is I multiply the 7 and the negative 9, and 00:02:17.460 --> 00:02:23.200 I get 3 to the minus 63. 00:02:23.200 --> 00:02:28.670 So, you see, it works just as easily with negative numbers. 00:02:28.670 --> 00:02:33.465 So now, I'm going to teach you one final exponent property. 00:02:36.970 --> 00:02:45.740 Let's say I have 2 times 9, and I raise that whole thing 00:02:45.740 --> 00:02:48.710 to the hundredth power. 00:02:48.710 --> 00:02:53.540 It turns out of this is equal to 2 to the hundredth power 00:02:53.540 --> 00:02:56.890 times 9 to the hundredth power. 00:02:56.890 --> 00:02:58.180 Now let's make sure that that makes sense. 00:02:58.180 --> 00:02:59.880 Let's do it with a smaller example. 00:02:59.880 --> 00:03:06.190 What if it was 4 times 5 to the third power? 00:03:06.190 --> 00:03:15.440 Well, that would just be equal to 4 times 5 times 4 times 5 00:03:15.440 --> 00:03:21.890 times 4 times 5, right, which is the same thing as 4 times 4 00:03:21.890 --> 00:03:26.410 times 4 times 5 times 5 times 5, right? 00:03:26.410 --> 00:03:28.820 I just switched the order in which I'm multiplying, which 00:03:28.820 --> 00:03:30.730 you can do with multiplication. 00:03:30.730 --> 00:03:33.030 Well, 4 times 4 times 4, well, that's just equal 00:03:33.030 --> 00:03:34.640 to 4 to the third. 00:03:34.640 --> 00:03:39.220 And 5 times 5 times 5 is equal to 5 to the third. 00:03:39.220 --> 00:03:42.110 Hope that gives you a good intuition of why this 00:03:42.110 --> 00:03:43.350 property here is true. 00:03:43.350 --> 00:03:46.170 And actually, when I had first learned exponent rules, I would 00:03:46.170 --> 00:03:48.480 always forget the rules, and I would always do this proof 00:03:48.480 --> 00:03:50.750 myself, or the other proofs. 00:03:50.750 --> 00:03:52.700 And a proof is just an explanation of why the rule 00:03:52.700 --> 00:03:56.660 works, just to make sure that I was doing it right. 00:03:56.660 --> 00:04:00.080 So given everything that we've learned to now-- actually, let 00:04:00.080 --> 00:04:03.810 me review all of the rules again. 00:04:03.810 --> 00:04:10.390 If I have 2 to the seventh times 2 to the third, 00:04:10.390 --> 00:04:13.366 well, then I can add the exponents, 2 to the tenth. 00:04:13.366 --> 00:04:20.080 If I have 2 the seventh over 2 the third, well, here I 00:04:20.080 --> 00:04:24.580 subtract the exponents, and I get 2 to the fourth. 00:04:24.580 --> 00:04:30.910 If I have 2 to the seventh to the third power, well, here 00:04:30.910 --> 00:04:32.310 I multiplied the exponents. 00:04:32.310 --> 00:04:35.180 That gives you 2 to the 21. 00:04:35.180 --> 00:04:42.690 And if I had 2 times 7 to the third power, well, that equals 00:04:42.690 --> 00:04:47.650 2 to the third times 7 to the third. 00:04:47.650 --> 00:04:52.260 Now, let's use all of these rules we've learned to actually 00:04:52.260 --> 00:04:55.500 try to do some, what I would call, composite problems that 00:04:55.500 --> 00:04:58.630 involve you using multiple rules at the same time. 00:04:58.630 --> 00:05:00.630 And a good composite problem was that problem that I had 00:05:00.630 --> 00:05:03.145 introduced you to at the end of that last seminar. 00:05:06.600 --> 00:05:20.330 Let's say I have 3 squared times 9 to the eighth power, 00:05:20.330 --> 00:05:25.780 and all of that I'm going to raise to the negative 2 power. 00:05:25.780 --> 00:05:27.250 So what can I do here? 00:05:27.250 --> 00:05:32.970 Well, 3 and 9 are two separate bases, but 9 can actually 00:05:32.970 --> 00:05:35.760 be expressed as an exponent of 3, right? 00:05:35.760 --> 00:05:37.540 9 is the same thing as 3 squared, so let's 00:05:37.540 --> 00:05:40.210 rewrite 9 like that. 00:05:40.210 --> 00:05:44.650 That's equivalent to 3 squared times-- 9 is the same thing as 00:05:44.650 --> 00:05:50.520 3 squared to the eighth power, and then all of that to the 00:05:50.520 --> 00:05:52.350 negative 2 power, right? 00:05:52.350 --> 00:05:54.330 All I did is I replaced 9 with 3 squared because 00:05:54.330 --> 00:05:57.610 we know 3 times 3 is 9. 00:05:57.610 --> 00:06:00.070 Well, now we can use the multiplication rule on 00:06:00.070 --> 00:06:01.630 this to simplify it. 00:06:01.630 --> 00:06:09.500 So this is equal to 3 squared times 3 to the 2 times 8, 00:06:09.500 --> 00:06:15.230 which is 16, and all of that to the negative 2. 00:06:15.230 --> 00:06:16.610 Now, we can use the first rule. 00:06:16.610 --> 00:06:18.960 We have the same base, so we can add the exponents, and 00:06:18.960 --> 00:06:23.280 we're multiplying them, so that equals 3 to the eighteen power, 00:06:23.280 --> 00:06:28.210 right, 2 plus 16, and all that to the negative 2. 00:06:28.210 --> 00:06:29.150 And now we're almost done. 00:06:29.150 --> 00:06:31.870 We can once again use this multiplication rule, and we 00:06:31.870 --> 00:06:36.580 could say 3-- this is equal to 3 to the eighteenth times 00:06:36.580 --> 00:06:42.220 negative 2, so that's 3 to the minus 36. 00:06:42.220 --> 00:06:46.040 So this problem might have seemed pretty daunting at 00:06:46.040 --> 00:06:49.380 first, but there aren't that many rules, and all you have to 00:06:49.380 --> 00:06:51.580 do is keep seeing, oh, wow, that little part of the 00:06:51.580 --> 00:06:52.970 problem, I can simplify it. 00:06:52.970 --> 00:06:55.520 Then you simplify it, and you'll see that you can keep 00:06:55.520 --> 00:06:59.215 using rules until you get to a much simpler answer. 00:06:59.215 --> 00:07:02.290 And actually the Level 1 problems don't even involve 00:07:02.290 --> 00:07:03.340 problems this difficult. 00:07:03.340 --> 00:07:06.380 This'll be more on the exponent rules, Level 2. 00:07:06.380 --> 00:07:07.850 But I think at this point you're ready 00:07:07.850 --> 00:07:10.125 to try the problems. 00:07:12.860 --> 00:07:15.550 I'm kind of divided whether I want you to memorize the rules 00:07:15.550 --> 00:07:18.800 because I think it's better to almost forget the rules and 00:07:18.800 --> 00:07:20.740 have to prove it to yourself over and over again to 00:07:20.740 --> 00:07:22.010 the point that you remember the rules. 00:07:22.010 --> 00:07:25.400 Because if you just memorize the rules, later on in life 00:07:25.400 --> 00:07:27.905 when you haven't done it for a couple of years, you might kind 00:07:27.905 --> 00:07:29.340 of forget the rules, and then you won't know how 00:07:29.340 --> 00:07:30.590 to get back to the rules. 00:07:30.590 --> 00:07:31.880 But it's up to you. 00:07:31.880 --> 00:07:35.160 I just hope you do understand why these rules work, and as 00:07:35.160 --> 00:07:36.616 long as you practice and you pay attention to the signs, 00:07:36.616 --> 00:07:40.510 you should have no problems with the Level 1 exercises. 00:07:40.510 --> 00:07:42.010 Have fun!

Exponent rules part 1

https://www.youtube.com/watch?v=kITJ6qH7jS0

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WEBVTT Kind: captions Language: en 00:00:01.230 --> 00:00:05.600 Welcome to the presentation on level one exponent rules. 00:00:05.600 --> 00:00:08.150 Let's get started with some problems. 00:00:08.150 --> 00:00:12.870 So if I were to ask you what 2 -- that's a little fatter than 00:00:12.870 --> 00:00:15.080 I wanted it to be, but let's keep it fat so it doesn't look 00:00:15.080 --> 00:00:20.260 strange -- 2 the third times -- and dot is another way of 00:00:20.260 --> 00:00:23.230 saying times -- if I were to ask you what 2 to the third 00:00:23.230 --> 00:00:27.820 times 2 to the fifth is, how would you figure that out? 00:00:27.820 --> 00:00:30.610 Actually, let me use a skinnier pen because that does look bad. 00:00:30.610 --> 00:00:35.120 So, 2 to the third times 2 to the fifth. 00:00:35.120 --> 00:00:37.610 Well there's one way that I think you do know how to do it. 00:00:37.610 --> 00:00:42.150 You could figure out that 2 to the third is 9, and 00:00:42.150 --> 00:00:45.380 that 2 to the fifth is 32. 00:00:45.380 --> 00:00:46.840 And then you could multiply them. 00:00:46.840 --> 00:00:54.010 And 8 times 32 is 240, plus it's 256, right? 00:00:54.010 --> 00:00:55.530 You could do it that way. 00:00:55.530 --> 00:00:58.550 That's reasonable because it's not that hard to figure out 2 00:00:58.550 --> 00:01:00.520 to the third is and what 2 to the fifth is. 00:01:00.520 --> 00:01:03.150 But if those were much larger numbers this method might 00:01:03.150 --> 00:01:04.770 become a little difficult. 00:01:04.770 --> 00:01:08.520 So I'm going to show you using exponent rules you can actually 00:01:08.520 --> 00:01:12.340 multiply exponentials or exponent numbers without 00:01:12.340 --> 00:01:15.715 actually having to do as much arithmetic or actually you 00:01:15.715 --> 00:01:18.120 could handle numbers much larger than your normal math 00:01:18.120 --> 00:01:20.780 skills might allow you to. 00:01:20.780 --> 00:01:23.060 So let's just think what 2 to the third times 00:01:23.060 --> 00:01:24.670 2 to the fifth means. 00:01:24.670 --> 00:01:32.940 2 to the third is 2 times 2 times 2, right? 00:01:32.940 --> 00:01:35.200 And we're multiplying that times 2 to the fifth. 00:01:35.200 --> 00:01:43.160 And that's 2 times 2 times 2 times 2 times 2. 00:01:43.160 --> 00:01:44.200 So what do we have here? 00:01:44.200 --> 00:01:47.870 We have 2 times 2 times 2, times 2 times 2 times 00:01:47.870 --> 00:01:49.780 2 times 2 times 2. 00:01:49.780 --> 00:01:52.640 Really all we're doing is we're multiplying 2 how many times? 00:01:52.640 --> 00:01:58.920 Well, one, two, three, four, five, six, seven, eight. 00:01:58.920 --> 00:02:03.410 So that's the same thing as 2 to the eighth. 00:02:03.410 --> 00:02:05.050 Interesting. 00:02:05.050 --> 00:02:08.200 3 plus 5 is equal to 8. 00:02:08.200 --> 00:02:12.360 And that makes sense because 2 to the 3 is 2 multiplying by 00:02:12.360 --> 00:02:15.400 itself three times, to the fifth is 2 multiplying by 00:02:15.400 --> 00:02:17.540 itself five times, and then we're multiplying the two, so 00:02:17.540 --> 00:02:19.980 we're going to multiply 2 eight times. 00:02:19.980 --> 00:02:22.720 I hope I achieved my goal of confusing you just now. 00:02:22.720 --> 00:02:23.580 Let's do another one. 00:02:26.130 --> 00:02:33.780 If I said 7 squared times 7 to the fourth. 00:02:33.780 --> 00:02:36.550 That's a 4. 00:02:36.550 --> 00:02:42.180 Well, that equals 7 times 7, right, that's 7 squared, 00:02:42.180 --> 00:02:44.430 times and now let's do 7 to the fourth. 00:02:44.430 --> 00:02:50.090 7 times 7 times 7 times 7. 00:02:50.090 --> 00:02:53.780 Well now we're multiplying 7 by itself six times, so 00:02:53.780 --> 00:02:56.590 that equal 7 to the sixth. 00:02:56.590 --> 00:03:00.130 So in general, whenever I'm multiplying exponents of the 00:03:00.130 --> 00:03:04.620 same base, that's key, I can just add the exponents. 00:03:04.620 --> 00:03:12.520 So 7 to the hundredth power times 7 to the fiftieth 00:03:12.520 --> 00:03:15.440 power, and notice this is an example now. 00:03:15.440 --> 00:03:17.750 It would be very hard without a computer to figure out what 00:03:17.750 --> 00:03:19.320 7 to the hundredth power is. 00:03:19.320 --> 00:03:22.190 And likewise, very hard without a computer to figure out what 00:03:22.190 --> 00:03:24.050 7 to the fiftieth power is. 00:03:24.050 --> 00:03:32.730 But we could say that this is equal to 7 to the 100 plus 50, 00:03:32.730 --> 00:03:37.790 which is equal to 7 to the 150. 00:03:37.790 --> 00:03:40.430 Now I just want to give you a little bit of warning, make 00:03:40.430 --> 00:03:41.630 sure that you're multiplying. 00:03:41.630 --> 00:03:49.150 Because if I had 7 to the 100 plus 7 to the 50, there's 00:03:49.150 --> 00:03:50.590 actually very little I could do here. 00:03:50.590 --> 00:03:54.440 I couldn't simplify this number. 00:03:54.440 --> 00:03:56.710 But I'll throw out one to you. 00:03:56.710 --> 00:04:04.810 If I had 2 to the 8 times 2 to the 20, we know we 00:04:04.810 --> 00:04:06.570 can add these exponents. 00:04:06.570 --> 00:04:12.500 So that gives you 2 to the 28, right? 00:04:12.500 --> 00:04:20.820 What if I had 2 to the 8 plus 2 to the 8? 00:04:20.820 --> 00:04:22.890 This is a bit of a trick question. 00:04:22.890 --> 00:04:25.030 Well I just said if we're adding, we can't 00:04:25.030 --> 00:04:26.900 really do anything. 00:04:26.900 --> 00:04:28.530 We can't really simplify it. 00:04:28.530 --> 00:04:30.670 But there's a little trick here that we actually have 00:04:30.670 --> 00:04:32.980 two 2 to the 8, right? 00:04:32.980 --> 00:04:35.080 There's 2 to the 8 times 1, 2 to the 8 times 2. 00:04:35.080 --> 00:04:41.240 So this is the same thing as 2 times 2 to the 8, isn't it? 00:04:41.240 --> 00:04:42.150 2 times 2 to the 8. 00:04:42.150 --> 00:04:44.940 That's just 2 to the 8 plus itself. 00:04:44.940 --> 00:04:49.030 And 2 times to the 8, well that's the same thing as 2 to 00:04:49.030 --> 00:04:53.170 the first times 2 to the 8. 00:04:53.170 --> 00:04:55.500 And 2 to the first times 2 to the 8 by the same rule we just 00:04:55.500 --> 00:04:59.040 did is equal to 2 to the 9. 00:04:59.040 --> 00:05:01.080 So I thought I would just throw that out to you. 00:05:01.080 --> 00:05:03.280 And it works even with negative exponents. 00:05:03.280 --> 00:05:13.840 If I were to say 5 to the negative 100 times 3 to the, 00:05:13.840 --> 00:05:18.370 say, 100 -- oh sorry, times 5 -- this has to be a 5. 00:05:18.370 --> 00:05:20.140 I don't know what my brain was doing. 00:05:20.140 --> 00:05:25.150 5 to the negative 100 times 5 to the 102, that would 00:05:25.150 --> 00:05:27.890 equal 5 squared, right? 00:05:27.890 --> 00:05:30.930 I just take minus 100 plus 102. 00:05:30.930 --> 00:05:31.940 This is a 5. 00:05:31.940 --> 00:05:35.080 I'm sorry for that brain malfunction. 00:05:35.080 --> 00:05:37.860 And of course, that equals 25. 00:05:37.860 --> 00:05:39.210 So that's the first exponent rule. 00:05:39.210 --> 00:05:40.760 Now I'm going to show you another one, and it kind of 00:05:40.760 --> 00:05:43.900 leads from the same thing. 00:05:43.900 --> 00:05:55.280 If I were to ask you what 2 to the 9 over 2 to the 10 equals, 00:05:55.280 --> 00:05:56.940 that looks like that could be a little confusing. 00:05:56.940 --> 00:06:00.720 But it actually turns out to be the same rule, because what's 00:06:00.720 --> 00:06:03.110 another way of writing this? 00:06:03.110 --> 00:06:08.360 Well, we know that this is also the same thing as 2 to the 9 00:06:08.360 --> 00:06:12.710 times 1 over 2 to the 10, right? 00:06:12.710 --> 00:06:14.460 And we know 1 over 2 to the 10. 00:06:14.460 --> 00:06:18.700 Well, you could re-write right this as 2 the 9 times 2 to 00:06:18.700 --> 00:06:20.850 the negative 10, right? 00:06:20.850 --> 00:06:25.270 All I did is I took 1 over 2 to the 10 and I flipped it and I 00:06:25.270 --> 00:06:26.990 made the exponent negative. 00:06:26.990 --> 00:06:28.330 And I think you know that already from 00:06:28.330 --> 00:06:30.660 level two exponents. 00:06:30.660 --> 00:06:33.090 And now, once again, we can just add the exponents. 00:06:33.090 --> 00:06:39.300 9 plus negative 10 equals 2 to the negative 1, or we could 00:06:39.300 --> 00:06:42.000 say that equals 1/2, right? 00:06:42.000 --> 00:06:44.730 So it's an interesting thing here. 00:06:44.730 --> 00:06:48.110 Whatever is the bottom exponent, you could put it in 00:06:48.110 --> 00:06:50.800 the numerator like we did here, but turn it into a negative. 00:06:50.800 --> 00:06:53.760 So that leads us to the second exponent rule, simplification 00:06:53.760 --> 00:06:59.860 is we could just say that this equals 2 to the 9 minus 10, 00:06:59.860 --> 00:07:02.190 which equals 2 to the negative 1. 00:07:02.190 --> 00:07:05.160 Let's do another problem like that. 00:07:05.160 --> 00:07:16.400 If I said 10 to the 200 over 10 to the 50, well that 00:07:16.400 --> 00:07:23.640 just equals 10 to the 200 minus 50, which is 150. 00:07:23.640 --> 00:07:30.870 Likewise, if I had 7 to the fortieth power over 7 to 00:07:30.870 --> 00:07:35.940 the negative fifth power, this will equal 7 to the 00:07:35.940 --> 00:07:41.420 fortieth minus negative 5. 00:07:41.420 --> 00:07:46.230 So it equals 7 to the forty-fifth. 00:07:46.230 --> 00:07:48.310 Now I want you to think about that, does that make sense? 00:07:48.310 --> 00:07:54.480 Well, we could have re-written this equation as 7 to the 00:07:54.480 --> 00:07:59.180 fortieth times 7 to the fifth, right? 00:07:59.180 --> 00:08:02.806 We could have taken this 1 over 7 to the negative 5 and turn it 00:08:02.806 --> 00:08:06.640 into 7 to the fifth, and that would also just be 7 00:08:06.640 --> 00:08:08.160 to the forty-five. 00:08:08.160 --> 00:08:10.810 So the second exponent rule I just taught you actually is no 00:08:10.810 --> 00:08:12.390 different than that first one. 00:08:12.390 --> 00:08:14.810 If the exponent is in the denominator, and of course, it 00:08:14.810 --> 00:08:18.210 has to be the same base and you're dividing, you subtract 00:08:18.210 --> 00:08:20.570 it from the exponent in the numerator. 00:08:20.570 --> 00:08:23.390 If they're both in the numerator, as in this case, 7 00:08:23.390 --> 00:08:26.580 to the fortieth times 7 to the fifth -- actually there's no 00:08:26.580 --> 00:08:29.370 numerator, but they're essentially multiplying by each 00:08:29.370 --> 00:08:32.420 other, and of course, you have to have the same base. 00:08:32.420 --> 00:08:35.690 Then you add the exponents. 00:08:35.690 --> 00:08:37.700 I'm going to add one variation of this, and actually this is 00:08:37.700 --> 00:08:40.360 the same thing but it's a little bit of a trick question. 00:08:40.360 --> 00:08:56.470 What is 2 to the 9 times 4 to the 100? 00:08:56.470 --> 00:08:58.190 Actually, maybe I shouldn't teach this to you, you have 00:08:58.190 --> 00:08:59.480 to wait until I teach you the next rule. 00:08:59.480 --> 00:09:01.900 But I'll give you a little hint. 00:09:01.900 --> 00:09:09.570 This is the same thing as 2 the 9 times 2 squared to the 100. 00:09:09.570 --> 00:09:12.320 And the rule I'm going to teach you now is that when you have 00:09:12.320 --> 00:09:15.610 something to an exponent and then that number raised to 00:09:15.610 --> 00:09:18.930 an exponent, you actually multiply these two exponents. 00:09:18.930 --> 00:09:24.980 So this would be 2 the 9 times 2 to the 200. 00:09:24.980 --> 00:09:27.020 And by that first rule we learned, this would 00:09:27.020 --> 00:09:29.760 be 2 to the 209. 00:09:29.760 --> 00:09:31.140 Now in the next module I'm going to cover 00:09:31.140 --> 00:09:31.940 this in more detail. 00:09:31.940 --> 00:09:34.650 I think I might have just confused you. 00:09:34.650 --> 00:09:37.050 But watch the next video and then after the next video I 00:09:37.050 --> 00:09:40.400 think you're going to be ready to do level one exponent rules. 00:09:40.400 --> 00:09:41.930 Have fun.

Mulitplication 8: Multiplying decimals (Old video)

https://www.youtube.com/watch?v=m5z6pOsxF_8

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en

WEBVTT Kind: captions Language: en 00:00:01.290 --> 00:00:04.830 Welcome to the presentation on multiplying decimals. 00:00:04.830 --> 00:00:07.230 Let's get started. 00:00:07.230 --> 00:00:10.120 So I think you'll find out that multiplying decimals is not a 00:00:10.120 --> 00:00:13.220 lot more difficult than just multiplying regular numbers. 00:00:13.220 --> 00:00:15.460 And I'll show you in a problem. 00:00:15.460 --> 00:00:16.940 Let me pick some random numbers. 00:00:16.940 --> 00:00:25.590 Let's say I had 7,518. 00:00:25.590 --> 00:00:29.590 Actually, let's make that's 75.18. 00:00:29.590 --> 00:00:31.700 Clearly you can tell I'm doing this on the fly. 00:00:31.700 --> 00:00:40.010 75.18 times 0.97. 00:00:40.010 --> 00:00:41.140 So first you look at this problem you're like, 00:00:41.140 --> 00:00:41.970 oh boy, that's tough. 00:00:41.970 --> 00:00:44.120 These decimals-- I don't even know how to approach it. 00:00:44.120 --> 00:00:45.490 Well this is what you do. 00:00:45.490 --> 00:00:47.960 You ignore the decimals when you start the problem and you 00:00:47.960 --> 00:00:50.640 pretend like it's just a regular multiplication problem. 00:00:50.640 --> 00:00:53.435 And if you ignore the decimals like I said, at 00:00:53.435 --> 00:00:57.290 the beginning, 7,518 on top and 97 on the bottom. 00:00:57.290 --> 00:00:58.470 And if that doesn't make sense let me just show you. 00:00:58.470 --> 00:01:00.440 I'm just going to ignore the decimals and do this like a 00:01:00.440 --> 00:01:02.320 normal multiplication problem. 00:01:02.320 --> 00:01:03.460 So normal multiplication. 00:01:03.460 --> 00:01:05.870 I'd start at the 1's place right here. 00:01:05.870 --> 00:01:07.430 I'd say 7 times 8. 00:01:07.430 --> 00:01:09.530 Well, 7 times 8 is 56. 00:01:09.530 --> 00:01:11.790 Carry the 5. 00:01:11.790 --> 00:01:13.890 7 times 1 is 7. 00:01:13.890 --> 00:01:17.020 Plus the 5 is 12. 00:01:17.020 --> 00:01:18.490 2 down here. 00:01:18.490 --> 00:01:20.300 Carry the 1. 00:01:20.300 --> 00:01:23.740 7 times 5 is 35. 00:01:23.740 --> 00:01:25.825 Plus the 1 is 36. 00:01:25.825 --> 00:01:27.800 Put the 6 here. 00:01:27.800 --> 00:01:29.970 Carry the 3. 00:01:29.970 --> 00:01:33.830 And then 7 times 7 is 49. 00:01:33.830 --> 00:01:36.700 Plus 2 is 52. 00:01:36.700 --> 00:01:39.370 So just put 52 here. 00:01:39.370 --> 00:01:41.430 So just like normal multiplication we just took the 00:01:41.430 --> 00:01:43.850 1's place right here, the 7, so it's actually not the 1's, but 00:01:43.850 --> 00:01:46.695 we're ignoring the decimals so if there were no decimals this 00:01:46.695 --> 00:01:48.030 would be the 1's place. 00:01:48.030 --> 00:01:50.080 And we're multiplying it by the top number. 00:01:50.080 --> 00:01:55.030 7 times 7,518 is equal to 52,626. 00:01:55.030 --> 00:01:57.230 And like regular multiplication, we 00:01:57.230 --> 00:01:58.050 do the 10's place. 00:01:58.050 --> 00:02:00.450 And this isn't really the 10's place, but if you ignore 00:02:00.450 --> 00:02:02.030 the decimals it would be. 00:02:02.030 --> 00:02:05.120 And let's cross all this stuff out since we're not using it. 00:02:05.120 --> 00:02:08.870 9 times 8 is 72. 00:02:08.870 --> 00:02:10.110 Carry the 7. 00:02:10.110 --> 00:02:12.260 9 times 1 is 9. 00:02:12.260 --> 00:02:15.450 Plus 7 is 16. 00:02:15.450 --> 00:02:16.960 Carry the 1. 00:02:16.960 --> 00:02:19.870 9 times 5 is 45. 00:02:19.870 --> 00:02:21.620 This is good practice for me too, I haven't done 00:02:21.620 --> 00:02:24.360 my multiplication tables in a long time. 00:02:24.360 --> 00:02:26.320 9 times 5 is 45. 00:02:26.320 --> 00:02:29.590 Plus 1 is 46. 00:02:29.590 --> 00:02:31.570 Carry the 4. 00:02:31.570 --> 00:02:35.070 9 times 7 is 63. 00:02:35.070 --> 00:02:36.710 Plus 4 is 67. 00:02:39.350 --> 00:02:40.780 Now we add. 00:02:40.780 --> 00:02:43.610 So you're probably thinking, boy, what do decimals have 00:02:43.610 --> 00:02:44.410 to do with this at all. 00:02:44.410 --> 00:02:46.090 I'm just doing a regular multiplication problem. 00:02:46.090 --> 00:02:46.530 And I'll show you. 00:02:46.530 --> 00:02:49.380 Actually the decimals only come in right at the very end. 00:02:49.380 --> 00:02:52.740 So what I do is now I just add like I do a regular level 00:02:52.740 --> 00:02:54.790 4 multiplication problem. 00:02:54.790 --> 00:02:57.880 So I say 6 plus 0 is 6. 00:02:57.880 --> 00:03:00.880 2 plus 2 is 4. 00:03:00.880 --> 00:03:03.440 6 plus 6 is 12. 00:03:03.440 --> 00:03:05.170 Carry the 1. 00:03:05.170 --> 00:03:08.640 1 plus 2 plus 6 is 9. 00:03:08.640 --> 00:03:10.990 5 plus 7 is 12. 00:03:10.990 --> 00:03:13.260 Carry the 1. 00:03:13.260 --> 00:03:15.370 1 plus 6 is 7. 00:03:15.370 --> 00:03:18.970 OK, so now here's where the decimals come into play. 00:03:18.970 --> 00:03:21.300 And your I think you're going to be shocked by how 00:03:21.300 --> 00:03:22.680 straightforward this is. 00:03:22.680 --> 00:03:24.770 What I do is I go back to the original problem and now I 00:03:24.770 --> 00:03:26.430 actually pay attention to the decimals. 00:03:26.430 --> 00:03:29.160 And I say, how many total numbers are behind 00:03:29.160 --> 00:03:30.610 the decimal point? 00:03:30.610 --> 00:03:34.150 Well, there's 1 number behind the decimal point, 2 numbers 00:03:34.150 --> 00:03:36.860 behind the decimal point, 3 numbers behind the decimal 00:03:36.860 --> 00:03:40.050 point, 4 numbers behind the decimal point. 00:03:40.050 --> 00:03:41.750 1, 2, 3, 4. 00:03:41.750 --> 00:03:43.860 So there are 4 numbers behind the decimal point in the 00:03:43.860 --> 00:03:46.330 problem I did, and I just count here. 00:03:46.330 --> 00:03:49.810 1, 2, 3, 4. 00:03:49.810 --> 00:03:52.930 The answer will also have 4 numbers behind the decimal 00:03:52.930 --> 00:03:54.620 point, and that's the answer. 00:03:54.620 --> 00:03:59.190 72.9246. 00:03:59.190 --> 00:04:00.750 Now let me ask you a question. 00:04:00.750 --> 00:04:08.870 If I had a 0 here, would that count as an extra number 00:04:08.870 --> 00:04:11.010 behind the decimal point? 00:04:11.010 --> 00:04:14.785 Well, it only would have been if you actually used the 00:04:14.785 --> 00:04:16.820 0 in the multiplication. 00:04:16.820 --> 00:04:18.770 Maybe that confuses you. 00:04:18.770 --> 00:04:21.220 What I would recommend if you have any trailing 0's 00:04:21.220 --> 00:04:22.660 with a decimal like this. 00:04:22.660 --> 00:04:25.000 you actually should just ignore those 0's and then do the 00:04:25.000 --> 00:04:26.610 problem just the way I did it. 00:04:26.610 --> 00:04:31.140 And when remember, that's only for trailing 0's. 00:04:31.140 --> 00:04:35.270 If this was the bottom number then that 0 would matter 00:04:35.270 --> 00:04:38.770 because it's not a trailing 0, it's actually part 00:04:38.770 --> 00:04:40.260 of the number. 00:04:40.260 --> 00:04:44.070 Let's do a couple more examples and I think that'll make sense. 00:04:44.070 --> 00:04:47.830 So let's say I had 5-- and I'm going to do a simpler 00:04:47.830 --> 00:04:49.600 example arithmetically. 00:04:49.600 --> 00:04:52.070 I think it'll help you with some principles. 00:04:52.070 --> 00:05:00.880 If I said 5.10 times 1.09. 00:05:00.880 --> 00:05:03.020 So there's two things we could do. 00:05:03.020 --> 00:05:05.560 We could just multiply it the way it is. 00:05:05.560 --> 00:05:07.840 Actually let's do it both ways and I'll show you you get the 00:05:07.840 --> 00:05:10.150 same answer whether or not you ignore that 0. 00:05:10.150 --> 00:05:14.460 So in the first case let's not ignore the 0. 00:05:14.460 --> 00:05:16.440 Let's use that 0, even though that trailing 0 in the 00:05:16.440 --> 00:05:19.600 decimal-- 5.10 is the same thing as 5.1. 00:05:19.600 --> 00:05:20.750 But let's use it. 00:05:20.750 --> 00:05:22.680 9 times 0 is 0. 00:05:22.680 --> 00:05:24.990 9 times 1 is 9. 00:05:24.990 --> 00:05:28.570 9 times 5 is 45. 00:05:28.570 --> 00:05:30.820 And in the 0's place you put a 0 and then 0 00:05:30.820 --> 00:05:32.885 times everything is 0. 00:05:32.885 --> 00:05:36.780 0 times 0, 0 times 1, 0 times 5. 00:05:36.780 --> 00:05:37.740 Put two 0's here. 00:05:37.740 --> 00:05:40.190 And then 1 times 0 is 0. 00:05:40.190 --> 00:05:41.880 1 times 1 is 1. 00:05:41.880 --> 00:05:44.550 And 1 times 5 is 5. 00:05:44.550 --> 00:05:46.420 And now we add it all. 00:05:46.420 --> 00:05:51.750 We get 0, 9, 5, 5, 5. 00:05:51.750 --> 00:05:53.430 And like we did before, we just count the decimals. 00:05:53.430 --> 00:05:55.650 1, 2, 3, 4. 00:05:55.650 --> 00:05:57.450 So 1, 2 3, 4. 00:05:57.450 --> 00:05:59.730 So the decimal will go here. 00:05:59.730 --> 00:06:03.470 So we got 5.5590 as the answer. 00:06:03.470 --> 00:06:05.140 Now what if we did like I was recommending, we 00:06:05.140 --> 00:06:05.950 actually ignored the 0? 00:06:08.880 --> 00:06:15.320 And I can actually rewrite it as 1.09 times 5.1. 00:06:15.320 --> 00:06:16.490 Because you know in multiplication order 00:06:16.490 --> 00:06:17.510 doesn't matter. 00:06:17.510 --> 00:06:19.940 a times b is the same thing as b times a. 00:06:19.940 --> 00:06:22.010 2 times 3 is the same thing as 3 times 2. 00:06:22.010 --> 00:06:27.695 So 1.09 times 5.1 is the same thing as 5.1 times 1.09. 00:06:27.695 --> 00:06:29.320 So let's just multiply this out. 00:06:29.320 --> 00:06:30.590 And notice, these are the same numbers. 00:06:30.590 --> 00:06:34.340 All I did is I took the 0 off. 00:06:34.340 --> 00:06:38.900 So first, I just ignore the decimals I say 1 times 9 is 9. 00:06:38.900 --> 00:06:41.110 1 times 0 is 0. 00:06:41.110 --> 00:06:43.200 1 times 1 is 1. 00:06:43.200 --> 00:06:44.910 Put a 0 here. 00:06:44.910 --> 00:06:48.240 5 times 9 is 45. 00:06:48.240 --> 00:06:49.720 Carry the 4. 00:06:49.720 --> 00:06:51.070 5 two 0 is 0. 00:06:51.070 --> 00:06:54.060 Plus 4 is 4. 00:06:54.060 --> 00:06:57.410 5 times 1 is 5. 00:06:57.410 --> 00:06:58.830 Now I add. 00:06:58.830 --> 00:07:04.290 9, 5, 5, 5. 00:07:04.290 --> 00:07:06.890 Now I'm at the point that I can actually pay attention 00:07:06.890 --> 00:07:09.040 to the decimal points. 00:07:09.040 --> 00:07:10.890 How many numbers are behind the decimals? 00:07:10.890 --> 00:07:14.010 Well, there's 1, 2, 3. 00:07:14.010 --> 00:07:18.330 So I go 1, 2, 3 and put the decimal point right here. 00:07:18.330 --> 00:07:20.520 Notice I got the same exact answer. 00:07:20.520 --> 00:07:23.610 The only difference is that this one had a trailing 0, 00:07:23.610 --> 00:07:25.630 which really doesn't make a number any different. 00:07:25.630 --> 00:07:27.400 I could add a hundred 0's here and the number's really 00:07:27.400 --> 00:07:30.490 not a different number. 00:07:30.490 --> 00:07:33.950 If you were a computer programmer or a statistician 00:07:33.950 --> 00:07:35.250 of some kind, this could be an important number. 00:07:35.250 --> 00:07:36.520 But ignore what I just said. 00:07:36.520 --> 00:07:41.360 And for your purposes, these trailing 0's mean nothing. 00:07:44.250 --> 00:07:46.700 Same way a leading 0 actually wouldn't mean nothing. 00:07:46.700 --> 00:07:47.500 No one ever does that. 00:07:50.350 --> 00:07:51.900 Well, let me see how much time I have. 00:07:51.900 --> 00:07:52.470 I have 2 more minutes. 00:07:52.470 --> 00:07:55.520 Let me do one more problem just to maybe hit the point home. 00:07:55.520 --> 00:08:02.770 You know, this is really no different than level 00:08:02.770 --> 00:08:03.820 4 multiplication. 00:08:03.820 --> 00:08:06.020 And at the end you just have to count the numbers 00:08:06.020 --> 00:08:07.620 behind the decimal point. 00:08:07.620 --> 00:08:11.040 So 5 times 5 is 25. 00:08:11.040 --> 00:08:12.390 Whoops. 00:08:12.390 --> 00:08:12.680 25. 00:08:12.680 --> 00:08:14.580 I'm already getting messy. 00:08:14.580 --> 00:08:15.910 Carry the 2. 00:08:15.910 --> 00:08:18.030 5 times 7 is 35. 00:08:18.030 --> 00:08:20.870 Plus 2 is 37. 00:08:20.870 --> 00:08:23.380 Bring down the 7, carry the 3. 00:08:23.380 --> 00:08:25.710 5 times 0 is 0. 00:08:25.710 --> 00:08:26.455 Plus 3. 00:08:26.455 --> 00:08:29.340 So it's 375, ignore that blob. 00:08:29.340 --> 00:08:30.485 I'm sorry for being so messy. 00:08:30.485 --> 00:08:32.010 And then you put a 0. 00:08:32.010 --> 00:08:34.620 1 times 5 is 5. 00:08:34.620 --> 00:08:36.870 1 times 7 is 7. 00:08:36.870 --> 00:08:38.270 Ignore that. 00:08:38.270 --> 00:08:39.114 Now we add. 00:08:39.114 --> 00:08:41.510 We say 5 plus 0 is 5. 00:08:41.510 --> 00:08:44.430 7 plus 5 is 12. 00:08:44.430 --> 00:08:47.060 1 plus 3 plus 7 is 11. 00:08:47.060 --> 00:08:48.320 So we got our answer, now we just have to 00:08:48.320 --> 00:08:49.820 count the decimals. 00:08:49.820 --> 00:08:55.610 So here we have 1, 2, 3, 4, 5 numbers behind 00:08:55.610 --> 00:08:56.860 the decimal point. 00:08:56.860 --> 00:09:00.100 But in our answer we only have 4 digits, so how can we get 5 00:09:00.100 --> 00:09:01.610 numbers behind the decimal point? 00:09:01.610 --> 00:09:03.090 Well, we start here. 00:09:03.090 --> 00:09:08.820 We say 1, 2, 3, 4 and we need one more number behind the 00:09:08.820 --> 00:09:10.770 decimal point, so we add a 0 here. 00:09:10.770 --> 00:09:13.090 And then we put the decimal point. 00:09:13.090 --> 00:09:14.380 See what I just did. 00:09:14.380 --> 00:09:18.030 We had to have 5 numbers behind the decimal point. 00:09:18.030 --> 00:09:20.080 And we only had 4 numbers in the answer. 00:09:20.080 --> 00:09:22.950 So I added a leading 0 and then put the decimal point. 00:09:22.950 --> 00:09:26.990 And now we have 5 numbers behind the decimal point. 00:09:26.990 --> 00:09:29.390 And I've shown you a very mechanical way of doing this. 00:09:29.390 --> 00:09:32.030 Hopefully in the future I can give you a seminar on actually 00:09:32.030 --> 00:09:35.510 why this method of counting the numbers behind the decimal 00:09:35.510 --> 00:09:37.070 points actually works. 00:09:37.070 --> 00:09:40.340 But I think you are ready to try some problems on 00:09:40.340 --> 00:09:41.890 multiplying decimals. 00:09:41.890 --> 00:09:43.400 Have fun.

Adding Decimals (Old)

https://www.youtube.com/watch?v=SxZUFA2SGX8

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en

WEBVTT Kind: captions Language: en 00:00:00.850 --> 00:00:04.660 Welcome to the presentation on adding decimals. 00:00:04.660 --> 00:00:07.440 Let's do some problems. 00:00:07.440 --> 00:00:21.840 So let's say I had 0.008-- that's an 8-- 5 plus-- and 00:00:21.840 --> 00:00:33.260 I'm writing it side by side on purpose-- 1.799. 00:00:33.260 --> 00:00:36.250 So at first you're like, these decimals, they 00:00:36.250 --> 00:00:38.110 confuse me, I give up. 00:00:38.110 --> 00:00:39.750 But I'm going to show you that it's actually very 00:00:39.750 --> 00:00:42.080 straightforward and it's actually no more difficult 00:00:42.080 --> 00:00:44.200 than doing normal addition. 00:00:44.200 --> 00:00:46.340 And there's only one thing you have to think about 00:00:46.340 --> 00:00:47.630 when you do decimals. 00:00:47.630 --> 00:00:50.350 You have to line up the decimal points. 00:00:50.350 --> 00:00:52.465 So let's start with the first number and let's rewrite it so 00:00:52.465 --> 00:00:54.750 that we can actually do the math. 00:00:54.750 --> 00:01:01.740 0.0085. 00:01:01.740 --> 00:01:05.050 Now for the second number let's put the decimal point right 00:01:05.050 --> 00:01:09.120 below where that first decimal is on the top number, and 00:01:09.120 --> 00:01:10.110 then rewrite the number. 00:01:10.110 --> 00:01:17.920 So it's 1.799. 00:01:17.920 --> 00:01:21.050 So all we did is we just rewrote both numbers with their 00:01:21.050 --> 00:01:22.230 decimal points lined up. 00:01:22.230 --> 00:01:23.320 And we could've flipped them around. 00:01:23.320 --> 00:01:27.570 We could've written the 1.799 first and the 0.0085 below it. 00:01:27.570 --> 00:01:30.860 What's key-- and if you get this point and you already know 00:01:30.860 --> 00:01:32.810 how to do addition, you've already figured out how to do 00:01:32.810 --> 00:01:36.350 adding decimals-- is that you line up the decimal points. 00:01:36.350 --> 00:01:37.810 So once we line up the decimal points we're 00:01:37.810 --> 00:01:40.990 ready to start adding. 00:01:40.990 --> 00:01:43.440 But hey Sal, there's something very strange 00:01:43.440 --> 00:01:44.600 here you might say. 00:01:44.600 --> 00:01:46.540 How can I add this 5 to a nothing? 00:01:46.540 --> 00:01:48.960 Well there's a very easy solution to that. 00:01:48.960 --> 00:01:50.870 Let's just put a 0 right here. 00:01:50.870 --> 00:01:52.160 And how can I do that? 00:01:52.160 --> 00:01:56.440 How can I just randomly augment a number with a 0? 00:01:56.440 --> 00:01:58.840 Well, I don't know if I've already shown you in another 00:01:58.840 --> 00:02:01.930 module, but when you add trailing 0's to decimals, it 00:02:01.930 --> 00:02:04.230 doesn't really change the value of the decimal. 00:02:04.230 --> 00:02:05.500 And I think that make sense to you. 00:02:05.500 --> 00:02:10.570 If I said 1.10 that that's the same thing as 1.1. 00:02:10.570 --> 00:02:14.120 Which is the same thing as 1.100. 00:02:14.120 --> 00:02:15.720 It's all 1 and 1/10. 00:02:15.720 --> 00:02:18.820 And these 0's just kind of add extra digits of precision, 00:02:18.820 --> 00:02:20.570 which actually, don't change the actual value. 00:02:20.570 --> 00:02:23.510 And maybe later when I'm teaching computer programming 00:02:23.510 --> 00:02:25.110 and computation, the precision might matter. 00:02:25.110 --> 00:02:26.140 But for now, it doesn't. 00:02:26.140 --> 00:02:27.210 It's just the value. 00:02:27.210 --> 00:02:29.480 So I added this 0 here and you can always do that. 00:02:29.480 --> 00:02:31.600 You can add trailing 0's without actually changing 00:02:31.600 --> 00:02:32.850 the value of the number. 00:02:32.850 --> 00:02:34.880 And now we're really ready to add. 00:02:34.880 --> 00:02:38.380 5 plus 0 is 5. 00:02:38.380 --> 00:02:40.375 8 plus 9 is 17. 00:02:43.290 --> 00:02:46.730 1 plus 0 plus 9 is 10. 00:02:46.730 --> 00:02:48.020 Carry the 1. 00:02:48.020 --> 00:02:53.110 1 plus 0 plus 7 is 8. 00:02:53.110 --> 00:02:55.690 And then bring down this 1 because there's 00:02:55.690 --> 00:02:56.830 nothing to add it to. 00:02:56.830 --> 00:02:58.350 And we could have even added a leading zero 00:02:58.350 --> 00:02:59.630 here if you wanted to. 00:02:59.630 --> 00:03:01.680 My wife's a doctor and she says it's key that you always add 00:03:01.680 --> 00:03:03.740 that leading 0 so that no one gives someone the wrong 00:03:03.740 --> 00:03:04.355 amount of medicine. 00:03:04.355 --> 00:03:07.850 But anyway, so now we are almost done. 00:03:07.850 --> 00:03:09.730 We've done the addition and now we just have to figure out 00:03:09.730 --> 00:03:10.470 where to put the decimal point. 00:03:10.470 --> 00:03:13.496 Well, the decimal point just drops straight down. 00:03:13.496 --> 00:03:16.020 And we're able to do that because we already lined 00:03:16.020 --> 00:03:16.870 up the decimal points. 00:03:16.870 --> 00:03:21.160 So the answer to this problem is 1.8075. 00:03:21.160 --> 00:03:25.150 It seems complicated, but all you have to do, line up the 00:03:25.150 --> 00:03:29.115 decimal points, add 0's where appropriate, and add. 00:03:29.115 --> 00:03:31.900 And if you know how to add you already know how to do this. 00:03:31.900 --> 00:03:32.760 Let's do another problem. 00:03:36.650 --> 00:03:49.360 58.75 plus 0.028. 00:03:49.360 --> 00:03:53.240 Now if you saw a problem like this written in this format, 00:03:53.240 --> 00:03:55.250 your temptation might be to immediately start adding. 00:03:55.250 --> 00:03:58.550 Add the 5 to the 8, the 7 to the 2, the 8 to the 0. 00:03:58.550 --> 00:04:01.090 And then just bring down the 5 or something of that nature. 00:04:01.090 --> 00:04:03.750 And you would be wrong. 00:04:03.750 --> 00:04:05.410 And I think you realize why you'd be wrong. 00:04:05.410 --> 00:04:06.970 Because you forgot the first step. 00:04:06.970 --> 00:04:09.360 The first step is line up the decimal points. 00:04:09.360 --> 00:04:12.930 When you're doing addition with decimals, the number one step 00:04:12.930 --> 00:04:14.740 is line up the decimal points. 00:04:14.740 --> 00:04:17.220 So let's line up the decimal points. 00:04:17.220 --> 00:04:20.010 So we can just rewrite that top number again. 00:04:20.010 --> 00:04:23.350 58.75. 00:04:23.350 --> 00:04:25.180 And now let's rewrite the bottom number so that the 00:04:25.180 --> 00:04:26.700 decimal points line up. 00:04:26.700 --> 00:04:29.300 So we'll put the decimal points right below it. 00:04:29.300 --> 00:04:34.270 And it's 0.028. 00:04:34.270 --> 00:04:37.570 And now we can add. 00:04:37.570 --> 00:04:39.280 And just like before you might say, how can I 00:04:39.280 --> 00:04:40.610 add nothing to this 8? 00:04:40.610 --> 00:04:44.150 We can add a trailing 0 to this top number because it really 00:04:44.150 --> 00:04:46.280 doesn't change the value of the number and now it kind of 00:04:46.280 --> 00:04:47.340 gives us a sense of security. 00:04:47.340 --> 00:04:49.840 We have something to add to the 8. 00:04:49.840 --> 00:04:50.900 So let's do that. 00:04:50.900 --> 00:04:53.910 0 plus 8 is 8. 00:04:53.910 --> 00:04:57.370 5 plus 2 is 7. 00:04:57.370 --> 00:05:01.280 7 plus 0 is 7. 00:05:01.280 --> 00:05:04.970 8 plus blank space is 8. 00:05:04.970 --> 00:05:06.720 And we could have added a 0 there and it would've been the 00:05:06.720 --> 00:05:08.290 same thing, had the leading 0. 00:05:08.290 --> 00:05:10.550 We wouldn't have given someone the wrong about of medicine. 00:05:10.550 --> 00:05:14.150 5 plus blank space is 5. 00:05:14.150 --> 00:05:17.990 And we just drop down that decimal point, and we're done. 00:05:17.990 --> 00:05:19.410 It's that straightforward. 00:05:19.410 --> 00:05:23.320 You line up the decimal points, add any 0's that might make you 00:05:23.320 --> 00:05:25.510 feel more comfortable because it's often more comfortable to 00:05:25.510 --> 00:05:28.720 add to a 0 than to add to a blank space. 00:05:28.720 --> 00:05:30.610 And then you do your addition and then you drop down 00:05:30.610 --> 00:05:31.260 the decimal point. 00:05:31.260 --> 00:05:34.370 The answer is 58.778. 00:05:34.370 --> 00:05:36.920 And it's always good to kind of do a reality check. 00:05:36.920 --> 00:05:39.060 To say, well, does my answer make sense? 00:05:39.060 --> 00:05:44.680 I have 58.75, so roughly 58, almost 59. 00:05:44.680 --> 00:05:45.750 58 and 3/4. 00:05:45.750 --> 00:05:48.030 And I'm adding a very small number to it. 00:05:48.030 --> 00:05:51.140 I'm adding 0.028. 00:05:51.140 --> 00:05:55.900 So my answer shouldn't change much I guess is the 00:05:55.900 --> 00:05:56.600 way you could view it. 00:05:56.600 --> 00:05:59.890 It should still be around 58 something because I'm only 00:05:59.890 --> 00:06:01.650 adding a very small amount. 00:06:01.650 --> 00:06:04.640 If you had done it this way and you had just started adding 00:06:04.640 --> 00:06:08.680 immediately, you would have gotten 59. 00:06:08.680 --> 00:06:09.140 something. 00:06:09.140 --> 00:06:12.500 And you're like, boy, but I only added 0.028 not 0.28. 00:06:12.500 --> 00:06:14.710 And you would've been a little bit suspicious of your answer. 00:06:14.710 --> 00:06:17.690 I hope I didn't confuse you, but it's really healthy to 00:06:17.690 --> 00:06:19.800 always think about the magnitude of your numbers and 00:06:19.800 --> 00:06:23.770 get a feel for just what you are adding and not always 00:06:23.770 --> 00:06:25.810 do it purely mechanical. 00:06:25.810 --> 00:06:30.200 Although sometimes it is fun just to chug through things. 00:06:30.200 --> 00:06:33.880 Now let's do one more problem just for good measure. 00:06:33.880 --> 00:06:35.605 I have some time left on the video. 00:06:38.630 --> 00:06:48.070 102.1 plus 2.56. 00:06:48.070 --> 00:06:49.360 So once again, think about magnitude. 00:06:49.360 --> 00:06:54.020 102.1 plus 2.56, that should be like 104. 00:06:54.020 --> 00:06:54.720 something. 00:06:54.720 --> 00:06:55.580 I won't do it in my head. 00:06:55.580 --> 00:06:57.640 We'll do it on paper. 00:06:57.640 --> 00:06:59.460 So first thing, we line up the decimal points. 00:06:59.460 --> 00:07:08.630 102.1 plus-- line up the decimal points-- 2.56. 00:07:08.630 --> 00:07:11.170 Now we could add a trailing 0 here and we're ready to add. 00:07:11.170 --> 00:07:13.510 0 plus 6 is 6. 00:07:13.510 --> 00:07:15.760 1 plus 5 is 6. 00:07:15.760 --> 00:07:18.260 2 plus 2 is 4. 00:07:18.260 --> 00:07:20.000 0 plus nothing is 0. 00:07:20.000 --> 00:07:22.380 1 plus nothing is 1. 00:07:22.380 --> 00:07:24.120 Bring down that decimal point. 00:07:24.120 --> 00:07:26.330 104.66. 00:07:26.330 --> 00:07:27.400 It was that easy. 00:07:27.400 --> 00:07:28.050 Hopefully it's easy. 00:07:28.050 --> 00:07:30.480 I don't want to make you think that it's easy if you're 00:07:30.480 --> 00:07:32.360 finding it hard because it actually is hard the 00:07:32.360 --> 00:07:33.510 first time you do it. 00:07:33.510 --> 00:07:37.390 But anyway, I think you're ready now to try the module 00:07:37.390 --> 00:07:38.500 on adding decimals. 00:07:38.500 --> 00:07:40.470 I hope you have fun.

Dividing decimals with hundredths example 3

https://www.youtube.com/watch?v=S0uuK7SQcA8

vtt

https://www.youtube.com/api/timedtext?v=S0uuK7SQcA8&ei=gGeUZY3qMcaxp-oP4eajMA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249840&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=28413ADA5D8B70132E7A097D5253AC7465541B78.3CAA5BD71C74C6FEB776BAA08E92E960655C3018&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.820 --> 00:00:03.560 Welcome to the presentation on dividing decimals. 00:00:03.560 --> 00:00:05.620 Let's get started with a problem. 00:00:05.620 --> 00:00:23.925 If I were to say how many times does 0.28 go into 23.828? 00:00:23.925 --> 00:00:26.320 So you're going to see that these dividing decimal problems 00:00:26.320 --> 00:00:28.480 are actually just like the level 4 division problems. 00:00:28.480 --> 00:00:30.730 You just have to figure out where to put the decimal. 00:00:30.730 --> 00:00:34.060 So what you do is you always want to take this decimal and 00:00:34.060 --> 00:00:37.180 move it over as many spaces as necessary to turn this 00:00:37.180 --> 00:00:39.380 number into a whole number. 00:00:39.380 --> 00:00:42.670 So in this case, we have to move it one space, two spaces 00:00:42.670 --> 00:00:44.680 over to put the decimal here. 00:00:44.680 --> 00:00:47.230 Well, if I did that with this number I have to do it 00:00:47.230 --> 00:00:48.350 with this number as well. 00:00:48.350 --> 00:00:50.595 So I moved it over two spaces to the right, so I have 00:00:50.595 --> 00:00:54.370 to move this decimal two spaces to the right-- 1, 2. 00:00:54.370 --> 00:00:56.950 Decimal goes here, and I put the decimal right 00:00:56.950 --> 00:00:58.450 above, right there. 00:00:58.450 --> 00:01:03.570 Now I can treat this 28 as a whole number. 00:01:03.570 --> 00:01:07.460 And if I want to, let me see if I could-- well, I want to erase 00:01:07.460 --> 00:01:09.210 the old decimal because if you were doing it with a pen you 00:01:09.210 --> 00:01:11.540 would kind of have the same problem I have. 00:01:11.540 --> 00:01:14.450 So now we do it just like a level 4 division problem. 00:01:14.450 --> 00:01:17.950 So we say, how many times does 28 go into 2? 00:01:17.950 --> 00:01:19.260 Well, no times. 00:01:19.260 --> 00:01:21.100 2 is smaller than 28. 00:01:21.100 --> 00:01:23.780 How many times does 28 go into 23? 00:01:23.780 --> 00:01:27.600 Once again, still, it goes into it zero times because 00:01:27.600 --> 00:01:29.820 23 is smaller than 28. 00:01:29.820 --> 00:01:30.110 How Many. 00:01:30.110 --> 00:01:34.570 Times does 28 go into 238? 00:01:34.570 --> 00:01:36.680 So let's think about that. 00:01:36.680 --> 00:01:39.320 28 is almost 30. 00:01:39.320 --> 00:01:44.170 238 is almost 240. 00:01:44.170 --> 00:01:47.940 So 30 goes into 240 eight times Because 3 goes 00:01:47.940 --> 00:01:49.630 into 24 eight times. 00:01:49.630 --> 00:01:54.480 So I'm going to guess that 28 goes into 238 eight times. 00:01:54.480 --> 00:01:55.690 And it literally it a guess. 00:01:55.690 --> 00:01:58.380 You have to try out some numbers sometimes. 00:01:58.380 --> 00:02:00.250 8 times 8 is 64. 00:02:03.200 --> 00:02:05.620 8 times 2 is 16. 00:02:05.620 --> 00:02:07.260 Plus 2 is 22. 00:02:10.470 --> 00:02:11.590 Subtract. 00:02:11.590 --> 00:02:15.390 I get 14. 00:02:15.390 --> 00:02:19.410 I guessed right because the remainder when I divide 28 into 00:02:19.410 --> 00:02:23.030 238 and I say it goes into it eight times is 14, 00:02:23.030 --> 00:02:24.480 which is less than 28. 00:02:24.480 --> 00:02:28.740 So 8 was the largest number of times that the 28 could go into 00:02:28.740 --> 00:02:31.120 238 without being larger. 00:02:31.120 --> 00:02:32.600 So now I bring down this 2. 00:02:32.600 --> 00:02:35.170 Once again, you recognize this is just purely a level 2 00:02:35.170 --> 00:02:39.110 division problem-- a level 4 division problem. 00:02:39.110 --> 00:02:42.180 So now I say, how many times does 28 go into 142? 00:02:42.180 --> 00:02:43.980 Well, once again, I'm going to approximate. 00:02:43.980 --> 00:02:47.360 28, it's almost 30. 00:02:47.360 --> 00:02:50.900 Let's see, 30 times 4 is 120. 00:02:50.900 --> 00:02:53.400 So yeah, I'll take a guess and I'll say let's say it 00:02:53.400 --> 00:02:54.490 goes into it four times. 00:02:54.490 --> 00:02:58.770 I could be wrong, but let's see if it works out. 00:02:58.770 --> 00:02:59.980 Let me get rid of this old 6. 00:02:59.980 --> 00:03:03.710 4 times 8 is 32. 00:03:03.710 --> 00:03:06.960 And 4 times 2 is 8. 00:03:06.960 --> 00:03:09.440 Plus 3 is 11. 00:03:12.790 --> 00:03:14.480 2 minus 2 is 0. 00:03:14.480 --> 00:03:16.830 4 minus 1 is 3. 00:03:16.830 --> 00:03:17.410 Huh. 00:03:17.410 --> 00:03:18.600 Interesting. 00:03:18.600 --> 00:03:22.640 So it turns out that my remainder here is larger than 00:03:22.640 --> 00:03:27.410 28, so I actually could have divided 28 into 142 00:03:27.410 --> 00:03:28.860 one more time. 00:03:28.860 --> 00:03:31.450 So let me go back and change that. 00:03:31.450 --> 00:03:34.130 See, it's not a mechanical thing. 00:03:34.130 --> 00:03:37.330 And if you feel unsure sometimes, you just have to try 00:03:37.330 --> 00:03:38.780 numbers and see if they work. 00:03:38.780 --> 00:03:41.360 And otherwise, you raise or lower the number accordingly. 00:03:41.360 --> 00:03:43.410 So let me erase that 4. 00:03:46.480 --> 00:03:47.640 I'm going to try not to mess up. 00:03:52.510 --> 00:03:54.130 Erase all this stuff down here. 00:03:56.790 --> 00:03:59.420 I probably should have tried it out on the side first before 00:03:59.420 --> 00:04:01.230 doing all this and I wouldn't have had to go back 00:04:01.230 --> 00:04:03.270 and erase it. 00:04:03.270 --> 00:04:06.510 And then let me get back to what I was doing. 00:04:06.510 --> 00:04:08.800 So when I went into it four times the remainder was too 00:04:08.800 --> 00:04:10.710 large, so let me try five now. 00:04:13.350 --> 00:04:17.590 5 times 8 is 40. 00:04:17.590 --> 00:04:18.890 5 times 2 is 10. 00:04:18.890 --> 00:04:19.910 Plus 4 is 14. 00:04:22.890 --> 00:04:25.530 142 minus 140 is 2. 00:04:25.530 --> 00:04:25.950 Good. 00:04:25.950 --> 00:04:27.900 2 is less than 28. 00:04:27.900 --> 00:04:29.200 This 5 is correct. 00:04:29.200 --> 00:04:32.910 Now I just bring down the 8. 00:04:32.910 --> 00:04:36.500 28 goes into 28 exactly one time. 00:04:36.500 --> 00:04:40.154 1 times 28 is 28. 00:04:40.154 --> 00:04:41.320 Remainder of 0. 00:04:41.320 --> 00:04:42.340 Done. 00:04:42.340 --> 00:04:49.610 So 28 goes into 2,382.8 85.1 times. 00:04:49.610 --> 00:05:03.550 Or you could say, 0.28 goes into 23.828 85.1 times. 00:05:03.550 --> 00:05:05.300 That's the answer we had gotten. 00:05:05.300 --> 00:05:06.540 And that makes sense. 00:05:06.540 --> 00:05:08.740 It's always good to do a reality check because if I took 00:05:08.740 --> 00:05:14.070 85.1 and I multiplied it by 0.28, it makes sense that 00:05:14.070 --> 00:05:16.110 I'd get a number around 23. 00:05:16.110 --> 00:05:18.770 0.28 is almost 1/3. 00:05:18.770 --> 00:05:22.060 So 23 is almost 1/3 of 85. 00:05:22.060 --> 00:05:23.820 So at least it makes sense in rough numbers. 00:05:23.820 --> 00:05:27.600 When you're doing decimals, if I had gotten 800 here instead 00:05:27.600 --> 00:05:30.530 of 85, I'd be like, oh, well, 0.28 times 800? 00:05:30.530 --> 00:05:32.820 I don't know if that equals 23. 00:05:32.820 --> 00:05:35.270 So it's always good to just do a reality check and get a sense 00:05:35.270 --> 00:05:39.710 for at least the magnitude of what your answer should be. 00:05:39.710 --> 00:05:40.660 Let's do another problem. 00:05:43.740 --> 00:05:58.130 Let's do 3.3 goes into 43.23. 00:05:58.130 --> 00:05:59.490 That's a 3. 00:05:59.490 --> 00:06:01.340 So first thing we want to do is move the decimal. 00:06:01.340 --> 00:06:03.230 We just have to move it one space here, so we move it 00:06:03.230 --> 00:06:04.870 once space here as well. 00:06:04.870 --> 00:06:06.300 Put the decimal right up here. 00:06:06.300 --> 00:06:09.200 And now it's just a level 4 division problem. 00:06:09.200 --> 00:06:13.480 33 goes into 4 zero times. 00:06:13.480 --> 00:06:16.880 33 goes into 43 one time. 00:06:16.880 --> 00:06:17.830 That's easy. 00:06:17.830 --> 00:06:21.680 1 times 33 is 33. 00:06:21.680 --> 00:06:22.540 Do the subtraction. 00:06:22.540 --> 00:06:25.400 43 minus 33 is 10. 00:06:25.400 --> 00:06:27.830 Bring down this 2. 00:06:27.830 --> 00:06:31.180 33 goes into 102? 00:06:31.180 --> 00:06:33.300 You could eyeball that one and say, about three times 00:06:33.300 --> 00:06:34.740 because 3 times 33 is 99. 00:06:37.520 --> 00:06:41.240 3 times 33 is 99. 00:06:41.240 --> 00:06:42.570 102 minus 99? 00:06:42.570 --> 00:06:43.380 Well, that's easy. 00:06:43.380 --> 00:06:44.760 That's 3. 00:06:44.760 --> 00:06:48.440 We just bring down this 3. 00:06:48.440 --> 00:06:52.500 33 goes into 33 one time. 00:06:52.500 --> 00:06:53.720 1 times 33 is 33. 00:06:57.360 --> 00:06:58.330 0. 00:06:58.330 --> 00:07:06.010 So 3.3 goes into 43.23 13.1 times. 00:07:06.010 --> 00:07:08.470 Or, if you move the decimal over, and when you move the 00:07:08.470 --> 00:07:11.230 decimal over to the right one spot, all you're doing is 00:07:11.230 --> 00:07:16.940 you're multiplying both the divisor and the dividend by 10. 00:07:16.940 --> 00:07:20.030 Which is fine as long as you multiply both of them by 10. 00:07:20.030 --> 00:07:27.830 It's also like saying 33 goes into 432.3 13.1 times. 00:07:27.830 --> 00:07:28.830 Let's do one more problem. 00:07:28.830 --> 00:07:30.390 I think I have time. 00:07:30.390 --> 00:07:32.290 YouTube puts a limit on this stuff. 00:07:32.290 --> 00:07:44.310 so let's say 2.5 goes into 0.3350 how many times? 00:07:44.310 --> 00:07:47.860 Well once again, let's move the decimal point over one here. 00:07:47.860 --> 00:07:49.780 So we move the decimal point over one here. 00:07:49.780 --> 00:07:51.660 Put the decimal here. 00:07:51.660 --> 00:07:54.140 So how many times does 25 go into 3? 00:07:54.140 --> 00:07:55.580 Well 0. 00:07:55.580 --> 00:07:58.790 So you could put a 0 here just for fun if you want. 00:07:58.790 --> 00:08:01.470 How many times does 25 go into 33? 00:08:01.470 --> 00:08:02.990 Well, it goes into it one time. 00:08:02.990 --> 00:08:06.950 1 times 25 is 25. 00:08:06.950 --> 00:08:09.120 33 minus 25 is 8. 00:08:09.120 --> 00:08:11.490 Bring down the five. 00:08:11.490 --> 00:08:12.730 25 goes into 85? 00:08:12.730 --> 00:08:16.440 Well, we know 25 times 3 is 75. 00:08:16.440 --> 00:08:18.920 So it'll go into it three times. 00:08:18.920 --> 00:08:20.080 3 times 25. 00:08:20.080 --> 00:08:23.000 We know that's 75. 00:08:23.000 --> 00:08:26.040 85 minus 75 is 10. 00:08:26.040 --> 00:08:27.890 Bring down the 0. 00:08:27.890 --> 00:08:30.220 Up here we had brought down the 5 before. 00:08:30.220 --> 00:08:33.300 And 25 goes into 100 four times. 00:08:33.300 --> 00:08:42.920 So our answer is 2.5 goes into 0.3350 0.134 times. 00:08:42.920 --> 00:08:45.680 So as you see, the only difference step between what 00:08:45.680 --> 00:08:49.360 we're doing when we're dividing decimals and when we're doing 00:08:49.360 --> 00:08:51.510 level 4 division is we just have to make sure we get the 00:08:51.510 --> 00:08:54.070 decimal in the right place. 00:08:54.070 --> 00:08:57.620 You shift the decimal here enough so that this becomes a 00:08:57.620 --> 00:09:00.450 whole number and you just have to shift the decimal here 00:09:00.450 --> 00:09:01.910 the same number of times. 00:09:01.910 --> 00:09:04.600 And once you do that it just becomes a level 00:09:04.600 --> 00:09:05.950 4 division problem. 00:09:05.950 --> 00:09:09.080 And the whole trick with level 4 division is always be willing 00:09:09.080 --> 00:09:12.330 to try numbers, and if the numbers don't work, 00:09:12.330 --> 00:09:13.240 adjust them accordingly. 00:09:13.240 --> 00:09:15.450 Don't feel that there should be a way that you can just always 00:09:15.450 --> 00:09:16.380 power through these problems. 00:09:16.380 --> 00:09:18.320 You have to do a little bit of trial and error and maybe use 00:09:18.320 --> 00:09:21.070 your eraser or do some work on side every now and then. 00:09:21.070 --> 00:09:24.140 But anyway, I think you're ready to do some dividing 00:09:24.140 --> 00:09:25.960 decimals problems. 00:09:25.960 --> 00:09:28.010 I hope you have some fun.

Level 4 division

https://www.youtube.com/watch?v=gHTH6PKfpMc

vtt

https://www.youtube.com/api/timedtext?v=gHTH6PKfpMc&ei=gmeUZbjsIc6fp-oP3-aYiAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249842&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=42541E1A351657DA8EEA615E8D74885C8BE0B534.B371E7AA538812CEB132898D43176F557AC224E9&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.250 --> 00:00:05.640 Welcome to the presentation on level 4 division. 00:00:05.640 --> 00:00:09.540 So what makes level 4 division harder than level 3 division is 00:00:09.540 --> 00:00:13.690 instead of having a one-digit number being divided into a 00:00:13.690 --> 00:00:15.700 multi-digit number, we're now going to have a two-addition 00:00:15.700 --> 00:00:17.890 number divided into a multi-digit number. 00:00:17.890 --> 00:00:21.540 So let's get started with some practice problems. 00:00:21.540 --> 00:00:25.140 So let's start with what I would say is a relatively 00:00:25.140 --> 00:00:26.210 straightforward example. 00:00:26.210 --> 00:00:27.595 The level 4 problems you'll see are actually a 00:00:27.595 --> 00:00:28.300 little harder than this. 00:00:28.300 --> 00:00:40.320 But let's say I had 25 goes into 6,250. 00:00:40.320 --> 00:00:46.660 So the best way to think about this is you say, OK, I have 25. 00:00:46.660 --> 00:00:48.520 Does 25 go into 6? 00:00:48.520 --> 00:00:49.300 Well, no. 00:00:49.300 --> 00:00:52.900 Clearly 6 is smaller than 25, so 25 does not go into 6. 00:00:52.900 --> 00:00:55.990 So then ask yourself, well, then if 25 doesn't go into 00:00:55.990 --> 00:00:58.990 6, does 25 go into 62? 00:00:58.990 --> 00:00:59.740 Well, sure. 00:00:59.740 --> 00:01:03.940 62 is larger than 25, so 25 will go into 62? 00:01:03.940 --> 00:01:05.230 Well, let's think about it. 00:01:05.230 --> 00:01:07.370 25 times 1 is 25. 00:01:07.370 --> 00:01:10.800 25 times 2 is 50. 00:01:10.800 --> 00:01:13.180 So it goes into 62 at least two times. 00:01:13.180 --> 00:01:15.700 And 25 times 3 is 75. 00:01:15.700 --> 00:01:16.900 So that's too much. 00:01:16.900 --> 00:01:21.300 So 25 goes into 62 two times. 00:01:21.300 --> 00:01:23.820 And there's really no mechanical way to go 00:01:23.820 --> 00:01:25.030 about figuring this out. 00:01:25.030 --> 00:01:27.170 You have to kind of think about, OK, how many times do 00:01:27.170 --> 00:01:28.375 I think 25 will go into 62? 00:01:28.375 --> 00:01:29.440 And sometimes you get it wrong. 00:01:29.440 --> 00:01:30.560 Sometimes you'll put a number here. 00:01:30.560 --> 00:01:33.180 Say if I didn't know, I would've put a 3 up here and 00:01:33.180 --> 00:01:34.530 then I would've said 3 times 25 and I would've 00:01:34.530 --> 00:01:35.930 gotten a 75 here. 00:01:35.930 --> 00:01:37.640 And then that would have been too large of a number, so 00:01:37.640 --> 00:01:39.840 I would have gone back and changed it to a 2. 00:01:39.840 --> 00:01:44.870 Likewise, if I had done a 1 and I had done 1 tmes 25, when I 00:01:44.870 --> 00:01:46.580 subtracted it out, the difference I would've gotten 00:01:46.580 --> 00:01:48.290 would be larger than 25. 00:01:48.290 --> 00:01:50.150 And then I would know that, OK, 1 is too small. 00:01:50.150 --> 00:01:51.970 I have to increase it to 2. 00:01:51.970 --> 00:01:53.490 I hope I didn't confuse you too much. 00:01:53.490 --> 00:01:56.060 I just want you to know that you shouldn't get nervous if 00:01:56.060 --> 00:01:58.520 you're like, boy, every time I go through the step it's kind 00:01:58.520 --> 00:02:01.170 of like- I kind of have to guess what the numbers is as 00:02:01.170 --> 00:02:02.800 opposed to kind of a method. 00:02:02.800 --> 00:02:04.540 And that's true; everyone has to do that. 00:02:04.540 --> 00:02:08.650 So anyway, so 25 goes into 62 two times. 00:02:08.650 --> 00:02:10.480 Now let's multiply 2 times 25. 00:02:10.480 --> 00:02:13.840 Well, 2 times 5 is 10. 00:02:13.840 --> 00:02:18.740 And then 2 times 2 plus 1 is 5. 00:02:18.740 --> 00:02:21.860 And we know that 25 times 2 is 50 anyway. 00:02:21.860 --> 00:02:23.020 Then we subtract. 00:02:23.020 --> 00:02:24.970 2 minus 0 is 2. 00:02:24.970 --> 00:02:27.630 6 minus 5 is 1. 00:02:27.630 --> 00:02:31.000 And now we bring down the 5. 00:02:31.000 --> 00:02:33.400 So the rest of the mechanics are pretty much just like a 00:02:33.400 --> 00:02:35.540 level 3 division problem. 00:02:35.540 --> 00:02:41.250 Now we ask ourselves, how many times does 25 go into 125? 00:02:41.250 --> 00:02:44.720 Well, the way I think about it is 25-- it goes into 100 about 00:02:44.720 --> 00:02:47.750 four times, so it will go into 125 one more time. 00:02:47.750 --> 00:02:49.560 It goes into it five times. 00:02:49.560 --> 00:02:51.650 If you weren't sure you could try 4 and then you would see 00:02:51.650 --> 00:02:53.110 that you would have too much left over. 00:02:53.110 --> 00:02:57.540 Or if you tried 6 you would see that you would actually get 6 00:02:57.540 --> 00:03:00.020 times 25 is a number larger than 125. 00:03:00.020 --> 00:03:02.380 So you can't use 6. 00:03:02.380 --> 00:03:07.520 So if we say 25 goes into 125 five times then we just 00:03:07.520 --> 00:03:10.270 multiply 5 times 5 is 25. 00:03:13.630 --> 00:03:18.090 5 times 2 is 10 plus 2, 125. 00:03:18.090 --> 00:03:19.400 So it goes in exact. 00:03:19.400 --> 00:03:21.922 So 125 minus 125 is clearly 0. 00:03:21.922 --> 00:03:25.150 Then we bring down this 0. 00:03:25.150 --> 00:03:28.270 And 25 goes into 0 zero times. 00:03:28.270 --> 00:03:29.430 0 times 25 is 0. 00:03:29.430 --> 00:03:30.990 Remainder is 0. 00:03:30.990 --> 00:03:39.500 So we see that 25 goes into 6,250 exactly 250 times. 00:03:39.500 --> 00:03:40.460 Let's do another problem. 00:03:45.820 --> 00:03:50.820 Let's say I had-- let me pick an interesting number. 00:03:50.820 --> 00:03:56.170 Let's say I had 15 and I want to know how many 00:03:56.170 --> 00:04:05.710 times it goes into 2,265. 00:04:05.710 --> 00:04:07.330 Well, we just do the same thing we did before. 00:04:07.330 --> 00:04:09.780 We say OK, does 15 go into 2? 00:04:09.780 --> 00:04:10.710 No. 00:04:10.710 --> 00:04:12.910 So does 15 go into 22? 00:04:12.910 --> 00:04:13.430 Sure. 00:04:13.430 --> 00:04:16.250 15 goes into 22 one time. 00:04:16.250 --> 00:04:18.220 Notice we wrote the 1 above the 22. 00:04:18.220 --> 00:04:20.800 If it go had gone into 2 we would've written the 1 here. 00:04:20.800 --> 00:04:23.260 But 15 goes into 22 one time. 00:04:23.260 --> 00:04:25.430 1 times 15 is 15. 00:04:28.070 --> 00:04:35.000 22 minus 15-- we could do the whole carrying thing-- 1, 12. 00:04:35.000 --> 00:04:36.380 12 minus 5 is 7. 00:04:36.380 --> 00:04:37.850 1 minus 1 is 0. 00:04:37.850 --> 00:04:40.120 22 minus 15 is 7. 00:04:40.120 --> 00:04:43.190 Bring down the 6. 00:04:43.190 --> 00:04:46.980 OK, now how many times does 15 go into 76? 00:04:46.980 --> 00:04:50.330 Once again, there isn't a real easy mechanical way to do it. 00:04:50.330 --> 00:04:53.210 You can kind of eyeball it and estimate. 00:04:53.210 --> 00:04:55.530 Well, 15 times 2 is 30. 00:04:55.530 --> 00:04:58.490 15 times 4 is 60. 00:04:58.490 --> 00:05:01.510 15 times 5 is 75. 00:05:01.510 --> 00:05:05.500 That's pretty close, so let's say 15 goes into 76 five times. 00:05:08.380 --> 00:05:11.820 So 5 times 5 once again, I already figured it out in 00:05:11.820 --> 00:05:13.760 my head, but I'll just do it again. 00:05:13.760 --> 00:05:15.200 5 times 1 is 5. 00:05:15.200 --> 00:05:17.920 Plus 7. 00:05:17.920 --> 00:05:20.740 Oh, sorry. 00:05:20.740 --> 00:05:22.210 5 times 5 is 25. 00:05:25.420 --> 00:05:26.620 5 times 1 is 5. 00:05:26.620 --> 00:05:27.425 Plus 2 is 7. 00:05:30.030 --> 00:05:32.040 Now we just subtract. 00:05:32.040 --> 00:05:34.640 76 minus 75 is clearly 1. 00:05:34.640 --> 00:05:37.770 Bring down that 5. 00:05:37.770 --> 00:05:42.090 Well, 15 goes into 15 exactly one time. 00:05:42.090 --> 00:05:44.330 1 times 15 is 15. 00:05:47.600 --> 00:05:50.340 Subtract it and we get a remainder of 0. 00:05:50.340 --> 00:05:57.410 So 15 goes into 2,265 exactly 151 times. 00:05:57.410 --> 00:06:01.370 So just think about what we're doing here and why it's a 00:06:01.370 --> 00:06:04.100 little bit harder than when you have a one-digit number here. 00:06:04.100 --> 00:06:05.950 Is that you have to kind of think about, well, how many 00:06:05.950 --> 00:06:08.580 times does this two-digit number go into this 00:06:08.580 --> 00:06:09.580 larger number? 00:06:09.580 --> 00:06:13.480 And since you don't know two-digit multiplication 00:06:13.480 --> 00:06:15.770 tables-- very few people do-- you have to do a little 00:06:15.770 --> 00:06:16.440 bit of guesswork. 00:06:16.440 --> 00:06:18.500 Sometimes you can look at this first digit and look at the 00:06:18.500 --> 00:06:20.600 first digit here and make an estimate. 00:06:20.600 --> 00:06:21.800 But sometimes it's trial and error. 00:06:21.800 --> 00:06:23.990 You'll try and when you multiply it out you might get 00:06:23.990 --> 00:06:25.510 it wrong on the first try. 00:06:25.510 --> 00:06:27.690 Let's do another problem. 00:06:27.690 --> 00:06:29.600 And actually, I'm going to pick numbers at random, so it might 00:06:29.600 --> 00:06:31.020 not have an easy remainder. 00:06:31.020 --> 00:06:32.300 But I think you'll get the point. 00:06:32.300 --> 00:06:34.800 I won't teach you decimals now, so I'll just leave the 00:06:34.800 --> 00:06:36.590 remainder if there is one. 00:06:36.590 --> 00:06:49.580 Let's say I had 67 going into 5,978. 00:06:49.580 --> 00:06:52.220 So I just picked these numbers randomly out of my head, so 00:06:52.220 --> 00:06:55.790 I'll show you that I also sometimes have to do a little 00:06:55.790 --> 00:06:57.680 bit of guesswork to figure out how many times one of these 00:06:57.680 --> 00:07:00.350 two-digit numbers go into a larger number. 00:07:00.350 --> 00:07:03.560 So 67 goes into 5 zero times. 00:07:03.560 --> 00:07:06.620 67 goes into 59 zero times. 00:07:06.620 --> 00:07:13.440 67 goes into 597-- so let's see. 00:07:13.440 --> 00:07:19.880 67 is almost 70 and 597 is almost 600. 00:07:19.880 --> 00:07:27.850 So if it was 70 goes into-- 70 times 9 to 630. 00:07:27.850 --> 00:07:29.820 Because 7 times 9 is 63. 00:07:29.820 --> 00:07:33.110 So I'm going to just eyball approximate. 00:07:33.110 --> 00:07:35.060 I'm going to say that it goes into it eight times. 00:07:35.060 --> 00:07:35.950 I might be wrong. 00:07:38.550 --> 00:07:40.980 And you can always check, but well, we're going to actually 00:07:40.980 --> 00:07:42.930 check in this step essentially. 00:07:42.930 --> 00:07:44.620 8 times 7-- well that's 56. 00:07:47.820 --> 00:07:51.160 And then 8 times 6 is 48. 00:07:51.160 --> 00:07:52.900 Plus 2 is 53. 00:07:57.330 --> 00:07:59.310 7 minus 6 is 1. 00:07:59.310 --> 00:08:01.740 9 minus 9 is 6. 00:08:01.740 --> 00:08:03.560 5 minus 5 is 0. 00:08:03.560 --> 00:08:04.650 61. 00:08:04.650 --> 00:08:05.170 So good. 00:08:05.170 --> 00:08:08.240 I got it right because if I got a number here that was larger 00:08:08.240 --> 00:08:13.560 than-- 67 or larger, than that means that this number up 00:08:13.560 --> 00:08:15.050 here wasn't large enough. 00:08:15.050 --> 00:08:17.770 But here, I got a number that's positive because 00:08:17.770 --> 00:08:20.530 536 is less than 597. 00:08:20.530 --> 00:08:24.070 And it's less than 67, so I did that step right. 00:08:24.070 --> 00:08:28.250 So now we bring down this 8. 00:08:28.250 --> 00:08:30.760 Now this one might be a little bit trickier this time. 00:08:30.760 --> 00:08:35.500 Once again, we have almost 70 and here we have almost 630. 00:08:35.500 --> 00:08:37.990 So maybe it will go into it 9 times. 00:08:37.990 --> 00:08:40.610 Well, let's give it a try and see if it does. 00:08:43.780 --> 00:08:46.215 9 times 7 is 63. 00:08:49.340 --> 00:08:51.720 9 times 6 is 54. 00:08:51.720 --> 00:08:53.870 Plus 6 is 60. 00:08:53.870 --> 00:08:54.700 Good. 00:08:54.700 --> 00:08:56.610 So it did actually go into it nine times because 00:08:56.610 --> 00:08:59.480 603 is less than 618. 00:08:59.480 --> 00:09:01.730 8 minus 3 is 5. 00:09:01.730 --> 00:09:04.580 1 minus 0 is 1. 00:09:04.580 --> 00:09:06.970 And 6 minus 6 is 0. 00:09:10.420 --> 00:09:13.480 We have a remainder of 15, which is smaller than 67. 00:09:13.480 --> 00:09:15.420 So I'm not going to teach you decimals right now, so we can 00:09:15.420 --> 00:09:16.970 just leave that remainder. 00:09:16.970 --> 00:09:23.690 So what we could say is that 67 goes into 5,978 89 times. 00:09:23.690 --> 00:09:25.980 And when it goes into it 89 times, you're left with 00:09:25.980 --> 00:09:28.670 a remainder of 15. 00:09:28.670 --> 00:09:31.850 hopefully you're ready now to try some level 00:09:31.850 --> 00:09:33.560 4 division problems. 00:09:33.560 --> 00:09:35.130 Have fun.

Level 4 Subtraction

https://www.youtube.com/watch?v=omUfrXtHtN0

vtt

https://www.youtube.com/api/timedtext?v=omUfrXtHtN0&ei=gGeUZfTWL4iBp-oPm6CC2AY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249840&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=652C2ABE24F6BB7E2DCB4FCA3EBAE40C52F11D4E.B65301EDD0B27147BA30E38A5B5A5EC13F94ECF7&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.090 --> 00:00:04.020 Welcome to the presentation on level 4 subtraction. 00:00:04.020 --> 00:00:06.640 Let's get started with some problems. 00:00:06.640 --> 00:00:22.970 First problem I have here is 33,220 minus 399. 00:00:22.970 --> 00:00:26.250 So this like we did with, I believe, that we also did 00:00:26.250 --> 00:00:29.140 borrowing in the level subtraction, we have to go 00:00:29.140 --> 00:00:31.920 through all of the digits in the top number starting with 00:00:31.920 --> 00:00:36.220 the top right digit and make sure that they are larger 00:00:36.220 --> 00:00:37.320 than the digit below it. 00:00:37.320 --> 00:00:42.080 Because you can only attract a smaller number from 00:00:42.080 --> 00:00:42.710 a larger number. 00:00:42.710 --> 00:00:44.630 You can't do it the other way, at least, until we 00:00:44.630 --> 00:00:45.850 learn negative numbers. 00:00:45.850 --> 00:00:47.710 So let's go through this and check to make sure all the 00:00:47.710 --> 00:00:48.660 top numbers are larger. 00:00:48.660 --> 00:00:49.640 Well immediately, we see no. 00:00:49.640 --> 00:00:52.650 Well this 0 is not larger than 9. 00:00:52.650 --> 00:00:56.300 So we have to borrow to make the 0 bigger. 00:00:56.300 --> 00:01:00.170 So what we do is we borrow 1-- well some people say 00:01:00.170 --> 00:01:01.010 we're borrowing a 1. 00:01:01.010 --> 00:01:03.450 Some people say, I mean, borrowing a 1 from the 00:01:03.450 --> 00:01:06.535 10's place is really like borrowing a 10. 00:01:06.535 --> 00:01:09.840 So for simplicity let's just say we're borrowing a 1. 00:01:09.840 --> 00:01:15.710 So you borrow a 1 from this 2 and this 10 will become a-- the 00:01:15.710 --> 00:01:18.580 0 sorry, will become a 10. 00:01:18.580 --> 00:01:23.040 And since we borrowed that 1, this 2 will become a 1. 00:01:23.040 --> 00:01:26.940 We took 1 away from this 2 and we gave it to the 0 to make 10. 00:01:26.940 --> 00:01:29.560 We actually took 10 away from this 2 because this 00:01:29.560 --> 00:01:30.230 2's in the 10's place. 00:01:30.230 --> 00:01:32.230 I don't want to confuse you too much. 00:01:32.230 --> 00:01:35.600 If just the mechanics of it are we took 1 away from the 2 and 00:01:35.600 --> 00:01:39.410 we put it in front of the 0 to make 10. 00:01:39.410 --> 00:01:40.320 Now let's keep checking. 00:01:40.320 --> 00:01:42.400 So now we have a 1 in this place. 00:01:42.400 --> 00:01:46.430 1 is smaller than 9, so we have to borrow again. 00:01:46.430 --> 00:01:50.400 So we borrow 1 from this 2 now. 00:01:50.400 --> 00:01:52.920 So this 2 now becomes 1. 00:01:52.920 --> 00:01:57.150 And this 1 will now become an 11. 00:01:57.150 --> 00:01:59.920 So now we have a 10 is larger than a 9. 00:01:59.920 --> 00:02:02.250 11 is larger than a 9. 00:02:02.250 --> 00:02:06.810 1 is not larger than 3, so we have to borrow again. 00:02:06.810 --> 00:02:08.920 This is a good problem. 00:02:08.920 --> 00:02:10.360 Maybe I should've warmed you all up a little bit more. 00:02:10.360 --> 00:02:11.540 It involves a lot of borrowing. 00:02:11.540 --> 00:02:13.790 So in order to borrow we do the same thing over again. 00:02:13.790 --> 00:02:17.630 This 1 will become 11. 00:02:17.630 --> 00:02:21.082 And it's going to borrow from this 3, which will become a 2. 00:02:21.082 --> 00:02:22.270 I think we're done now. 00:02:22.270 --> 00:02:25.900 10 is larger than 9, 11 is larger than 9, 11 is larger 00:02:25.900 --> 00:02:28.620 than 3, 2 is larger than nothing, 3 is larger 00:02:28.620 --> 00:02:29.800 than nothing. 00:02:29.800 --> 00:02:30.915 So now we're ready to subtract. 00:02:30.915 --> 00:02:32.550 This is the easy part. 00:02:32.550 --> 00:02:35.220 10 minus 9 is 1. 00:02:35.220 --> 00:02:39.230 11 minus 9 is 2. 00:02:39.230 --> 00:02:45.950 11 minus 3 is 8. 00:02:45.950 --> 00:02:48.560 2 minus nothing is 2. 00:02:48.560 --> 00:02:51.160 3 minus nothing is 3. 00:02:51.160 --> 00:02:56.930 So we get 32,821. 00:02:56.930 --> 00:02:59.480 The only thing that makes this harder than just normal 00:02:59.480 --> 00:03:01.470 subtraction is that you have to know how to do the borrowing. 00:03:01.470 --> 00:03:03.600 And the way I do the borrowing might be different than the way 00:03:03.600 --> 00:03:05.490 you learned in school, but I think it's easier because you 00:03:05.490 --> 00:03:08.680 do all of the borrowing at once instead of switching back and 00:03:08.680 --> 00:03:11.250 forth between borrowing and subtracting. 00:03:11.250 --> 00:03:14.040 So all we did here, we said that 0 is less than 9. 00:03:14.040 --> 00:03:15.320 Let's borrow 1. 00:03:15.320 --> 00:03:19.100 The 0 becomes a 10 because we got this 1 right here. 00:03:19.100 --> 00:03:22.630 We got this 1 from this 2 and this 2 became a 1. 00:03:22.630 --> 00:03:25.390 I think you might see the pattern if we do a couple 00:03:25.390 --> 00:03:26.460 of more problems. 00:03:26.460 --> 00:03:29.370 So let's do a couple of more. 00:03:29.370 --> 00:03:47.571 If I had 25,633 minus 578. 00:03:47.571 --> 00:03:49.860 So name drill. 00:03:49.860 --> 00:03:52.970 Start at the top right and we make sure that the digits on 00:03:52.970 --> 00:03:55.630 top are larger than the digit below it. 00:03:55.630 --> 00:04:00.530 Immediately we see 3 is smaller than 8, so we have to borrow. 00:04:00.530 --> 00:04:02.445 So this 3 will become 13. 00:04:05.820 --> 00:04:11.430 And then we borrow from this 3, which will now become 2. 00:04:11.430 --> 00:04:13.695 We took a 1 away from this 3, it became a 2, and 00:04:13.695 --> 00:04:16.250 this 1 is right here. 00:04:16.250 --> 00:04:19.790 13 is now larger than 8, but 2 is now smaller than 7. 00:04:19.790 --> 00:04:21.120 So we have to borrow again. 00:04:21.120 --> 00:04:24.640 This 2 becomes a 12. 00:04:24.640 --> 00:04:29.290 And this 6 will become a 5. 00:04:29.290 --> 00:04:33.920 13 is larger than 8, 12 is larger than 7, 5 is the same 00:04:33.920 --> 00:04:37.420 as 5, so you can actually do the subtraction. 00:04:37.420 --> 00:04:39.560 Because 5 minus 5 is 0. 00:04:39.560 --> 00:04:41.430 As long as the top number's not smaller than the 00:04:41.430 --> 00:04:42.180 number below it. 00:04:42.180 --> 00:04:44.290 And then obviously this 5 is larger than this 0 and 00:04:44.290 --> 00:04:46.740 this 2 is larger than this nothing here. 00:04:46.740 --> 00:04:48.450 So now we're ready to subtract. 00:04:48.450 --> 00:04:53.890 13 minus 8 is 5. 00:04:53.890 --> 00:04:58.320 12 minus 7 is 5. 00:04:58.320 --> 00:05:01.530 5 minus 5 is 0. 00:05:01.530 --> 00:05:04.780 5 minus nothing is 5. 00:05:04.780 --> 00:05:05.710 Bring down the 2. 00:05:05.710 --> 00:05:13.090 So the answer is 25,055. 00:05:13.090 --> 00:05:16.470 So let's do a problem now that I think will confuse you a 00:05:16.470 --> 00:05:19.900 little bit more because the borrowing isn't as easy. 00:05:19.900 --> 00:05:22.480 You have to actually borrow from a couple places. 00:05:22.480 --> 00:05:39.800 Let's say I had 37,002 minus-- let's say I had 155. 00:05:39.800 --> 00:05:42.090 So the same drill. 00:05:42.090 --> 00:05:45.130 So this 2 is less than 5, so we have to borrow. 00:05:45.130 --> 00:05:48.820 So this 2 will become a 12. 00:05:48.820 --> 00:05:51.930 But huh, there's a 0 here, so you can't borrow from the 0. 00:05:51.930 --> 00:05:53.940 Some people will let you borrow from the 0, but I think that 00:05:53.940 --> 00:05:56.103 just confuses things because you can't borrow from the 00:05:56.103 --> 00:05:57.450 0, there's nothing there. 00:05:57.450 --> 00:05:59.710 So instead of borrowing from the 0 we look to this 0. 00:05:59.710 --> 00:06:01.470 Well, there's still nothing there. 00:06:01.470 --> 00:06:04.250 So now we look, oh, there's a 7 here. 00:06:04.250 --> 00:06:07.620 So what we do is instead of borrowing a 1 from the 0, which 00:06:07.620 --> 00:06:15.490 is hard to do, we borrow 1 from this 700, from this whole 700. 00:06:15.490 --> 00:06:18.330 And what is 700 minus 1? 00:06:18.330 --> 00:06:19.480 Right, it's 699. 00:06:19.480 --> 00:06:26.710 So that 700 becomes 699. 00:06:26.710 --> 00:06:28.770 Cross all of this out. 00:06:28.770 --> 00:06:31.320 And now let's check our numbers again. 00:06:31.320 --> 00:06:36.870 12 is larger than 5, nine is larger than 5, 9 is larger than 00:06:36.870 --> 00:06:40.500 1, 6 is larger than nothing, and 3 is larger than nothing, 00:06:40.500 --> 00:06:42.410 so we're ready to subtract. 00:06:42.410 --> 00:06:45.670 12 minus 5 is 7. 00:06:45.670 --> 00:06:48.730 9 minus 5 is 4. 00:06:48.730 --> 00:06:51.606 9 minus 1 is 8. 00:06:51.606 --> 00:06:54.180 6 minus nothing is 6. 00:06:54.180 --> 00:06:55.795 3 minus nothing is 3. 00:06:55.795 --> 00:06:57.630 So there, we're done. 00:06:57.630 --> 00:07:02.100 The answer is 36,847. 00:07:02.100 --> 00:07:05.851 I think we could have time for one more problem. 00:07:05.851 --> 00:07:21.870 Let's say I had 3,201 minus-- let's say it's 502. 00:07:21.870 --> 00:07:22.950 Same drill. 00:07:22.950 --> 00:07:27.040 1 is less than 2, so we have to borrow. 00:07:27.040 --> 00:07:28.370 Turn that into an 11. 00:07:28.370 --> 00:07:30.380 But you can't borrow from the 0, so you're going to have to 00:07:30.380 --> 00:07:33.000 borrow from this entire 20. 00:07:33.000 --> 00:07:34.800 Well, what's 20 minus 1? 00:07:34.800 --> 00:07:37.060 Right, it's 19. 00:07:37.060 --> 00:07:38.080 This becomes a 19. 00:07:40.660 --> 00:07:41.660 So let's check again. 00:07:41.660 --> 00:07:43.480 11 is greater than 2. 00:07:43.480 --> 00:07:44.320 Check. 00:07:44.320 --> 00:07:45.570 9 is greater than 0. 00:07:45.570 --> 00:07:46.620 Check. 00:07:46.620 --> 00:07:46.970 Uh-oh. 00:07:46.970 --> 00:07:49.310 1 is not greater than 5. 00:07:49.310 --> 00:07:50.840 So we have to borrow again. 00:07:50.840 --> 00:07:53.000 This 1 becomes an 11. 00:07:53.000 --> 00:07:56.450 We borrowed from this 3, which becomes a 2. 00:07:56.450 --> 00:08:00.630 11 is greater than 2, 9 is greater than 0, 11 is greater 00:08:00.630 --> 00:08:04.200 than 5, 2 is obviously greater than nothing below it. 00:08:04.200 --> 00:08:06.100 So we're ready to subtract. 00:08:06.100 --> 00:08:09.130 11 minus 2 is 9. 00:08:09.130 --> 00:08:12.020 9 minus 0 is 9. 00:08:12.020 --> 00:08:14.470 11 minus 5 is 6. 00:08:14.470 --> 00:08:17.590 And 2 minus nothing is 2. 00:08:17.590 --> 00:08:26.480 So 3,201 minus 502 is equal to 2,699. 00:08:26.480 --> 00:08:28.340 I think you're now ready to try some of the level 4 00:08:28.340 --> 00:08:29.110 subtraction problems. 00:08:29.110 --> 00:08:31.530 You just always have to remember, do you're 00:08:31.530 --> 00:08:32.950 borrowing first. 00:08:32.950 --> 00:08:36.030 Make sure all the numbers on top are larger than, or at 00:08:36.030 --> 00:08:38.620 least as large as all the numbers on the bottom. 00:08:38.620 --> 00:08:40.720 And then you can just do your subtraction like a normal 00:08:40.720 --> 00:08:42.010 subtraction problem. 00:08:42.010 --> 00:08:44.500 I hope you have some fun doing this. 00:08:44.500 --> 00:08:46.090 Talk to you later.

Subtracting decimals (old)

https://www.youtube.com/watch?v=0mOH-qNGM7M

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https://www.youtube.com/api/timedtext?v=0mOH-qNGM7M&ei=gGeUZZ3YLdDMp-oPx_mlgA8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249840&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=78E3E9224D1C3CC19EE5A29AC84529F8A9CB2CA1.8F1A5B3AF4A9236A37D43F6849D213AB3B257579&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.060 --> 00:00:04.490 Welcome to the presentation on subtracting decimal numbers. 00:00:04.490 --> 00:00:07.390 Let's get started with some problems. 00:00:07.390 --> 00:00:23.470 The first problem I have here says 5.73 minus 00:00:23.470 --> 00:00:30.670 0.0821 equals who knows? 00:00:30.670 --> 00:00:33.260 So the first thing you always want to do with a decimal like 00:00:33.260 --> 00:00:35.680 this, and I actually kind of inadvertently did this, is that 00:00:35.680 --> 00:00:37.920 you want to line up the decimals. 00:00:37.920 --> 00:00:40.250 so you actually want this decimal to be right 00:00:40.250 --> 00:00:41.540 above this decimal. 00:00:41.540 --> 00:00:43.870 I almost did that when I did it, it must have been my 00:00:43.870 --> 00:00:44.870 subconscious doing it. 00:00:44.870 --> 00:00:46.290 But let me just do it a little bit neater. 00:00:46.290 --> 00:00:52.630 So it's 5.73, and I'll put the decimal here. 00:00:55.840 --> 00:01:00.060 0.0821. 00:01:00.060 --> 00:01:02.580 And some people say it's good to always put a zero in 00:01:02.580 --> 00:01:03.350 front of the decimal. 00:01:03.350 --> 00:01:05.600 My wife's a doctor and she says it's critical otherwise 00:01:05.600 --> 00:01:08.310 you might give someone the wrong amount of medicine. 00:01:08.310 --> 00:01:10.510 So, we've lined up the decimals and now we're 00:01:10.510 --> 00:01:11.175 ready to subtract. 00:01:13.960 --> 00:01:16.150 So one thing that you have to think about when you do 00:01:16.150 --> 00:01:20.830 decimals is we're going to have to subtract this 21 00:01:20.830 --> 00:01:23.890 ten-thousandths or this 2 and this 1 from something. 00:01:23.890 --> 00:01:26.470 We can't just subtract it from this blank space. 00:01:26.470 --> 00:01:30.910 So we have to add two 0s here. 00:01:30.910 --> 00:01:33.270 And as you know, with the decimal when you add 0s to the 00:01:33.270 --> 00:01:34.870 end of it, it really doesn't change the value 00:01:34.870 --> 00:01:35.980 of the decimal. 00:01:35.980 --> 00:01:38.650 So at this point, we just view this like a level 00:01:38.650 --> 00:01:40.660 four subtraction problem. 00:01:40.660 --> 00:01:44.610 So the first thing we do in any subtraction problem is see if 00:01:44.610 --> 00:01:46.970 any of the numbers on top are smaller than any of the 00:01:46.970 --> 00:01:47.770 numbers on the bottom. 00:01:47.770 --> 00:01:49.350 Well in this case there are a lot of them. 00:01:49.350 --> 00:01:53.040 So this 0 is less than this 1, this 0 is less than this 2, 00:01:53.040 --> 00:01:54.410 this 3 is less than this 8. 00:01:54.410 --> 00:01:56.780 So we're going to have to borrow. 00:01:56.780 --> 00:01:59.990 Some people will like to do their borrowing and 00:01:59.990 --> 00:02:02.250 subtracting, they kind of alternate between the two. 00:02:02.250 --> 00:02:04.510 I like to do all of my borrowing ahead of time. 00:02:04.510 --> 00:02:06.540 So what I do is I start in the top right and I say 00:02:06.540 --> 00:02:09.990 OK, 0 is less than 1. 00:02:09.990 --> 00:02:13.670 So that 0 becomes a 10. 00:02:13.670 --> 00:02:15.710 But in order to become a 10 I would have had to 00:02:15.710 --> 00:02:17.620 borrow 1 from some place. 00:02:17.620 --> 00:02:19.580 I look to the left of that 0 and I say well, can 00:02:19.580 --> 00:02:21.500 I borrow the 1 from 0? 00:02:21.500 --> 00:02:22.570 Well, no. 00:02:22.570 --> 00:02:23.440 This is just the way I do it. 00:02:23.440 --> 00:02:25.610 There are people who would actually let you borrow the 1 00:02:25.610 --> 00:02:27.630 from the 0, but I say no, instead of borrowing the 1 00:02:27.630 --> 00:02:30.790 from the 0, I borrow the 1 from this entire 30. 00:02:30.790 --> 00:02:34.470 So this 30 -- see, there's a 30 so I'm going to borrow 1 00:02:34.470 --> 00:02:36.650 from it and it becomes 29. 00:02:39.590 --> 00:02:45.710 So we borrowed 1 from this 30 to get a 10 here, and now let's 00:02:45.710 --> 00:02:49.700 check again to see if all of our numbers on top are larger 00:02:49.700 --> 00:02:50.900 than all the numbers on the bottom. 00:02:50.900 --> 00:02:56.550 Well 10 is larger than 1, 9 is larger than 2, 2 00:02:56.550 --> 00:02:58.090 is not larger than 8. 00:02:58.090 --> 00:02:59.230 So we have to borrow again. 00:02:59.230 --> 00:03:05.040 So if we're going to borrow, the 2 becomes a 12, and the 7 00:03:05.040 --> 00:03:08.100 -- we borrowed 1 from that -- becomes a 6. 00:03:08.100 --> 00:03:08.970 So let's check again. 00:03:08.970 --> 00:03:13.500 10 is larger than 1, 9 is larger than 2, 12 is larger 00:03:13.500 --> 00:03:17.280 than 8, 6 is larger than 0, and 5 is larger than 0. 00:03:17.280 --> 00:03:19.330 So now we've done all of our borrowing and we're ready to 00:03:19.330 --> 00:03:22.010 do some subtraction, and this is the easy part. 00:03:22.010 --> 00:03:25.630 10 minus 1 is 9. 00:03:25.630 --> 00:03:29.460 9 minus 2 is 7. 00:03:29.460 --> 00:03:33.370 12 minus 8 is 4. 00:03:33.370 --> 00:03:36.420 6 minus 0 is 6. 00:03:36.420 --> 00:03:39.920 5 minus 0 is 5. 00:03:39.920 --> 00:03:42.640 And we just bring down the decimal point. 00:03:42.640 --> 00:03:43.850 So there's our answer. 00:03:43.850 --> 00:03:55.130 5.73 minus 0.0821 is equal to 5.6479. 00:03:55.130 --> 00:03:56.530 There you go. 00:03:56.530 --> 00:03:58.655 I probably confused you, so let's do some more problems. 00:04:01.880 --> 00:04:05.260 Here's another one. 00:04:05.260 --> 00:04:09.795 8 -- let me leave some space on top to do the borrowing 00:04:09.795 --> 00:04:25.880 -- 8.25 minus 0.0105. 00:04:25.880 --> 00:04:29.100 So what was that first step that I always have to do? 00:04:29.100 --> 00:04:30.030 Right. 00:04:30.030 --> 00:04:31.190 To line up the decimals. 00:04:31.190 --> 00:04:32.720 So let me do that. 00:04:32.720 --> 00:04:46.080 So it's 8.25 and 0.0105. 00:04:46.080 --> 00:04:50.300 Notice I lined up this decimal right below this decimal. 00:04:50.300 --> 00:04:54.330 Now I add the 0s, just because this 0 and this 5 need to be 00:04:54.330 --> 00:04:56.040 subtracted from something. 00:04:56.040 --> 00:04:57.630 Now let me do my borrowing. 00:04:57.630 --> 00:05:00.190 So once again, all I do is check to see whether the 00:05:00.190 --> 00:05:03.180 top number is larger than the number below it. 00:05:03.180 --> 00:05:08.550 Well, this 0 is smaller than 5, so I'm going to have to borrow. 00:05:08.550 --> 00:05:09.350 So I'm going to borrow. 00:05:09.350 --> 00:05:11.400 I can't borrow from this 0, I have to borrow 00:05:11.400 --> 00:05:12.990 from this entire 50. 00:05:12.990 --> 00:05:16.900 So this 50, if I borrow 1 from 50 I get 49. 00:05:19.820 --> 00:05:22.840 And this 0 will then become a 10, right? 00:05:22.840 --> 00:05:26.060 I borrowed 1 from 50 to get a 10. 00:05:26.060 --> 00:05:27.560 Now, am I done? 00:05:27.560 --> 00:05:29.440 10 is larger than 5. 00:05:29.440 --> 00:05:31.550 9 is larger than 0. 00:05:31.550 --> 00:05:33.970 4 is larger than 1. 00:05:33.970 --> 00:05:35.910 2 is larger than 0. 00:05:35.910 --> 00:05:37.530 8 is larger than 0. 00:05:37.530 --> 00:05:39.610 So I think I'm ready to subtract. 00:05:39.610 --> 00:05:44.130 10 minus 5, well that's 5. 00:05:44.130 --> 00:05:47.240 9 minus 0 is 9. 00:05:47.240 --> 00:05:51.140 4 minus 1 is 3. 00:05:51.140 --> 00:05:54.710 2 minus 0 is 2. 00:05:54.710 --> 00:05:57.960 8 minus 0 is 8. 00:05:57.960 --> 00:06:01.730 And I bring down the decimal point. 00:06:01.730 --> 00:06:05.690 So if you mastered level four subtraction, the decimal 00:06:05.690 --> 00:06:08.450 problems really are just about lining up the decimal point, 00:06:08.450 --> 00:06:10.060 adding the 0s and then just doing a normal 00:06:10.060 --> 00:06:11.440 subtraction problem. 00:06:11.440 --> 00:06:14.400 In general with subtraction I think most people have the most 00:06:14.400 --> 00:06:17.260 trouble with the borrowing. 00:06:17.260 --> 00:06:19.230 The way I do it I think is a little bit different than is 00:06:19.230 --> 00:06:20.100 taught in a lot of schools. 00:06:20.100 --> 00:06:22.140 A lot of schools they'll do the subtraction and they'll 00:06:22.140 --> 00:06:23.670 borrow alternatively. 00:06:23.670 --> 00:06:27.140 But I find this easier when I just borrow ahead of time, and 00:06:27.140 --> 00:06:30.320 I also, like for example in this problem, when I had to 00:06:30.320 --> 00:06:32.930 make this 0 into a 10, instead of borrowing from the 0, which 00:06:32.930 --> 00:06:35.820 is not intuitive because I can't really borrow from the 0, 00:06:35.820 --> 00:06:40.720 I borrowed from this entire 50 and I made that into a 49. 00:06:40.720 --> 00:06:42.040 Let's do one more problem. 00:06:47.000 --> 00:06:58.500 If I have 2.64 minus 0.0486. 00:06:58.500 --> 00:07:01.670 So once again, let's line up the decimal points. 00:07:01.670 --> 00:07:11.720 2.64 and it's .0486. 00:07:11.720 --> 00:07:16.620 Lined up the decimal points, the 0s on top. 00:07:16.620 --> 00:07:19.380 You're going to have a 0 here, so I have to borrow. 00:07:19.380 --> 00:07:20.490 Becomes a 10. 00:07:20.490 --> 00:07:22.260 Can't borrow from the 0, so I have to borrow 00:07:22.260 --> 00:07:23.920 from this entire 40. 00:07:23.920 --> 00:07:27.166 So this 40 becomes a 39. 00:07:27.166 --> 00:07:29.440 I think I'm running out of space. 00:07:29.440 --> 00:07:31.450 So 10 is larger than 6. 00:07:31.450 --> 00:07:33.020 9 is larger than 8. 00:07:33.020 --> 00:07:34.420 3 is not larger than 4. 00:07:34.420 --> 00:07:36.750 So this 3 I'm going to have to borrow. 00:07:36.750 --> 00:07:38.480 So 3 becomes a 13. 00:07:38.480 --> 00:07:41.000 I apologize, I'm becoming scrunched. 00:07:41.000 --> 00:07:43.396 And this 6 becomes a 5. 00:07:43.396 --> 00:07:46.040 This is really bad, I shouldn't do it so messy. 00:07:46.040 --> 00:07:49.040 But now we say the 10 is larger than 6, the 9 is larger than 00:07:49.040 --> 00:07:52.560 the 8, this 13, this 13 should be on top of that 3. 00:07:52.560 --> 00:07:55.550 The 13 is larger than 4, and 5 is larger than 0. 00:07:55.550 --> 00:07:57.120 So we're ready to subtract. 00:07:57.120 --> 00:07:59.270 10 minus 6 is 4. 00:07:59.270 --> 00:08:01.590 9 minus 8 is 1. 00:08:01.590 --> 00:08:04.320 13 minus 4 is 9. 00:08:04.320 --> 00:08:07.800 5 minus 0 is 5. 00:08:07.800 --> 00:08:09.880 2 minus nothing is 2. 00:08:09.880 --> 00:08:12.180 Bring down the decimal point. 00:08:12.180 --> 00:08:21.530 So 2.64 minus 0.0486 is equal to 2.5914. 00:08:21.530 --> 00:08:23.200 Hope I didn't confuse you too much. 00:08:23.200 --> 00:08:26.740 But I think you're ready now to try the 00:08:26.740 --> 00:08:28.680 subtraction of decimals. 00:08:28.680 --> 00:08:30.250 Have fun.

Unit conversion within the metric system

https://www.youtube.com/watch?v=w0nqd_HXHPQ

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WEBVTT Kind: captions Language: en 00:00:00.880 --> 00:00:03.310 Welcome to the presentation on units. 00:00:03.310 --> 00:00:05.390 Let's get started. 00:00:05.390 --> 00:00:12.770 So if I were to ask you, or if I were to say, I have traveled 00:00:12.770 --> 00:00:20.910 0.05 kilometers-- some people say KIL-ometers 00:00:20.910 --> 00:00:22.020 or kil-O-meters. 00:00:24.850 --> 00:00:28.250 If I have traveled 0.05 kilometers, how many 00:00:28.250 --> 00:00:30.865 centimeters have I traveled? 00:00:30.865 --> 00:00:32.590 That's question mark centimeters. 00:00:35.880 --> 00:00:38.390 So before we break into the math, it's important to just 00:00:38.390 --> 00:00:41.730 know what these prefixes centi and kilo mean. 00:00:41.730 --> 00:00:44.650 And it's good to memorize this, or when you're first starting 00:00:44.650 --> 00:00:46.130 to do these problems, you can just write them down on a piece 00:00:46.130 --> 00:00:48.080 of paper, just so you have a reference. 00:00:48.080 --> 00:01:06.460 So kilo means 1,000, hecto means 100, deca means 10. 00:01:06.460 --> 00:01:09.840 You might recognize that from decade, 10 years. 00:01:09.840 --> 00:01:13.980 And then, of course, you have no prefix, means 1. 00:01:13.980 --> 00:01:15.580 No prefix. 00:01:15.580 --> 00:01:18.710 No prefix equals 1. 00:01:18.710 --> 00:01:28.100 deci is equal to 0.1 or 1/10. 00:01:28.100 --> 00:01:32.510 centi-- I keep changing between cases. 00:01:32.510 --> 00:01:38.590 centi is equal to 0.01, or 1/100. 00:01:38.590 --> 00:01:45.110 And then milli is equal to 0.001, and that's the 00:01:45.110 --> 00:01:48.710 same thing as 1/1,000. 00:01:48.710 --> 00:01:52.810 And the way I remember, I mean, centi, if you think of a 00:01:52.810 --> 00:01:54.740 centipede, it has a 100 feet. 00:01:54.740 --> 00:01:58.360 A millipede, I'm not sure if a millipede has 1,000 feet, but 00:01:58.360 --> 00:02:00.850 that's the implication when someone says a millipede 00:02:00.850 --> 00:02:02.550 because pede means feet. 00:02:02.550 --> 00:02:03.810 So let's go back to the problem. 00:02:03.810 --> 00:02:08.490 If I have 0.05 kilometers, how many centimeters do I have? 00:02:08.490 --> 00:02:10.350 Whenever I do a problem like this, I like to actually 00:02:10.350 --> 00:02:12.930 convert my number to meters because that's 00:02:12.930 --> 00:02:14.290 very easy for me. 00:02:14.290 --> 00:02:18.150 And actually, I'm going to abbreviate this is km, and 00:02:18.150 --> 00:02:21.530 we can abbreviate this as cm for centimeters. 00:02:21.530 --> 00:02:28.480 So let's say 0.05 km. 00:02:28.480 --> 00:02:32.500 Well, if I want to convert this into meters, is it going to be 00:02:32.500 --> 00:02:37.150 more than 0.05 meters or less than 0.05? 00:02:37.150 --> 00:02:40.820 Well, a kilometer is a very large distance, so in terms 00:02:40.820 --> 00:02:43.430 of meters, it's going to be a much bigger number. 00:02:43.430 --> 00:02:52.600 So we can multiply this times 1,000 meters, and I'll do 00:02:52.600 --> 00:02:53.880 it over 1, per kilometer. 00:02:56.490 --> 00:02:58.050 And what does that get? 00:02:58.050 --> 00:03:04.890 Well, 0.05 times 1,000 is equal to 50, right? 00:03:04.890 --> 00:03:07.670 I just multiplied 0.05 times 1,000. 00:03:07.670 --> 00:03:12.610 And with the units, I now have kilometers times 00:03:12.610 --> 00:03:16.260 meters over kilometers. 00:03:16.260 --> 00:03:18.290 And the kilometers cancel out. 00:03:18.290 --> 00:03:22.260 And just so you're familiar with this, you can treat units 00:03:22.260 --> 00:03:24.640 exactly the same way that you would treat numbers 00:03:24.640 --> 00:03:25.670 or variables. 00:03:25.670 --> 00:03:28.990 As long as you have the same unit in the numerator and the 00:03:28.990 --> 00:03:30.970 denominator, you can cancel them out, assuming that you're 00:03:30.970 --> 00:03:33.490 not adding units, you're multiplying units. 00:03:33.490 --> 00:03:36.540 So you have kilometers times meters divided by kilometers, 00:03:36.540 --> 00:03:40.030 and that equals 50 meters. 00:03:40.030 --> 00:03:43.890 And it's good to always do a reality check after every step. 00:03:43.890 --> 00:03:45.700 Usually when you do these types of problems, you know, OK, if I 00:03:45.700 --> 00:03:48.780 want to go from kilometers to meters, I'm going to use the 00:03:48.780 --> 00:03:51.010 number 1,000, because that's the relationship between 00:03:51.010 --> 00:03:52.170 a kilometer and a meter. 00:03:52.170 --> 00:03:54.830 And you're always confused, well, do I multiply by 1,000, 00:03:54.830 --> 00:03:56.250 or do I divide by 1,000? 00:03:56.250 --> 00:03:58.720 And you always have to say, well, if I'm going from 00:03:58.720 --> 00:04:03.050 kilometers to meters, I'm going-- 1 kilometer is 00:04:03.050 --> 00:04:05.550 1,000 meters, right? 00:04:05.550 --> 00:04:07.820 So I'm going to be multiplying by 1,000. 00:04:07.820 --> 00:04:09.130 I'm going to get a bigger number. 00:04:09.130 --> 00:04:12.490 So that's why I went from 0.05, and I multiplied it 00:04:12.490 --> 00:04:14.600 by 1,000, and I got 50. 00:04:14.600 --> 00:04:16.100 So let's get back to the problem. 00:04:16.100 --> 00:04:19.400 0.05 kilometers is equal to 50 meters. 00:04:19.400 --> 00:04:20.210 We're not done yet. 00:04:20.210 --> 00:04:23.280 Now, you need to convert those 50 meters into centimeters. 00:04:23.280 --> 00:04:25.540 Well, we do the same thing. 00:04:25.540 --> 00:04:32.730 50 meters times-- how many-- so what's the relationship between 00:04:32.730 --> 00:04:33.740 meters and centimeters? 00:04:33.740 --> 00:04:36.320 Well, if we look at the chart, we see it's 100. 00:04:36.320 --> 00:04:38.310 And the question I'm going to ask you, am I going to multiply 00:04:38.310 --> 00:04:41.580 by 100, or am I going to divide by 100? 00:04:41.580 --> 00:04:42.520 Well, it's the same thing. 00:04:42.520 --> 00:04:45.320 We're going from a bigger unit to a smaller unit, so one of 00:04:45.320 --> 00:04:48.240 a bigger unit is equal to a bunch of the smaller units. 00:04:48.240 --> 00:04:50.330 So we're going to multiply. 00:04:50.330 --> 00:04:56.590 So we say times 100 centimeters per meter, right? 00:04:56.590 --> 00:04:57.330 And that just makes sense. 00:04:57.330 --> 00:04:59.710 There's 100 centimeters per meter. 00:04:59.710 --> 00:05:02.840 So 50 meters times 100 centimeters per meter is equal 00:05:02.840 --> 00:05:12.625 to 50 times 100 is 5,000, and then the meters cancel out, 00:05:12.625 --> 00:05:15.260 and you get centimeters. 00:05:15.260 --> 00:05:21.330 So what we have here is that 0.05 kilometers is equal 00:05:21.330 --> 00:05:24.850 to 5,000 centimeters. 00:05:24.850 --> 00:05:26.090 Let's do another problem. 00:05:26.090 --> 00:05:28.580 I think the more examples you see, it'll make them 00:05:28.580 --> 00:05:28.940 a little more sense. 00:05:28.940 --> 00:05:31.360 And always try to visualize what we're doing, the scale. 00:05:31.360 --> 00:05:32.830 Otherwise, it's very confusing whether you 00:05:32.830 --> 00:05:36.020 should multiply or divide. 00:05:36.020 --> 00:05:41.100 Let's say I have 422 decigrams. 00:05:47.110 --> 00:05:49.530 Grams are a measure of mass. 00:05:49.530 --> 00:05:51.470 One gram is actually a very small amount. 00:05:51.470 --> 00:05:53.710 That's what you measure-- I guess in the metric 00:05:53.710 --> 00:05:56.770 system, they measure gold in terms of grams. 00:05:56.770 --> 00:06:03.180 And I want to convert this into milligrams. 00:06:03.180 --> 00:06:07.200 So before we start the problem, let's just do a reality check. 00:06:07.200 --> 00:06:09.700 Am I going from a bigger unit to a smaller unit, or a smaller 00:06:09.700 --> 00:06:10.870 unit to a bigger unit? 00:06:10.870 --> 00:06:18.870 Well, decigrams, that's 1/10 of a gram, and I'm going 00:06:18.870 --> 00:06:23.110 to 1/1,000 of a gram. 00:06:25.750 --> 00:06:26.960 So there's two ways of doing this. 00:06:26.960 --> 00:06:29.940 We can convert to grams and then convert to the other unit. 00:06:29.940 --> 00:06:32.350 That sometimes make things easy. 00:06:32.350 --> 00:06:35.910 Or we could say, well, how many milligrams is 00:06:35.910 --> 00:06:37.920 equal to one decigram? 00:06:37.920 --> 00:06:42.140 Well, a milligram, as we see here, is 100 00:06:42.140 --> 00:06:43.270 times smaller, right? 00:06:43.270 --> 00:06:47.330 To go from 1/10 to 1/1,000, you have to decrease 00:06:47.330 --> 00:06:48.780 in size by 100. 00:06:48.780 --> 00:07:02.570 So we could just say 422 decigrams times 100 00:07:02.570 --> 00:07:08.230 milligrams per decigram. 00:07:12.390 --> 00:07:17.620 And then the decigrams will cancel out, and I'll get 422 00:07:17.620 --> 00:07:24.405 times 100, 42,200 milligrams. 00:07:29.950 --> 00:07:31.790 Now, another way you could have done it is the way we just 00:07:31.790 --> 00:07:32.810 did that last problem. 00:07:32.810 --> 00:07:36.880 We could say 422 decigrams, we could convert that to grams. 00:07:36.880 --> 00:07:41.620 We could say 422-- I'm just going to say dg. 00:07:41.620 --> 00:07:45.500 That's not really a familiar unit-- decigrams. 00:07:45.500 --> 00:07:48.150 And how many decigrams are there per gram? 00:07:48.150 --> 00:07:51.260 If we're going to gram, 422 is going to be a smaller 00:07:51.260 --> 00:07:53.360 number of grams, right? 00:07:53.360 --> 00:07:59.110 So we could say times 1 decigram is equal 00:07:59.110 --> 00:08:01.230 to how many grams? 00:08:01.230 --> 00:08:07.200 Well, 1 decigram is equal to-- no, sorry. 00:08:07.200 --> 00:08:09.800 1 gram is equal to how many decigrams? 00:08:09.800 --> 00:08:13.650 Well, 1 gram is equal to 10 decigrams. 00:08:13.650 --> 00:08:16.560 And the reason why this makes sense is if we have a decigram 00:08:16.560 --> 00:08:18.290 in the numerator here, we want a decigram in the 00:08:18.290 --> 00:08:19.790 denominator here. 00:08:19.790 --> 00:08:26.480 So if we have decigrams cancel out, 422 decigrams will equal-- 00:08:26.480 --> 00:08:33.240 that divided by 10 is equal to 42.2 grams. 00:08:33.240 --> 00:08:35.770 And now we can just go from grams to milligrams. 00:08:35.770 --> 00:08:37.120 Well, that's an easy one. 00:08:37.120 --> 00:08:41.150 1 gram is equal to 1,000 milligrams, so we could say 00:08:41.150 --> 00:08:48.620 times 1,000 milligram per gram. 00:08:48.620 --> 00:08:53.680 The grams cancel out, and we're left with 42,200 00:08:53.680 --> 00:08:55.310 milligrams, right? 00:08:55.310 --> 00:08:59.380 42.2 times 1,000. 00:08:59.380 --> 00:09:01.530 Hopefully, that doesn't confuse you too much. 00:09:01.530 --> 00:09:04.330 The important thing is to always take a step back and 00:09:04.330 --> 00:09:06.960 really visualize and think about, should I be getting a 00:09:06.960 --> 00:09:09.310 larger number or a smaller number than the one 00:09:09.310 --> 00:09:10.920 I started off with? 00:09:10.920 --> 00:09:13.840 I think you're ready to now try some problems. 00:09:13.840 --> 00:09:15.320 Have fun!