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With an understanding that vectors are simply mathematical equivalents of mythological hybrid beasts, we will investigate their parts to see if we can understand them better. Surely, with the help of the minotaur and griffin vectors, the calculus...
Please purchase the full module to see the rest of this course
Purchase the Points, Vectors, and Functions Pass and get full access to this Calculus chapter. No limits found here. |
PG Calculator Standard for Android is a powerful scientific calculator. It works in Algebraic and RPN (Reverse Polish Notation) modes and allows to use the following mathematical functions: addition, subtraction, multiplication, and division, power and roots, trigonometrical functions: sin, cos, tan, asin, acos, and atan, natural and decimal logarithms, factorials, and octal, binary, and hexadecimal number formats.
What's new in this version: Version 1.5.4 fixes minor issue with window leakage while changing the skin |
This interactive resource, produced by the University of Leicester, is designed to enable students to explore transformations of shapes including translation by a vector, stretches, rotations around a point, reflection in the axes and reflections in the lines y=x and y=-x.
The first activity draws a parallelogram which is translated…
This interactive resource, produced by the University of Leicester, is designed to enable students to explore transformations of functions including translations, stretches, reflection in the axes and rotations.
The first activity uses function notation to explore translations parallel to the x axis and translations parallel…
These resources commisioned by the Qualifications and Curriculum Agency (QCDA) are to support the linked pair of mathematics GCSEs piloted from September 2010, which were developed in response to Professor Adrian Smith's inquiry into post-14 mathematics education in 2004. Together they assess the National Curriculum mathematics… |
Synopsis
This is the first comprehensive text on African Mathematics that can be used to address some of the problematic issues in this area. These issues include attitudes, curriculum development, educational change, academic achievement, standardized and other tests, performance factors, student characteristics, cross-cultural differences and studies, literacy, native speakers, social class and differences, equal education, teaching methods, knowledge level, educational guidelines and policies, transitional schools, comparative education, other subjects such as physics and social studies, surveys, talent, educational research, teacher education and qualifications, academic standards, teacher effectiveness, lesson plans and modules, teacher characteristics, instructional materials, program effectiveness, program evaluation, African culture, African history, Black studies, class activities, educational games, number systems, cognitive ability, foreign influence, and fundamental concepts. What unifies the chapters in this book can appear rather banal, but many mathematical insights are so obvious and so fundamental that they are difficult to absorb, appreciate, and express with fresh clarity. Some of the more basic insights are isolated by accounts of investigators who have earned their contemporaries' respect. Winner of the 2012 Cecil B. Currey Book |
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Mathematics for Economics and Business
Rebecca Taylor Simon Hawkins
ISBN: 0077107861 Copyright year: 2008
Welcome to the Online Learning Centre for Mathematics for Economics and Business, 1st European edition
Mathematics for Economics and Business by Rebecca Taylor and Simon Hawkins is an introductory level text aimed at undergraduates requiring an understanding of mathematics. For many students embarking on an economics or business course, the level of mathematics required to
understand key topics can at first seem daunting. This student friendly text takes a step-by-step approach to explaining mathematical principles and applying these to an economic and business context. The range of study tools employed throughout the text caters for different learning styles and levels of understanding, encouraging students to take an active role in their learning of the subject. This edition features:
Coverage of core mathematical principles found in economics and business courses, assuming little prior knowledge of the subject
Student notes provided in boxes within each chapter as a useful quick reference
tool, summarising key terms and providing tips to help understanding
Worked examples, to consolidate learning and demonstrate the mathematical
principles as applied in an economic and business context
Quick Problem boxes to test understanding and application of the mathematical
principles taught in each chapter. Answers are provided at the end of the chapter so that students can check progress
The book is accompanied by a range of supplementary resources designed to provide students with the support they need to gain an understanding of mathematical principles.
On this OLC you can find a host of information about the book, as well as a range of downloadable supplements for students and lecturers.
THE STUDENT CENTRE
The Student Centre contains material to accompany the study of Mathematics for Economics and Business. This material includes:
Additional exercises for students
Web links
Excel-based exercises
Self test questions
Click on the menu to the left of this page to view these resources.
Chapter-by-chapter resources may be viewed by clicking on the drop-down list.
THE LECTURER CENTRE
The Lecturer Centre for this title contains a host of downloadable material
for lecturers who adopt Mathematics for Economics and Business. The material found in the Lecturer Centre includes:
Seminar exercises
Teaching suggestions
Solutions to additional exercises in the student centre
PowerPoints
Solutions to questions in the book
Mock exam with answers
Accessing the lecturer centreRequest lecturer copy If you are considering using Mathematics for Economics and Business for course adoption, you can request a complementary
Click on the link at the base of this page to return to the Information Centre. |
Did You Know??
Integrating Visual Learning into Mathematics - Conclusion
Geogebra is a visual teaching of 2D geometry. It is an interactive geometrical engine that allows students to modify the values of each independent variable as how they want to. As you can see from the screenshot of the program taken below, as values are entered into the column of variables on the left, the geometrical shape and lines formed on the right are altered accordingly. This allows students to have a very clear understanding of exactly how each different variable affects the outcome of the graph.
Being able to modify the values at will allows students to better understand the relationship between individual variables and the overall graph.
In stark contrast, the conventional method of hand-drawing or sketching geographical figures with pen and paper are much less convenient. Students cannot change the values of each variable independently and have to mentally calculate the equation of each line before being able to plot the points on the graph paper. It is also much harder for students to mentally visualise the relationship between the variables and the geometrical graph, impeding their progress in the topic. |
Sunday, March 13, 2011
Functions can be a hard concept to teach to primary and secondary students! Math Is Fun gives as good of a review as anyone. One of the hardest concepts for students to grasp in when a function is really a function and when it isn't. If we are lucky enough to have a graph then we may simply use the vertical line test but normally this is not the case, so, let me offer an analogy
Using text messaging for our example here are our definitions using inputs and outputs.
Function: Cell Phone
Input: Cell phone numbers in our address (a)
Output: Sent text messages (TM)
Our Function is thus
TM = f(a)
Text massages are a function of what address we input.
Now we have a platform to teach our function concept. For additional questions, download the worksheet titled What is A Function in the math downloads section
Hey John, The worksheet is titled "When is a Function" and is in the Math Downloads section to the bottom right (if you can't find it, send me an e-mail and"ill mail it to you).
You're right that the concept can get slightly tricky to teach. The way i'm thinking is that you may send the same message to multiple people (and it's still a function), but you cannot send a message to input mom and it be received by dad. If you think about listings in your phonebook as the input and the receiver as the output I think it will work for an analogy. Let me know how it goes :) |
Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature. Around 200 additional exercises, and a full solutions manual for instructors, available via
A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.
This volume introduces techniques and theorems of Riemannian geometry, and opens the way to advanced topics. The text combines the geometric parts of Riemannian geometry with analytic aspects of the theory, and reviews recent research. The updated second edition includes a new coordinate-free formula that is easily remembered (the Koszul formula in disguise); an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus integrated into the text, allowing for an early treatment of the Sphere theorem using a forgotten proof by Berger; recent results regarding manifolds with positive curvature.
The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.
This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles, and Chern forms that are helpful for a deeper understanding of both classical and modern physics and engineering. It is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study.
A main addition introduced in this Third Edition is the inclusion of an Overview, which can be read before starting the text. This appears at the beginning of the text, before Chapter 1. Many of the geometric concepts developed in the text are previewed here and these are illustrated by their applications to a single extended problem in engineering, namely the study of the Cauchy stresses created by a small twist of an elastic cylindrical rod about its axis.
The
Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.
Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.
Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at |
Other Information
MATH135: Mathematics IA
This is the first mainstream mathematics unit. It is essential for students in science and technology, and recommended for students in many other areas who wish to enhance their mathematical skills. Apart from some brief discussion on complex numbers and congruencies, the main topic in the algebra half of this unit concerns linearity and the interplay between algebra and geometry. Plane geometry is first used to motivate the study of systems of linear equations. Algebraic techniques involving matrices and determinants are then developed to study these problems further. The algebraic machinery developed is then used to study geometrical problems in 3-dimensional space. The notion of a limit is developed to a more sophisticated level than in secondary school mathematics, and this is used to study the differential and integral calculus involving functions of one real variable to a far greater depth than before. Some simple numerical techniques on integration are also discussed. |
Summary: This manual is organized to follow the sequence of topics in the text, and provides an easy-to-follow, step-by-step guide with worked-out examples to help students fully understand and get the most out of their graphing calculator. Compatible models include the popular TI-83/84 Plus and MathPrint. This manual will be packaged with every text.
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Authors
Why a liberal arts mathematics book with a quantitative literacy focus?
How do you engage students with the study of math? Crauder, Evans, Johnson, and Noell have found the answer: Help them become intelligent consumers of the quantitative data to which they are exposed every day—in the news, on TV, and on the Internet.
In an age of record credit card debt, opinion polls, and questionable statistics, too few students have mastered the basic mathematical concepts required to think about and evaluate data. Quantitative Literacy: Thinking Between the Lines develops the idea of rates of change as a key concept in helping students make good personal, financial, and political decisions.
The goal of Quantitative Literacy is a more informed generation of college students who think critically about the data provided to them, the images shown to them, the facts presented to them, and the offers made to them. Quantitative Literacy shows students the mathematics that matters to them: their bank account, their medical tests, their daily news feed. It also develops their mathematical thinking, helping them to understand the difference between truthful and misleading mathematical reporting.
It's All in the Examples…
After taking your course and working with Quantitative Literacy, students will be equipped to think about and answer all of the following questions:
Will the Atkins Diet really help me lose weight?
How do I use logic to get the best results from a Google search?
Is the local carpet store trying to fool me into thinking their prices are lower because they quote price by the square foot instead of the square yard?
How far can I go on this tank of gas?
How do I interpret the results of my medical tests?
How can businesspeople and politicians use graphs and charts to mislead me?
Will inflation affect my savings and the age at which I can retire?
How do I avoid getting tricked by a Ponzi scheme?
I want to buy a new car in two years. How much do I need to save each month to achieve my goal? How much car can I really afford?
Why are games of chance so financially risky?
Does the golden rectangle explain the beauty of some paintings and architecture?
LearningCurve A research-based breakthrough in adaptive quizzing available in MathPortal. For more productive classtime and better grades. Simple to assign and simple to use. See for yourself. |
Matrix Calculator Pro is a professional windows software which can calculate matrix with real numbers and complex numbers. The complex number support for Polar format. Complex and Polar format can be... |
Key Curriculum Releases IMP Year 4, 2nd Edition
IMP is four-year core mathematics curriculum and is aligned with Common Core State Standards. Adoption of the IMP curriculum includes implementation strategies, supplemental materials, blackline masters, calculator guides, and assessment tools.
Year 4 covers topics such as statistical sampling, computer graphics and animation, an introduction to accumulation and integrals, and an introduction to sophisticated algebra, including transformations and composition.
The second edition of Year 4 includes a new student textbook, 2 new unit books, and three updated unit books |
The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses visit MIT OpenCourseWare at ocw.mit.edu.
PROFESSOR STRANG: Finally we get to positive definite matrices. I've used the word and now it's time to pin it down. And so this would be my thank you for staying with it while we do this important preliminary stuff about linear algebra. So starting the next lecture we'll really make a big start on engineering applications. But these matrices are going to be the key to everything. And I'll call these matrices K and positive definite, I will only use that word about a symmetric matrix. So the matrix is already symmetric and that means it has real eigenvalues and many other important properties, orthogonal eigenvectors. And now we're asking for more. And it's that extra bit that is terrific in all kinds of applications. So if I can do this bit of linear algebra.
So what's coming then, my review session this afternoon at four, I'm very happy that we've got, I think, the best MATLAB problem ever invented in 18.085 anyway. So that should get onto the website probably by tomorrow. And Peter Buchak is like the MATLAB person. So his review sessions are Friday at noon. And I just saw him and suggested Friday at noon he might as well just stay in here. And knowing that that isn't maybe a good hour for everybody. So you could see him also outside of that hour. But that's the hour he will be ready for all kinds of questions about MATLAB or about the homeworks. Actually you'll be probably thinking more also about the homework questions on this topic.
Ready for positive definite? You said yes, right? And you have a hint about these things. So we have a symmetric matrix and the beauty is that it brings together all of linear algebra. Including elimination, that's when we see pivots. Including determinants which are closely related to the pivots. And what do I mean by upper left? I mean that if I have a three by three symmetric matrix and I want to test it for positive definite, and I guess actually this would be the easiest test if I had a tiny matrix, three by three, and I had numbers then this would be a good test. The determinants, by upper left determinants I mean that one by one determinant. So just that first number has to be positive. Then the two by two determinant, that times that minus that times that has to be positive. Oh I've already been saying that. Let me just put in some letters. So a has to be positive. This is symmetric, so a times c has to be bigger than b squared. So that will tell us. And then for two by two we finish. For three by three we would also require the three by three determinant to be positive.
But already here you're seeing one point about a positive definite matrix. Its diagonal will have to be positive. And somehow its diagonal has to be not just above zero, but somehow it has to defeat b squared. So the diagonal has to be somehow more positive than whatever negative stuff might come from off the diagonal. That's why I would need a*c > b squared. So both of those will be positive and their product has to be bigger than the other guy.
And then finally, a third test is that all the eigenvalues are positive. And of course if I give you a three by three matrix, that's probably not the easiest test since you'd have to find the eigenvalues. Much easier to find the determinants or the pivots. Actually, just while I'm at it, so the first pivot of course is a itself. No difficulty there. The second pivot turns out to be the ratio of a*c - b squared to a. So the connection between pivots and determinants is just really close. Pivots are ratios of determinants if you work it out. The second pivot, maybe I would call that d_2, is the ratio of a*c - b squared over a. In other words it's (c - b squared)/a. Determinants are positive and vice versa. Then it's fantastic that the eigenvalues come into the picture.
So those are three ways, three important properties of a positive definite matrix. But I'd like to make the definition something different. Now I'm coming to the meaning. If I think of those as the tests, that's done. Now the meaning of positive definite. The meaning of positive definite and the applications are closely related to looking for a minimum. And so what I'm going to bring in here, so it's symmetric. Now for a symmetric matrix I want to introduce the energy. So this is the reason why it has so many applications and such important physical meaning is that what I'm about to introduce, which is a function of x, and here it is, it's x transpose times A, not A, I'm sticking with K for my matrix, times x. I think of that as some f(x).
And let's just see what it would be if the matrix was two by two, [a, b; b, c]. Suppose that's my matrix. We want to get a handle on what, this is the first time I've ever written something that has x's times x's. So it's going to be quadratic. They're going to be x's times x's. And x is a general vector of the right size so it's got components x_1, x_2. And there it's transpose, so it's a row. And now I put in the [a, b; b, c]. And then I put in x again. This is going to give me a very nice, simple, important expression. Depending on x_1 and x_2. Now what is, can we do that multiplication? Maybe above I'll do the multiplication of this pair and then I have the other guy to bring in. So here, that would be ax_1+bx_2. And this would be bx_1+cx_2. So that's the first, that's this times this. What am I going to get? What shape, what size is this result going to be? This K is n by n. x is a column vector. n by one. x transpose, what's the shape of x transpose? One by n? So what's the total result? One by one. Just a number. Just a function. It's a number. But it depends on the x's and the matrix inside.
Can we do it now? So I've got this to multiply by this. Do you see an x_1 squared showing up? Yes, from there times there. And what's it multiplied by? The a. The first term is this times the ax_1 is a(x_1 squared). So that's our first quadratic. Now there'd be an x_1, x_2. Let me leave that for a minute and find the x_2 squared because it's easy. So where am I going to get x_2 squared? I'm going to get that from x_2, second guy here times second guy here. There's a c(x_2 squared).
So you're seeing already where the diagonal shows up. The diagonal a, c, whatever is multiplying the perfect squares. And it'll be the off-diagonal that multiplies the x_1, x_2. We might call those the crossterms. And what do we get for that then? We have x_1 times this guy. So that's a crossterm. bx_1*x_2, right? And here's another one coming from x_2 times this guy. And what's that one? It's also bx_1*x_2. So x_1, multiply that, x_2 multiply that, and so what do we have for this crossterm here? Two of them. 2bx_1*x_2. In other words, that b and that b came together in the 2bx_1*x_2. So here's my energy. Can I just loosely call it energy? And then as we get to applications we'll see why.
So I'm interested in that because it has important meaning. Well, so now I'm ready to define positive definite matrices. So I'll call that number four. But I'm going to give it a big star. Even more because it's the sort of key. So the test will be, you can probably guess it, I look at this expression, that x transpose Ax. And if it's a positive definite matrix and this represents energy, the key will be that this should be positive. This one should be positive for all x's. Well, with one exception, of course. All x's except, which vector is it? x=0 would just give me-- See, I put K. My default for a matrix, but should be, it's K. Except x=0, except the zero vector. Of course. If x_1 and x_2 are both zero.
Now that looks a little maybe less straightforward, I would say, because it's a statement about this is true for all x_1 and x_2. And we better do some examples and draw a picture. Let me draw a picture right away. So here's x_1 direction. Here's x_2 direction. And here is the x transpose Ax, my function. So this depends on two variables. So it's going to be a sort of a surface if I draw it. Now, what point do we absolutely know? And I put A again. I am so sorry. Well, we know one point. It's there whatever that matrix might be. It's there. Zero, right? You just told me that if both x's are zero then we automatically get zero.
Now what do you think the shape of this curve, the shape of this graph is going to look like? The point is, if we're positive definite now. So I'm drawing the picture for positive definite. So my definition is that the energy goes up. It's positive, right? When I leave, when I move away from that point I go upwards. That point will be a minimum. Let me just draw it roughly. So it sort of goes up like this. These cheap 2-D boards and I've got a three-dimensional picture here. But you see it somehow? What word or what's your visualization? It has a minimum there. That's why minimization, which was like, the core problem in calculus, is here now. But for functions of two x's or n x's. We're up the dimension over the basic minimum problem of calculus. It's sort of like a parabola It's cross-sections cutting down through the thing would be just parabolas because of the x squared.
I'm going to call this a bowl. It's a short word. Do you see it? It opens up. That's the key point, that it opens upward. And let's do some examples. Tell me some positive definite. So positive definite and then let me here put some not positive definite cases. Tell me a matrix. Well, what's the easiest, first matrix that occurs to you as a positive definite matrix? The identity. That passes all our tests, its eigenvalues are one, its pivots are one, the determinants are one. And the function is x_1 squared plus x_2 squared with no b in it. It's just a perfect bowl, perfectly symmetric, the way it would come off a potter's wheel.
Now let me take one that's maybe not so, let me put a nine there. So I'm off to a reasonable start. I have an x_1 squared and a nine x_2 squared. And now I want to ask you, what could I put in there that would leave it positive definite? Well, give me a couple of possibilities. What's a nice, not too big now, that's the thing. Two. Two would be fine. So if I had a two there and a two there I would have a 4x_1*x_2 and it would, like, this, instead of being a circle, which it was for the identity, the plane there would cut out a ellipse instead. But it would be a good ellipse. Because we're doing squares, we're really, the Greeks understood these second degree things and they would have known this would have been an ellipse.
How high can I go with that two or where do I have to stop? Where would I have to, if I wanted to change the two, let me just focus on that one, suppose I wanted to change it. First of all, give me one that's, how about the borderline. Three would be the borderline. Why's that? Because at three we have nine minus nine for the determinant. So the determinant is zero. Of course it passed the first test. One by one was ok. But two by two was not, was at the borderline. What else should I think? Oh, that's a very interesting case. The borderline. You know, it almost makes it. But can you tell me the eigenvalues of that matrix? Don't do any quadratic equations.
How do I know, what's one eigenvalue of a matrix? You made it singular, right? You made that matrix singular. Determinant zero. So one of its eigenvalues is zero. And the other one is visible by looking at the trace. I just quickly mentioned that if I add the diagonal, I get the same answer as if I add the two eigenvalues. So that other eigenvalue must be ten. And this is entirely typical, that ten and zero, the extreme eigenvalues, lambda_max and lambda_min, are bigger than, these diagonal guys are inside. They're inside, between zero and ten and it's these terms that enter somehow and gave us an eigenvalue of ten and an eigenvalue of zero.
I guess I'm tempted to try to draw that figure. Just to get a feeling of what's with that one. It always helps to get the borderline case. So what's with this one? Let me see what my quadratic would be. Can I just change it up here? Rather than rewriting it. So I'm going to, I'll put it up here. So I have to change that four to what? Now that I'm looking at this matrix. That four is now a six. Six. This is my guy for this one. Which is not positive definite.
Let me tell you right away the word that I would use for this one. I would call it positive semi-definite because it's almost there, but not quite. So semi-definite allows the matrix to be singular. So semi-definite, maybe I'll do it in green what semi-definite would be. Semi-def would be eigenvalues greater than or equal zero. Determinants greater than or equal zero. Pivots greater than zero if they're there or then we run out of pivots. You could say greater than or equal to zero then. And energy, greater than or equal to zero for semi-definite. And when would the energy, what x's, what would be the like, you could say the ground states or something, what x's, so greater than or equal to zero, emphasize that possibility of equal in the semi-definite case.
Suppose I have a semi-definite matrix, yeah, I've got one. But it's singular. So that means a singular matrix takes some vector x to zero. Right? If my matrix is actually singular, then there'll be an x where Kx is zero. And then, of course, multiplying by x transpose, I'm still at zero. So the x's, the zero energy guys, this is straightforward, the zero energy guys, the ones where x transpose Kx is zero, will happen when Kx is zero. If Kx is zero, and we'll see it in that example.
Let's see it in that example. What's the x for which, I could say in the null space, what's the vector x that that matrix kills? , right? The vector . gives me . That's the vector that, so I get 3-3, the zero, 9-9, the zero. So I believe that this thing will be-- Is it zero at three, minus one? I think that it has to be, right? If I take x_1 to be three and x_2 to be minus one, I think I've got zero energy here. Do I? x_1 squared will be at the nine and nine x_2 squared will be nine more. And what will be this 6x_1*x_2? What will that come out for this x_1 and x_2? Minus 18. Had to, right? So I'd get nine from there, nine from there, minus 18, zero.
So the graph for this positive semi-definite will look a bit like this. There'll be a direction in which it doesn't climb. It doesn't go below the base, right? It's never negative. This is now the semi-definite picture. But it can run along the base. And it will for the vector x_1=3, x_2=-1, I don't know where that is, one, two, three, and then maybe minus one. Along some line here the graph doesn't go up. It's sitting, can you imagine that sitting in the base? I'm not Rembrandt here, but in the other direction it goes up. Oh, the hell with that one. Do you see, sort of? It's like a trough, would you say? I mean, it's like a, you know, a bit of a drainpipe or something. It's running along the ground, along this direction and in the other directions it does go up. So it's shaped like this with the base not climbing. Whereas here, there's no bad direction. Climbs every way you go. So that's positive definite and that's positive semi-definite.
Well suppose I push it a little further. Let me make a place here for a matrix that isn't even positive semi-definite. Now it's just going to go down somewhere. I'll start again with one and nine and tell me what to put in now. So this is going to be a case where the off-diagonal is too big, it wins. And prevents positive definite. So what number would you like here? Five? Five is certainly plenty. So now I have [1, 5; 5, 9]. Let me take a little space on a board just to show you. Sorry about that. So I'm going to do the [1, 5; 5, 9] just because they're all important, but then we're coming back to positive definite. So if it's [1, 5; 5, 9] and I do that usual x, x transpose Kx and I do the multiplication out, I see the one x_1 squared and I see the nine x_2 squareds. And how many x_1*x_2's do I see? Five from there, five from there, ten. And I believe that can be negative. The fact of having all nice plus signs is not going to help it because we can choose, as we already did, x_1 to be like a negative number and x_2 to be a positive. And we can get this guy to be negative and make it, in this case we can make it defeat these positive parts.
What choice would do it? Let me take x_1 to be minus one and tell me an x_2 that's good enough to show that this thing is not positive definite or even semi-definite, it goes downhill. Take x_2 equal? What do you say? 1/2? Yeah, I don't want too big an x_2 because if I have too big an x_2, then this'll be important. Does 1/2 do it? So I've got 1/4, that's positive, but not very. 9/4, so I'm up to 10/4, but this guy is what? Ten and the minus is minus five. Yeah. So that absolutely goes, at this one I come out less than zero. And I might as well complete.
So this is the case where I would call it indefinite. Indefinite. It goes up like if x_2 is zero, then it's just got x_1 squared, that's up. If x_1 is zero, it's only got x_2 squared, that's up. But there are other directions where it goes downhill. So it goes either up, it goes both up in some ways and down in others. And what kind of a graph, what kind of a surface would I now have for x transpose for this x transpose, this indefinite guy? So up in some ways and down in others. This gets really hard to draw. I believe that if you ride horses you have an edge on visualizing this. So it's called, what kind of a point's it called? Saddle point, it's called a saddle point. So what's a saddle point? That's not bad, right? So this is a direction where it went up. This is a direction where it went down. And so it sort of fills in somehow.
Or maybe, if you don't, I mean, who rides horses now? Actually maybe something we do do is drive over mountains. So the path, if the road is sort of well-chosen, the road will go, it'll look for the, this would be-- Yeah, here's our road. We would do as little climbing as possible. The mountain would go like this, sort of. So this would be like, the bottom part looking along the peaks of the mountains. But it's the top part looking along the driving direction. So driving, it's a maximum, but in the mountain range direction it's a minimum. So it's a saddle point. So that's what you get from a typical symmetric matrix. And if it was minus five it would still be the same saddle point, would still be 9-25, it would still be negative and a saddle.
Positive guys are our thing. Alright. So now back to positive definite. With these four tests and then the discussion of semi-definite. Very key, that energy. Let me just look ahead a moment. Most physical problems, many, many physical problems, you have an option. Either you solve some equations, either you find the solution from our equations, Ku=f, typically. Matrix equation or differential equation. Or there's another option of minimizing some function. Some energy. And it gives the same equations. So this minimizing energy will be a second way to describe the applications.
Now can I get a number five? There's an important number five and then you know really all you need to know about symmetric matrices. This gives me, about positive definite matrices, this gives me a chance to recap. So I'm going to put down a number five. Because this is where the matrices come from. Really important. And it's where they'll come from in all these applications that chapter two is going to be all about, that we're going to start. So they come, these positive definite matrices, so this is another way to, it's a test for positive definite matrices and it's, actually, it's where they come from. So here's a positive definite matrix. They come from A transpose A. A fundamental message is that if I have just an average matrix, possibly rectangular, could be a square but not symmetric, then sooner or later, in fact usually sooner, you end up looking at A transpose A. We've seen that already. And we already know that A transpose A is square, we already know it's symmetric and now we're going to know that it's positive definite. So matrices like A transpose A are positive definite or possibly semi-definite. There's that possibility. If A was the zero matrix, of course, we would just get the zero matrix which would be only semi-definite, or other ways to get a semi-definite.
So I'm saying that if K, if I have a matrix, any matrix, and I form A transpose A, I get a positive definite matrix or maybe just semi-definite, but not indefinite. Can we see why? Why is this positive definite or semi-? So that's my question. And the answer is really worth, it's just neat and worth seeing. So do I want to look at the pivots of A transpose A? No. They're something, but whatever they are, I can't really follow those well. Or the eigenvalues very well, or the determinants. None of those come out nicely. But the real guy works perfectly. So look at x transpose Kx. So I'm just doing, following my instinct here.
So if K is A transpose A, my claim is, what am I saying then about this energy? What is it that I want to discover and understand? Why it's positive. Why does taking any matrix, multiplying by its transpose produce something that's positive? Can you see any reason why that quantity, which looks kind of messy, I just want to look at it the right way to see why that should be positive, that should come out positive. So I'm not going to get into numbers, I'm not going to get into diagonals and off-diagonals. I'm just going to do one thing to understand that particular combination, x transpose A transpose Ax. What shall I do? Anybody see what I might do? Yeah, you're seeing here if you look at it again, what are you seeing here? Tell me again. If I take Ax together, then what's the other half? It's the transpose of Ax. So I just want to write that as, I just want to think of it that way, as Ax. And here's the transpose of Ax. Right? Because transposes of Ax, so transpose guys in the opposite order, and the multiplication--
This is the great. I call these proof by parenthesis because I'm just putting parentheses in the right place, but the key law of matrix multiplication is that, that I can put (AB)C is the same as A(BC). That rule, which is just multiply it out and you see that parentheses are not needed because if you keep them in the right order you can do this first, or you can do this first. Same answer. What do I learn from that? What was the point? This is some vector, I don't know especially what it is times its transpose. So that's the length squared. What's the key fact about that? That it is never negative. It's always greater than zero or possibly equal.
When does that quantity equal zero? When Ax is zero. When Ax is zero. Because this is a vector. That's the same vector transposed. And everybody's got that picture. When I take any y transpose y, I get y_1 squared plus y_2 squared through y_n squared. And I get a positive answer except if the vector is zero. So it's zero when Ax is zero. So that's going to be the key. If I pick any matrix A, and I can just take an example, but chapter, the applications are just going to be full of examples. Where the problem begins with a matrix A and then A transpose shows up and it's the combination A transpose A that we work with. And we're just learning that it's positive definite.
Unless, shall I just hang on since I've got here, I have to say when is it, have to get these two possibilities. Positive definite or only semi-definite. So what's the key to that borderline question? This thing will be only semi-definite if there's a solution to Ax=0. If there is an x, well, there's always the zero vector. Zero vector I can't expect to be positive. So I'm looking for if there's an x so that Ax is zero but x is not zero, then I'll only be semi-definite. That's the test. If there is a solution to Ax=0.
When we see applications that'll mean there's a displacement with no stretching. We might have a line of springs and when could the line of springs displace with no stretching? When it's free-free, right? If I have a line of springs and no supports at the ends, then that would be the case where it could shift over by the vector. So that would be the case where the matrix is only singular. We know that. The matrix is now positive semi-definite. We just learned that. So the free-free matrix, like B, both ends free, or C. So our answer is going to be that K and T are positive definite. And our other two guys, the singular ones, of course, just don't make it. B at both ends, the free-free line of springs, it can shift without stretching. Since Ax will measure the stretching when it just shifts rigid motion, the Ax is zero and we see only positive definite. And also C, the circular one. There it can displace with no stretching because it can just turn in the circle. So these guys will be only positive semi-definite.
Maybe I better say this another way. When is this positive definite? Can I use just a different sentence to describe this possibility? This is positive definite provided, so what I'm going to write now is to remove this possibility and get positive definite. This is positive definite provided, now, I could say it this way. The A has independent columns. So I just needed to give you another way of looking at this Ax=0 question. If A has independent columns, what does that mean? That means that the only solution to Ax=0 is the zero solution. In other words, it means that this thing works perfectly and gives me positive. When A has independent columns.
Let's just remember our K, T, B, C. So here's a matrix, so let me take the T matrix, that's this one, this guy. And then the third column is . Those three columns are independent. They point off. They don't lie in a plane. They point off in three different directions. And then there are no solutions to, no x's that's go Kx=0. So that would be a case of independent columns. Let me make a case of dependent columns. So and I'm going to make it B now. Now the columns of that matrix are dependent. There's a combination of them that give zero. They all lie in the same plane. There's a solution to that matrix times x equal zero. What combination of those columns shows me that they are dependent? That some combination of those three columns, some amount of this plus some amount of this plus some amount of that column gives me the zero vector. You see the combination. What should I take? again. No surprise. That's the vector that we know is in the everything shifting the same amount, nothing stretching.
Talking fast here about positive definite matrices. This is the key. Let's just ask a few questions about positive definite matrices as a way to practice. Suppose I had one. Positive definite. What about its inverse? Is that positive definite or not? So I've got a positive definite one, it's not singular, it's got positive eigenvalues, everything else. It's inverse will be symmetric, so I'm allowed to think about it. Will it be positive definite? What do you think? Well, you've got a whole bunch of tests to sort of mentally run through. Pivots of the inverse, you don't want to touch that stuff. Determinants, no. What about eigenvalues? What would be the eigenvalues if I have this positive definite symmetric matrix, its eigenvalues are one, four, five. What can you tell me about the eigenvalues of the inverse matrix? They're the inverses. So those three eigenvalues are? 1, 1/4, 1/5, what's the conclusion here? It is positive definite. Those are all positive, it is positive definite. So if I invert a positive definite matrix, I'm still positive definite.
All the tests would have to pass. It's just I'm looking each time for the easiest test. Let me look now, for the easiest test on K_1+K_2. Suppose that's positive definite and that's positive definite. What if I add them? What do you think? Well, we hope so. But we have to say which of my one, two, three, four, five would be a good way to see it. Would be a good way to see it. Good question. Four? We certainly don't want to touch pivots and now we don't want to touch eigenvalues either. Of course, if number four works, others will also work. The eigenvalues will come out positive. But not too easy to say what they are. Let's try test number four. So K_1. What's the test? So test number four tells us that this part, x transpose K_1*x, that that part is positive, right? That that part is positive. If we know that's positive definite. Now, about K_2 we also know that for every x, you see it's for every x, that helps, don't let me put x_2 there, for every x this will be positive.
And now what's the step I want to take? To get some information on the matrix K_1+K_2. I should add. If I add these guys, you see that it just, then I can write that as, I can write that this way. And what have I learned? I've learned that that's positive, even greater than, except for the zero vector. Because this was greater than, this is greater than. If I add two positive numbers, the energies are positive and the energies just add. The energies just add. So that definition four was the good way, just nice, easy way to see that if I have a couple of positive definite matrices, a couple of positive energies, I'm really coupling the two systems. This is associated somehow. I've got two systems, I'm putting them together and the energy is just even more positive. It's more positive either of these guys because I'm adding.
As I'm speaking here, will you allow me to try test number five, this A transpose A business? Suppose K_1 was A transpose A. If it's positive definite, it will. Be And suppose K_2 is B transpose B. If it's positive definite, it will be. Now I would like to write the sum somehow as, in this something transpose something. And I just do it now because I think it's like, you won't perhaps have thought of this way to do it. Watch. Suppose I create the matrix [A; B]. That'll be my new matrix. Say, call it C. Am I allowed to do that? I mean, that creates a matrix? These A and B, they had the same number of columns, n. So I can put one over the other and I still have something with n columns. So that's my new matrix C. And now I want C transpose. By the way, I'd call that a block matrix. You know, instead of numbers, it's got two blocks in there. Block matrices are really handy.
Now what's the transpose of that block matrix? You just have faith, just have faith with blocks. It's just like numbers. If I had a matrix [1; 5] then I'd get a row one, five. But what do you think? This is worth thinking about even after class. What would be, if this C matrix is this block A above B, what do you think for C transpose? A transpose, B transpose side by side. Just put in numbers and you'd see it. And now I'm going to take C transpose times C. I'm calling it C now instead of A because I've used the A in the first guy and I've used B in the second one and now I'm ready for C. How do you multiply block matrices? Again, you just have faith. What do you think? Tell me the answer. A transpose, I multiply that by that just as if they were numbers. And I add that times that just as if they were numbers. And what do I have? I've got K_1+K_2. So I've written K_1, this is K_1+K_2 and this is in my form C transpose C that I was looking for, that number five was looking for. So it's done it. It's done it. The fact of getting A, K_1 in this form, K_2 in this form. And I just made a block matrix and I got K_1+K_2. That's not a big deal in itself, but block matrices are really handy. It's good to take that step with matrices. Think of, possibly, the entries as coming in blocks and not just one at a time.
Well, thank you, okay. I swear Friday we'll start applications in all kinds of engineering problems and you'll have new applications |
More than 250,000 professionals worldwide are using Mathcad to perform, document, manage and share calculation and design work. The unique Mathcad visual format and easy-to-use whiteboard interface integrate live, standard mathematical notation, text and graphs into a single worksheet–making Mathcad ideal for knowledge capture, calculation reuse, and engineering collaboration.
Mathcad lets you type equations just as you would write them on a blackboard or in a reference book. There is no difficult syntax to learn; you simply type in your equations and then see the results. You can use Mathcad equations to solve just about any math problem you can think of, symbolically or numerically. You can place text anywhere on the worksheet to document your work.
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Summary
Widely known for incorporating interesting, relevant, and realistic applications, this text offers many real applications citing current data sources. There are a wide variety of opportunities for use of technology, allowing for increased visualization and a better understanding of difficult concepts. MyMathLab, a complete online course, will be available with this text. For the first time, a comprehensive series of lectures on video will be available.
Table of Contents
R. Algebra Reference
Polynomials
Factoring
Rational Expressions
Equations
Inequalities
Exponents
Radicals
Linear Functions
Slope and Equations of a Line
Linear Functions
Linear Mathematical Models
Constructing a Mathematical Model
Review Exercises
Extended Application: Using Marginal Cost to Estimate the Cost of Educating Immigrants |
Objective: On completion of the lesson the student will be able to calculate the gradient of a line given any two points on the line and also be capable of checking whether 3 or more points lie on the same line and what an unknown point will make to parallel lines.
Objective: On completion of the lesson the student will be able to draw a line which passes through the origin of the form y=mx and comment on its gradient compared to the gradients of other lines through the origin and use the information to solve problems.
Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the number from which the logarithm evolves.
Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the base from which the number came.
Objective: On completion of this lesson the student will be able to define basic logarithmic functions and describe the relationship between logarithms and exponents including graph logarithmic functions. The student will understand the relationship between logarit
Objective: On completion of the lesson the student will be able to test if a given sequence is an Arithmetic Progression or not and be capable of finding a formula for the nth term, find any term in the A.P. and to solve problems involving these concepts.
Objective: On completion of the lesson the student will be able to make an arithmetic progression between two given terms. This could involve finding one, two, or even larger number of arithmetic means |
Letts and Lonsdale's Success Revision Guides offer accessible content to help students manage their revision and prepare for exams efficiently. The content is broken into manageable sections and advice is offered to help build students' confidence. Exam tips and techniques are provided to support students throughout the revision process. These new titles are specific to the Edexcel exam board. This book is for the current GCSE Maths curriculum to be examined in 2010/2011 only. Other Letts revision titles are available for the new GCSE curriculum starting in 2010. |
Description
By connecting applications, modeling, and visualization, Gary Rockswold motivates students to learn mathematics in the context of their experiences. In order to both learn and retain the material, students must see a connection between the concepts and their real-lives. In this new edition, connections are taken to a new level with "See the Concept" features, where students make important connections through detailed visualizations that deepen understanding.
Rockswold is also known for presenting the concept of a function as a unifying theme, with an emphasis on the rule of four (verbal, graphical, numerical, and symbolic representations). A flexible approach allows instructors to strike their own balance of skills, rule of four, applications, modeling, and technology. Additionally, incorporating technology with this edition has never been so exciting! Within the MyMathLab® course, new Interactive Figures help students visualize difficult topics. Also, Getting Ready integrated review allows students to remediate "just-in-time," by providing review of prerequisite material when needed to help them succeed in the course.
Table of Contents
1. Introduction to Functions and Graphs
1.1 Numbers, Data, and Problem Solving
1.2 Visualizing and Graphing Data
Checking Basic Concepts for Sections 1.1 and 1.2
1.3 Functions and Their Representations
1.4 Types of Functions and Their Rates of Change
Checking Basic Concepts for Sections 1.3 and 1.4
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Extended and Discovery Exercises
2. Linear Functions and Equations
2.1 Equations of Lines
2.2 Linear Equations
Checking Basic Concepts for Sections 2.1 and 2.2
2.3 Linear Inequalities
2.4 More Modeling with Functions
Checking Basic Concepts for Sections 2.3 and 2.4
2.5 Absolute Value Equations and Inequalities
Checking Basic Concepts for Section 2.5
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Extended and Discovery Exercises
Chapters 1-2 Cumulative Review Exercises
3. Quadratic Functions and Equations
3.1 Quadratic Functions and Models
3.2 Quadratic Equations and Problem Solving
Checking Basic Concepts for Sections 3.1 and 3.2
3.3 Complex Numbers
3.4 Quadratic Inequalities
Checking Basic Concepts for Sections 3.3 and 3.4
3.5 Transformations of Graphs
Checking Basic Concepts for Section 3.5
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Extended and Discovery Exercises
4. More Nonlinear Functions and Equations
4.1 More Nonlinear Functions and Their Graphs
4.2 Polynomial Functions and Models
Checking Basic Concepts for Sections 4.1 and 4.2
4.3 Division of Polynomials
4.4 Real Zeros of Polynomial Functions
Checking Basic Concepts for Sections 4.3 and 4.4
4.5 The Fundamental Theorem of Algebra
4.6 Rational Functions and Models
Checking Basic Concepts for Sections 4.5 and 4.6
4.7 More Equations and Inequalities
4.8 Radical Equations and Power Functions
Checking Basic Concepts for Sections 4.7 and 4.8
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Extended and Discovery Exercises
Chapters 1-4 Cumulative Review Exercises
5. Exponential and Logarithmic Functions
5.1 Combining Functions
5.2 Inverse Functions and Their Representations
Checking Basic Concepts for Sections 5.1 and 5.2
5.3 Exponential Functions and Models
5.4 Logarithmic Functions and Models
Checking Basic Concepts for Sections 5.3 and 5.4
5.5 Properties of Logarithms
5.6 Exponential and Logarithmic Equations
Checking Basic Concepts for Sections 5.5 and 5.6
5.7 Constructing Nonlinear Models
Checking Basic Concepts for Section 5.7
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Extended and Discovery Exercises
6. Trigonometric Functions
6.1 Angles and Their Measure
6.2 Right Triangle Trigonometry
Checking Basic Concepts for Sections 6.1 and 6.2
6.3 The Sine and Cosine Functions and Their Graphs
6.4 Other Trigonometric Functions and Their Graphs
Checking Basic Concepts for Sections 6.3 and 6.4
6.5 Graphing Trigonometric Functions
6.6 Inverse Trigonometric Functions
Checking Basic Concepts for Sections 6.5 and 6.6
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Extended and Discovery Exercises
Chapters 1-6 Cumulative Review Exercises
7. Trigonometric Identities and Equations
7.1 Fundamental Identities
7.2 Verifying Identities
Checking Basic Concepts for Sections 7.1 and 7.2
7.3 Trigonometric Equations
7.4 Sum and Difference Identities
Checking Basic Concepts for Sections 7.3 and 7.4
7.5 Multiple-Angle Identities
Checking Basic Concepts for Section 7.5
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Extended and Discovery Exercises
8. Further Topics in Trigonometry
8.1 Law of Sines
8.2 Law of Cosines
Checking Basic Concepts for Sections 8.1 and 8.2
8.3 Vectors
8.4 Parametric Equations
Checking Basic Concepts for Sections 8.3 and 8.4
8.5 Polar Equations
8.6 Trigonometric Form and Roots of Complex Numbers
Checking Basic Concepts for Sections 8.5 and 8.6
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Extended and Discovery Exercises
Chapters 1-8 Cumulative Review Exercises
9. Systems of Equations and Inequalities
9.1 Functions and Systems of Equations in Two Variables
9.2 Systems of Inequalities in Two Variables
Checking Basic Concepts for Sections 9.1 and 9.2
9.3 Systems of Linear Equations in Three Variables
9.4 Solutions to Linear Systems Using Matrices
Checking Basic Concepts for Sections 9.3 and 9.4
9.5 Properties and Applications of Matrices
9.6 Inverses of Matrices
Checking Basic Concepts for Sections 9.5 and 9.6
9.7 Determinants
Checking Basic Concepts for Section 9.7
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Extended and Discovery Exercises
Chapters 1-9 Cumulative Review Exercises
10. Conic Sections
10.1 Parabolas
10.2 Ellipses
Checking Basic Concepts for Sections 10.1 and 10.2
10.3 Hyperbolas
Checking Basic Concepts for Section 10.3
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Extended and Discovery Exercises
11. Further Topics in Algebra
11.1 Sequences
11.2 Series
Checking Basic Concepts for Sections 11.1 and 11.2
11.3 Counting
11.4 The Binomial Theorem
Checking Basic Concepts for Sections 11.3 and 11.4
11.5 Mathematical Induction
11.6 Probability
Checking Basic Concepts for Sections 11.5 and 11.6
Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Extended and Discovery Exercises
Chapters 1-11 Cumulative Review Exercises
R. Reference: Basic Concepts from Algebra and Geometry
R.1 Formulas from Geometry
R.2 Integer Exponents
R.3 Polynomial Expressions
R.4 Factoring Polynomials
R.5 Rational Expressions
R.6 Radical Notation and Rational Exponents
R.7 Radical Expressions
Appendix A: Using the Graphing Calculator
Appendix B: A Library of Functions
Appendix C: Partial Fractions
Appendix D: Percent Change and Exponential Functions
Appendix E: Rotation of Axes
Bibliography
Answers to Selected Exercises
Photo Credits |
The Algebra 2 Tutor - 6 Hour Course - 2 DVD Set - Learn by Examples!
by Tutoring Services on August 13, 2011
Product Description The Algebra 2 Tutor is the easiest way to improve your grades in Algebra! How does a baby learn to speak? By being immersed in everyday conversation. What is the best way to learn Algebra? By being immersed Algebra! During this course the instructor will work out hundreds of examples with each step fully narrated so no one gets lost! See why thousands have discovered that the easiest way to learn Algebra is to learn by examples!
NOTE: It is highly recommended that you use this DVD along with the "Algebra 2 Tutor Companion Worksheet CD" which is also available on Amazon.com. |
1. The over-arching goal in this course is to learn to read and write mathematical proofs.
2. A second major goal is to be able to communicate mathematical concepts and mathematical proofs, using appropriate mathematical notation and language.
3. While this course is more about process than content, it is still a goal of this course to consider content which is important to further study of mathematics: logic, set theory, relations and functions, mathematical induction, number theory, and cardinality.
Course Objectives:
Proof:
1. Students shall demonstrate a basic understanding of axiomatic-deductive systems.
2. Students shall understand proofs and be able to judge the correctness of an argument.
3. Students shall demonstrate the ability to reason inductively.
4. Students shall demonstrate the ability to reason deductively.
Problem Solving:
5. Students shall demonstrate the ability to apply appropriate mathematical tools and methods to novel or non-routine problems.
6. Students shall demonstrate the ability to use various approaches to problem solving, and to see connections between these varied mathematical areas.
Communication
7. Students shall use the language of mathematics accurately and appropriately.
8. Students shall present mathematical content and argument orally.
9. Students shall present mathematical content and argument in written form.
Course Procedures:
This course is a very significant step for every mathematics major, and probably more than any other single course, certainly up to this point in your course work, lets you see what it is like to be a mathematician and whether you can find fulfillment as, or have the ability to be, a mathematician. I certainly have vivid memories of my equivalent course.
All mathematics educators believe in the adage, "Mathematics is not a spectator sport!" In this course, more than in most, however, your contributions will MAKE the course. I intend to allow a healthy fraction of our class time to consist of student presentations of attempted proofs. In addition to learning to read or listen to proofs, with comprehension, as well as learning to problem solve and write proofs, it is important to acquire a disinterested objectivity - like a scientist, we must strive to view a proof in its own terms and not take corrections personally. The "craft" of mathematics is in one sense very personal but at the same time your work must be held to the highest standard of objectivity.
I have chosen this textbook largely for its philosophy; it is designed to be used in a student-oriented class. I will make assignments virtually on a daily basis and it is very important that you work on these as best you can. I hope that at least once a week, on average, each of will get a chance to present some of your work. It is important that we not miss classes; this will be almost a "seminar" course - the size of the group and the philosophy will allow us to do that.
Because it is important that we learn to WRITE proofs which are correct and clear, as well as to present them orally, I will collect most of the daily assignments and grade them. There will also be three in-class, written, exams - one at the one-third mark, one at the two-thirds point, and one during the final exam time slot - I do believe in seeing what students can do on their own and "under pressure". At the same time, there will be frequent take-home assignments, both daily homework and other problem sets.
Grading: I will use the traditional 90-80-70-60 scale as a framework for assigning grades. I do realize the artificiality of such a scale - but I usually try to make the assignments reasonable enough so that students can earn an appropriate grade. It is harder in this course in particular to make things seem "objective", since writing proofs is something like writing essays in an English course - there is "correct" and "complete", but there is also "elegant" and "insightful". I will try to keep you informed along the way about your progress in the course.
Portfolios: Because this course is all about developing skills, albeit subtle and substantial skills, it is especially important to monitor your development. In fact, we are trying to compile a portfolio of the work of each mathematics major as they work their way through the courses in the major. I would like to ask you to turn in a collection of your best efforts during the course, probably five problems, which can be added to your portfolio. These portfolios should not only give you a chance to pay attention to your progress, but they also will give the department a picture of how well our students are learning what we want them to learn. You should include a brief write-up indicating why you chose the particular problem in your portfolio; generally it will be because the problem led to some sort of breakthrough in your mathematical thinking. This portfolio will be collected at the end of the semester.
Journals: I realize that sometimes "journaling" is overdone in this era of Journal-as-a-verb, but I do think it is important that each of you have a vehicle for thinking about your performance in the course and for letting me know how things are going. Consequently, I'd like to ask you each to turn in a journal entry, at least a few paragraphs, on the last class day of each month - I don't want to make the task too onerous but I also want to stay on top of things. So let's aim for the following: Jan30, Feb 27, Mar 31, Apr 30. I'll award 5 points for each of these.
The Schedule: For many courses I teach, I construct a fairly detailed schedule, with assignments and material to be covered each day throughout the semester. For several reasons, however, in this course we will be playing it a bit more by ear. It is difficult for me to know just how much text material we can comfortably cover (I do hope to at least cover 7 chapters). The emphasis I want to give to looking at your work also requires a certain flexibility in schedule - and it is difficult to say just what effect this will have. I do think it is much more important that you LEARN the material than that we "cover" it, "depth of understanding" rather than "breadth". |
The Midland Mathematical Experiment, in which a number of schools collaborated with the aim of developing a new approach to teaching mathematics in the Grammar School, developed a series of books to cover their 'A'-level syllabus. Sets, Mappings, Relations and Operations is split into sections, each section containingThe Midland Mathematical Experiment, in which a number of schools collaborated with the aim of developing a new approach to teaching mathematics in the Grammar School, developed a series of books to cover their 'A'-level syllabus. Groups, Rings and Fields is split into sections, each section containing explanations, examples parts to this book:
• resource, published by G Bell and Sons, was intended to complete the two-year course for those studying Pure Mathematics and also to extend the course to give adequate preparation for the 'S' papers of the GCE and similar examinations.
CHAPTER V - SYSTEMS OF CIRCLES
Power of a point with respect to a circle;… |
Where did math come from? Who thought up all those algebra symbols, and why? What's the story behind ?
This important addition to the New Mathematical Library series pays careful attention to applications of game theory in a wide variety of disciplines. The applications are treated in considerable depth. The book assumes only high school algebra, yet gently builds to mathematical thinking of some sophistication. Game Theory and Strategy might serve as an introduction to both axiomatic mathematical thinking and the fundamental process of mathematical modelling. It gives insight into both the nature of pure mathematics, and the way in which mathematics can be applied to real problems.
Number Theory Through Inquiry is an innovative textbook that leads students on a guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics.
In Biscuits of Number Theory, the editors have chosen articles that are exceptionally well written and that can be appreciated by anyone who has taken (or is taking) a first course in number theory. This book could be used as a textbook supplement for a number theory course, especially one that requires students to write papers or do outside reading.
Professor H. S. Wall wrote Creative Mathematics with the intention of leading students to develop their mathematical abilities, to help them learn the art of mathematics, and to teach them to create mathematical ideas. Creative Mathematics, according to Wall, "is not a compendium of mathematical facts and inventions to be read over as a connoisseur of art looks over paintings. It is, instead, a sketchbook in which readers try their hands at mathematical discovery." The book is self contained, and assumes little formal mathematical background on the part of the reader. Wall is earnest about developing mathematical creativity and independence in students. The student who has worked through Creative Mathematics will come away with heightened mathematical maturity.
Sink or Float: Thought Problems in Math and Physics is a collection of problems drawn from mathematics and the real world. Its multiple-choice format forces the reader to become actively involved in deciding upon the answer. The book's aim is to show just how much can be learned by using everyday common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem's solution, with explanation, appears in the answer section at the end of the book.
Resources for Teaching Discrete Mathematics presents nineteen classroom tested projects complete with student handouts, solutions, and notes to the instructor. Topics range from a first day activity that motivates proofs to applications of discrete mathematics to chemistry, biology, and data storage. Other projects provide: supplementary material on classic topics such as the towers of Hanoi and the Josephus problem, how to use a calculator to explore various course topics, how to employ Cuisenaire rods to examine the Fibonacci numbers and other sequences, and how you can use plastic pipes to create a geodesic dome.
A major aspect of mathematical training and its benefit to society is the ability to use logic to solve problems. The American Mathematics Competitions (AMC) have been given for more than fifty years to millions of high school students. This book considers the basic ideas behind the solutions to the majority of these problems, and presents examples and exercises from past exams to illustrate the concepts. Anyone taking the AMC exams or helping students prepare for them will find many useful ideas here.
Mathematical Interest Theory gives an introduction to how investments grow over time in a mathematically precise manner. The emphasis is on practical applications that give the reader a concrete understanding of why the various relationships should be true. Among the modern financial topics introduced are: arbitrage, options, futures, and swaps. The content of the book, along with an understanding of probability, will provide a solid foundation for readers embarking on actuarial careers.
Among the many beautiful and nontrivial theorems in geometry found here are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed. |
She completed the 2B placement exam but took 3 times as much time to complete it as was recommended. I thought better to start her slightly below her level to build confidence, learn the rod diagrams, and build speed and fluency with her facts and basic procedures.
We also used Intensive Practice books 2B, 3A, 3B, and part of 4A (not every problem though)
I made the decison to use Singapore because through my research 2 titles kept appearing over and over: Saxon and Singapore. Saxon is expensive and did not seem to be a good fit for my youngest daughter. Singapore seemed to be the best one to try first, since I wouldn't be out a lot of money if it flopped! Not very scientific or glamorous but the truth. [ed: Saxon at Home School Center may not be more expensive; I'll check.]
Once I worked with the program and saw the children's response to it I was sold.
I am average in my math ability and studied through Trig in college. I think at first Singapore can be intimidating, but after working with it, I find it is fairly straightforward.
I used the Instructor Guide for 2B and have not really used it since.
I try to work out all the rod diagrams, and boy am I getting good at them. [ed: oh! are these what I call 'bar models'? If so, I'm getting incredibly good at them myself.]
Jenny, at the Singapore Forum board, is a great help if I am hopelessly stuck. All problems at this level can be solved without using algebra and Jenny is very helpful for teaching people how to set up the rod diagrams. (singaporemath.com)
I also am learning much along with my daughters. [ed. note: based in my own experience, I think it's a good idea for parents to learn & re-learn elementary maths along with their children.]
I think Saxon is also a great program and a few of my homeschooling friends' kids are doing very well with it. |
The Calculus Lifesaver: All the Tools You Need to Excel at Calculus Adrian Banner
"Banner's book is a chatty, user-friendly guide to calculus that will be a useful addition to the resources available to students. Banner does an exceptionally thorough job while maintaining an engaging style."--Gerald B. Folland, author of Advanced Calculus
"This is an engaging read. Each page engenders at least one smile, often a chuckle, occasionally a belly laugh."--Charles R. MacCluer, author of Honors Calculus
"This book is significant. The author's attempt to give an 'inner monologue' into the thought process that is needed to solve calculus problems rather than just providing worked examples is novel and is in line with his purpose of helping the reader get a deeper understanding of calculus. The book is well written and the author's examples are clear and complete."--Thomas Seidenberg, Phillips Exeter Academy |
Many calculators have features that are suitable to handle Algebra, mathematical graphs, problem solving and Calculus. Some include features that are unique to engineering students. Very few calculators reach out to students studying science. That is disappointing because many students have to purchase multiple calculators to use in different classes.
The Texas Instruments TI-83-Plus Silver Edition contains features that useful for students studying algebra, calculus, statistics, biology, chemistry, or physics. The TI-83-Plus calculator is the one and only calculator that students will need to meet their requirements while in school.
Features
·TI-Graph Link for Windows and unit-to-unit cable
·1.5MB of memory
·Store up to 94 applications
·Split screen to display graph and editor or graph and table
·Pre-loaded with popular Handheld Software Applications
·Interactive equation solver
The first thing you will notice about the TI-83-Plus calculator is it sleek design and appearance. Most other calculators carry a very staid black or dark color appearance. The TI-83-Plus stands out from the crowd with the silver casing.
Once you turn on the TI-83-Plus calculator you will be amazed at the wide range of features that it has. It can easily handle a matrix, graphing, or equation solver. For science students there is a pre-programmed periodic table. The mathematic tools are substantial, including viewing equations, graphs, and coordinate simultaneously; advanced statistics and regression analysis, graphical analysis, and data analysis; and tools for engineering, financial, logarithm, trigonometry, and hyperbolic functions.
Texas Instruments has included several applications on the TI-83-Plus calculator. There is a spreadsheet, organizer for phone numbers, puzzle pack, probability simulation, and CBL/CBR applications
The Texas Instruments TI-83-Plus Silver edition calculator has many improvements over the standard TI-83-Plus calculator. The processor is 2.5 times faster, and it has over 1.5 MB of memory – 10 times that of the TI-83 Plus. The improvement in memory makes the TI-83-Plus Silver Edition faster and allows you to store more equations and programs.
The screen on the TI-83-Plus silver edition is very large LCD display that has 64 x 96 pixel resolution. The display can show eight lines of 16 characters each.
A graph link cable can be used to download applications that are specific to different subjects such as physics and geometry. You can use the graph link cable to download applications, games, data and more.
This calculator is also approved to be used on the SAT and ACT college entrance exams.
Every math student in high school or college needs a graphing calculator. A graphing calculator will allow you to easily solve complex mathematical equations. These equations can then easily be graphed on the calculator. You can zoom in to any point on the graph or be able to trace the graph itself.
The Texas Instruments Ti-nspire graphing calculator is clearly a revolutionary idea in the world of graphing calculators. Kids today are well versed in how to use a computer including how to use drop down menus, click and drag interfaces, mouse, documents and folders. The Texas Instruments Ti-nspire graphing calculator brings all those features to this graphing calculator.
Features
·Easy glide touchpad
·High resolution grayscale display
·20 MB Flash Rom
·USB connectivity
·View multiple representations of a problem on a single screen
·Create, save and review work in electronic documents
·Grab a graphed function and move it to see the effect
·"Link" representations
Most students find it difficult to use a graphing calculator and most of the time you see other students taking time to show them how to use their own calculator. The Texas Instruments Ti-nspire graphing calculator is radically different. If you have ever used a computer you will instinctively know how to use the Ti-inspire graphing calculator. The Ti-nspire graphing calculator is an in-your-hand computer used to learn math.
The Ti-nspire graphing calculator allows the student to see what impact a change in the equation will have on the results. You can use the touchpad to move a graph and easily see how this will change the answer.
There is a "linking" feature built into the Ti-nspire graphing calculator. You can make a change in one item in either the equation or the graph and instantly see how it would impact the other. This allows the students to easily understand the relationship.
The Ti-inspire can be used to easily "crack" the SAT or ACT exams. This calculator has been approved to be used on both exams and you will see many students carrying the Ti-nspire graphing calculator with them to take these exams.
Texas Instruments includes a 1 year warranty.
You can easily add a free TI-84 keyboard to this calculator. The only problem is that you must either mail in a card to Texas Instruments or request it online and then they will send it to you. It would be better if this was actually included with the product.
A graphing calculator is a learning tool designed to help students visualize and better understand concepts in mathematics and science. It allows students to make real-world connections in a variety of subjects. As they gain a deeper understanding of the material, students acquire the critical thinking and problem-solving skills they need to attain greater academic success.
Without argument, the graphing calculator reduces the time and effort required to perform cumbersome mathematical tasks. The Texas Instruments TI-89 Titanium Graphing Calculator is capable of providing multiple representations of mathematical concepts.
Features
·Motorola 68000 32-bit microprocessor running
·Algebraic factoring of expressions
·Algebraic simplification (CAS)
·Evaluation of trigonometric expressions to exact values
·Equation solving for a certain variable (CAS)
·Finding limits of functions
·Symbolic differentiation and integration
·Directly programmable in TI-BASIC
·Can run third-party applications
·Large 100 x 160 pixel display for split-screen views
·188 KB RAM and 2.7 MB flash memory for speed
Flash Technology
Most calculators are static. If new technology becomes available in a future product, you had to buy the new product to get it. The TI-89 titanium graphing calculator has 2.7 MB of flash memory. This allows you to upgrade to the latest software version without having to invest in buying a new calculator.
The TI-89 has 188K of RAM. This is more than enough RAM to be able to store functions, programs and data created by the user. The combination of the 188K of RAM and 2.7MB of flash memory allow the TI-89 to be very responsive in completing calculations.
Program Editor
Texas Instruments has included a program editor on the TI-89 titanium graphing calculator that lets you write custom applications. By building tables, tracing along curves, and zooming in on critical points, students may be able to process information in a more varied and meaningful way.
Computer Algebra System
The Texas Instruments TI-89 titanium graphing calculator contains a computer algebra system or CAS. The CAS feature on the TI-89 allows you to solve equations both for a numerical solution like x=4, but also for an answer where you want x in terms of y (like y=2x+3).
Large Display
The TI-89 graphing calculator has a very large 100 x 160 pixel display. You can actually display data in a split screen mode if you want. You can zoom in on graphs to look at an individual point. The display on the TI-89 is LCD. The display can be adjusted so that it can be easily viewable in any lighting condition.
Hard Shell Case
The TI-89 graphing calculator comes with a hard shell case. The cover can be flipped off and attached to the bottom when in use. When not in use, the cover easily protects the TI-89 from abuse in a backpack or if it is accidently dropped.
USB Connection
The TI-89 includes an USB connection to allow you to connect to your PC. You can also use the input/output port to sync with other TI-89 calculators.
Many high school students will have a graphing calculator that they use in Algebra and possibly pre-calculus when they are in high school. When they go on to college they soon discover that their graphing calculator cannot handle the requirements for college mathematics, science or engineering classes.
The Texas Instruments TI-86 graphing calculator includes a function evaluation table, deep entry recall, seven different graph styles with multiple line and shading options, and slope and direction fields for differential equations. The TI-86 can show numeric output for function, polar, parametric, and differential equation modes.
Features
·Graphing functions
·128k RAM with 96K user available
·Function evaluation table
·Input / Output port
·Zilog Z80 microprocessor
The TI-86 is a step ahead of its predecessor the TI-85. The TI-86 graphing calculator includes the same matrix features found in its predecessor but includes a new matrix editor. The new editor on the TI-86 allows students to view and edit matrices in two dimensions.
The TI-86 includes a new function evaluation table. The TI-86 graphing calculator can calculate minimums, maximums, integration, derivatives, arcs and roots. If you need to know what points are in a function, enter the function into the graph, and select table. The TI-86 will present you with an x,y table in a nice graphical format. All the points in the equation can be displayed in the table.
The screen on the TI -86 64 x 128 pixels which can display eight lines of 21 characters. There are two levels of display menus. It is high contrast so it is very readable in different lighting conditions.
The TI-86 graphing calculator is programmable. It was introduced in 1997. The fact that this calculator is still very popular 15 years after its introduction is a testimony to its power, features and ease of use. Over 80% of the product reviews for this product rate it as either 4 or 5 stars.
The TI-86 calculator runs on 4 alkaline batteries. You will find that you will easily drain the life out of the batteries. I would recommend getting four 750mAH NiMH batteries because they will last longer than the alkaline batteries.
The only negative is that the statistics package that is found on the TI-83 is not automatically included with the TI-86. You can easily download it though from and install it using the link cable.
Any student in an advanced math class in high school or calculus class in College will need a graphing calculator. A graphing calculator will allow you to solve an equation and then graph the results. A person is better able to learn the material when they are able to visually see the answer than by looking at an equation.
Texas Instruments has been making precision equipment for decades. The TI-84 Plus graphing calculator has become a common sight around both high school and college campuses due to the combination of its feature sets and its price.
Features
·24Kb of RAM
·480KB of Flash ROM
·13 pre loaded applications included
·8 lines by 16 character display
·Trace graphs
·Kickstand slide case
·USB connectivity
The predecessor to the TI-84 Plus was the TI-83 Plus which might be the best selling calculator ever. The TI-84 Plus graphing calculator builds on the success of its predecessor by offering more features. You can work mathematical calculations, graph them and easily understand the answer with the TI-84 Plus graphing calculator.
The TI-84 Plus graphing calculator runs off of 4 AAA alkaline batteries and one silver oxide battery. The silver oxide battery is used for backup. The 4 AAA alkaline batteries are not included so I recommending you purchase AAA batteries with the TI-84 Plus graphing calculator so you can use it as soon as you receive it. The TI-84 has an automatic shutoff to conserve battery life.
The TI-84 Plus graphing calculator has a very unique kickstand flip case. Most calculators have a flip case. The TI-84 Plus goes a step further to include a kickstand that holds the calculator upright so that you can easily read it while in use. The TI-84 Plus case is very durable which is very important because school age kids will abuse the calculator and the case will stand up to this abuse. It can easily be thrown into a backpack and dropped on the ground and not be damaged.
The included USB cable with the TI-84 Plus makes it very easy to transfer data. The calculator already comes with 11 included applications. You can install new applications on your TI-84 Plus by downloading them from your computer. Some of the preloaded applications include Transformation graphing, Conic Graphing, Inequality graphing, Topics in Algebra chapter 1, chapter 2, chapter 3, chapter 4, and chapter 5. Cabri, CBL, StudyCards, Science Tools, simultaneous probability, and TimeSpan are the other applications that you will notice in your new TI 84 Plus graphing calculator.
The biggest complaint against this calculator is that is very large when compared to its competitors. You could get a smaller calculator but you would get fewer features and pay more for it.
generally skyrocket in the last few years with more challenging textbooks to purchase and additional after school activities. <a href='In this way, both students will benefit from additional stimulation and extra socialization.
Are you searching for the right graphing calculator? If so the Texas Instruments TI-84 Plus Graphing Calculator has everything that you're looking for. Why purchase the TI-84 Plus? It's an awesome unit and handles everything from calculus, engineering, trigonometric, and also financial functions plus unlike other graphing calculators the Texas Instruments TI-84 comes with USB on-the-go technology for file sharing with other calculators. You can connect this calculator to your PC as well. The TI-84 plus is great for complex math and statistics, because it can display answers in the form of graphs. If you're a parent you should consider getting this calculator for your student because it will help them to successfully solve their mathematics and science material. Students can easily share their work on the TI-84 Plus because the built in USB port makes data transfer to computers and between hand held's very easy.They are also able to perform all the complex algebraic or geometric calculations of the most expensive and high-tech calculators on the market, and they can do it all for free. Free online calculators can also be found very easily. In addition to a simple Google search, any number of math blogs on the Web contain link to some great online calculators that are sure to serve whatever purpose you may have for them. While it is true that the best option for someone who has the money and scientific or mathematical knowledge would be to purchase an expensive and sophisticated calculator for the store, those of us who just need to make a quick calculation or conversion can do well with one of the many free calculators that can be found online.
Applying for a mortgage loan is a huge financial and emotional decision that needs to be taken with utmost concern and understanding and the monthly repayment is again the biggest outlay of every month especially when you will see that you are biting off more than you can chew. If the venture is not affordable then the payment of each month to repay the loan becomes a huge responsibility.
Thus to get complete relaxation it is important to seek advises from an expert so that they plan out the best type of loan for you with the minimum interest rates. These mortgage brokers have various options to make life simple and easy and one of the options which are readily available is the online mortgage calculator. Before the advent of the Internet the calculation related to loan were done by loan specialist or accountants and borrowers often had confusion in understanding the concept and the calculations involved in it, but as the online system is considered to be a boon these days so definitely the online mortgage calculator is also a big relief to the borrowers.Different annuity calculator takes different rates of inflation to work out the actual buying power of your estimated pension income at the time of your retirement. In most of the annuity calculator four variables are used and values of the three variables are to be filled up by you and the calculator works out the value of the unknown variable. As a whole, annuity calculator is an extremely useful tool to let you know how accurately your present investments can be fitted into tomorrow's world.
The internet is a booming marketplace. Online automotive lending is an industry that has begun to boom. There are several benefits of getting an automobile loan online, but there are some tips you should follow to fully utilize those benefits.
Online Credit Score The internet is a quick and hassle free place where you can purchase goods/services and acquire useful information.This calculator can prevent you from falling victim to this type of scam.
Compare quotes The internet provides a perfect venue for you to quickly and efficiently compare auto lender quotes. A useful tip for comparing is to use online sites that encourage lenders to compete for your business. This competition leads to lower interest rates and possibly shorter auto loan terms.
The internet is a great resource for individuals looking for an auto loan. If online features, such as credit scores, payment calculators, and competition sites, are used to their fullest, the borrower will always win.
Multi-Level Marketing (MLM), also known as Network MarketingThere are two types:Subscription or Money Plans and Product Based MLM. Subscription or Money PlansSubscription or Money plans are generally illegal and are not really Network Marketing or MLM. False credibilityThe promoters often call them that to give themselves some credibility They generally work by getting you to pay a subscription. Most of which the person or company running the plan gets and pay a percentage out to the person who introduced you to the plan.
You then go out and recruit as many people as you can so that you will get as much commission as possible. There is generally no product or service offered, or if there is it is generally worthless and just incidental. These plans are illegal, although are still often appearing on the doormat, coming in from countries out with the UK.RegistrationTo become part of it's sales force you have to become what they usually refer to as an agent or distributor. For this privilege it will normally cost anything between 45 and 200. 200 is currently the maximum set down by the DSA. This will normally provide you with a company training manual and operating procedures and a few samples of sales brochures, DVD's and stationery.
Now the general idea of network marketing is to get a lot of people retailing a little or in the case of one of the most well known organisations, getting a lot of people to retail a little but also to use all the consumables themselves. It is also sometimes possible to treat the business purely as a retailing opportunity. In fact for over two years I retailed just one of the product range from one MLM company as my sole source of income.Cash flowOften these companies will also give you credit so that you can order and then pay when the invoice arrives which is hopefully when you have collected your customers payments. If you do not have this facility then get a credit card and you will never have cash flow problems. In fact, I run two credit cards, one solely for business use and one for my own personal use.
Provided that you use a credit card wisely, you actually have extra rights under the 1974 Consumer Credit Act. Provided that you carefully checked the terms when opening your credit card account, then you will never incur expenses if you pay off the full amount before the final 'pay by' date. You should be able to achieve this if you are running your business correctly.Payments on interest-only mortgages, of course, are a lot easier to calculate – involving the multiplication of the amount borrowed, by the number of years, by the interest paid. The mortgage repayment calculator really comes into its own, of course, when you have some serious decisions to make about your mortgage. If it is your first, then you will want to know down to the last penny just how much the monthly repayments will be for the interest rate you are quoted. You may also probably want to compare the shorter- and longer-term costs of a repayment mortgage against an interest only mortgage. The calculator will help you compare the offers available from competing mortgage lenders. If you already have a mortgage, you might be interested in the effects of any rise or reduction in interest rate.
Burlington
BurlingtonEven KT our highest Sales $ Sales Rep has half of the customers he calls on earning less than negative $30! DH, while lower in Sales $, has more customers in positive profit territory. Of course some Sales Reps are much worse. Half of DB's customers lose $469 each year. From this data table and graph we can see that high and low performing Sales Reps have many unprofitable customers.
In fact, the best Sales Reps "hide" their bad customers because they have some very good customers. Improvement Action ItemsFrom this data we learned that all Sales Reps have work to do. Everyone has unprofitable customers. If you go back to the formula that we used to calculate profitability, you can understand what the Sales Reps had to do. The driver for these problems is the number of orders.This company's customers place orders too frequently. Instead of ordering a bulk delivery once a month, they have customers ordering barrels two-times a week. The Sales Reps had to go back to their customers and work together to understand the benefit to both companies of reducing order frequency. (If it costs us $105 to process an order, it also costs the customers to process purchase orders and invoices.)This change has allowed the business to grow, adding customers and volume, while reducing the number of people working in order processing.(They are also working on reducing order-processing labor costs through Lean and information technology.
Excel is perhaps the most important computer software program used in the workplace today. That's why so many workers and prospective employees are required to learn Excel in order to enter or remain in the workplace. From the viewpoint of the employer, particularly those in the field of information systems, the use of Excel as an end-user computing tool is essential.
Not only are many business professionals using Excel to perform everyday functional tasks in the workplace, an increasing number of employers rely on Excel for decision support. In general, Excel dominates the spreadsheet product industry with a market share estimated at 90 percent. Excel 2007 has the capacity for spreadsheets of up to a million rows by 16,000 columns, enabling the user to import and work with massive amounts of data and achieve faster calculation performance than ever before.Standard model cosmologists now play that expanding Universe 'film' in reverse. Travel back in time and the Universe is contracting, ever contacting. Alas, where do you stop that contraction? Well the standard model says when the Universe achieves a volume tinier than the tiniest subatomic particle! When (according to some texts) the Universe has achieved infinite density in zero volume – okay, maybe as close to infinite density and as close to zero volume as makes no odds. Translated, in the beginning the Universe was something within the realm of quantum physics! Now just because you can run the clock backwards to such extremes, doesn't mean that that reflects reality. How any scientist can say with a straight face that you can cram the entirety of not only the observable Universe, but the entire Universe (which is quite a bit larger yet again) into the volume smaller than the most fundamental of elementary particles is beyond me.What kind of physics is that? Curiouser and curiouser. Any and all miracles, Biblical or otherwise, are explainable as easily as saying "run program". More down to earth, you have multi-observations of things like the Loch Ness Monster, those highly geometrically complex crop circles, and ghosts, yet there's no real adequate theory, pro or con, that can account for their observed existence or creation. All up, perhaps some cosmic computer programmer/software writer whiz with a wicked sense of humour (a trickster 'god'?) is laughing its tentacles off since we haven't been able to figure it (our virtual reality) out. Of course maybe the minute we do, the fun's over and 'Dr. It' hits the delete key and that's the way the Universe ends – not with a Big Crunch, nor with a Heat Death, but with a "are you sure you want to delete this?
If you're a scientist, you probably don't think you need an iPhone app to help you with your endeavors, but many may actually help. And, if you are an aspiring scientist or man of science, or if science simply titillates and excites you, a number of iPhone applications were made for you! There are a lot of apps related to science for the iPhone, but a few stand above the rest. Why waste your time (and money) on somewhat lacking iPhone applications when you can get the best? Here are the top iPhone applications for scientists and those who want to unleash the scientist in them:JottNo scientist can conduct an experiment or study without a recording tool. This app turns your iPhone into a recorder. It actually records your voice and turns it into text. Talk about a hands-free way to create a pile of notes! Grafly and SolutionsThese two iPhone applications are so useful that many scientists use them and have raved about them.Since there is no multiple answers the children cannot guess. Kid Calc Elementary Math Help with Flash CardsThis math app is actually four apps all put together. It focuses on preschool and elementary students. It uses cool graphics along with fun games to keep young children engaged. There are animated flash cards, counting games, addition and subtraction math drills, and an animated calculator.
Ben Spratling Math TouchIf you are in a science class or an engineering student, this is considered the best math app. It automatically converts units and vector coordinate systems. It has an astronomical observation database and an equation database. Math Ref FreeSome math formulas are hard to remember but Math Ref Free takes the guessing out of it, helping you to understand about a particular formula and to find formulas.
There are also all kinds of tips to make understanding the formulas easier. Free Graphing CalculatorThe iPhone comes with different calculators to help you with your calculations. The free graphing calculator is a very powerful math app and it's also very flexible.Memory is usually allowed on graphing calculators by the organizations administering standardized tests. However, students with memory-enabled calculators are not allowed to bring stored examples into the exam, or take out the exam questions afterwards. This means that the memory must be cleared both before and after the exam. Calculator Tips for Exams:* Bring your calculator, even if you may not need it, they are not usually available at test centres.* Practice using your calculator on sample SAT mathematics questions before the test.* Don't buy an expensive, sophisticated calculator just to take the test – the problems simply do not require it.* Don't try to use a calculator on every question. First, decide how to solve the problem, and then decide whether to use the calculator. The calculator is meant to help, not get in the way.For those looking for a more standard scientific calculator, Powerone LE fits the bill. It's got a very intuitive interface that you'll pick up right away. Powereone also features unit conversion and a currency converter that stays current to international currency exchange rates. Additionally, its got a simple statistics calculator that finds the mean of a series of numbers. It could be better if the scientific features used a two-line interface.
GRAB A BUSTY BEER WITH FIND ME A GIFT'S SEXY BEER BOTTLE HOLDER! Find Me a Gift lets you slip your hands around a womanly waist without the fear of a slap! Findmeagift. com is an online gift company based in the Midlands. They stock a bevy of gifts, gadgets, gizmos and gift experience packages. You can be sure to get your hands around something unusual! The Sexy Beer Bottle Holder is a bottle holder in the shape of a sexy, bikini-clad body! The colours do vary, and the titillating two-piece fashioned holder is available in Red, Pink, Black and White. The Sexy Beer Bottle Holder measures approximately 14 cm x 10.5 cm x 9 cm so will snuggle nicely around any standard-sized beer bottle! After a hard day at work or relaxing on a summer's evening, you can't beat an ice-cold beer!When others raise a glass, you can lift the Sexy Beer Bottle Holder up high too, in true Richard Gere style! So hold on tight, don't let this beauty slip through your fingers!
So whether you are looking for a naughty birthday gift, a fun stag night accessory or simply for a novel sleeve to slip your beer into, the Sexy Beer Bottle Holders are just what the barman ordered. Forget women shaped like 'hour glasses'! The Sexy Beer Bottle Holders are making 'beer bottled' shaped women hotter. For a super sexy and ice cool way to sup your favourite bottled beverage, get your hands around one now! Find Me a Gift offers everything online without the need for people to spend money on petrol, parking and inflated prices.They have screens that can be navigated by touchpads, the ability to save documents, and have such a high resolution that users can upload pictures. My graphing calculator review is that many schools think that these calculators are a little too advanced. Some of these calculators can even connect to the Internet. This makes it much easier for students to cheat on tests.
Hence, schools continue to make students to purchase the old TI-83 and TI-84′s – calculators that are more expensive, and thirteen years older than some of the more advanced, contemporary calculators. While writing this graphing calculator review, it occurred to me that all of these models (including the TI-83 and 84) are great learning tools that can make math easier and fun.As well, tips for college visits before, during, and after the application process will be provided. |
Book
Review: How to Solve It: A New Aspect of
Mathematical Method
G. Polya. Princeton, NJ: Princeton University Press, 1957, Second Edition. Reviewed
by Jennifer Norton, Graduate Student Associate, TRC
Polya's How
to Solve It details the motives and procedures that lead to solutions
in mathematical problem solving and shows teachers how to help their students
learn how to solve problems. The interactive approach illustrated
in this text is designed to help students with their problem- solving
skills, while making sure they perform a reasonable amount of the work.
Teachers use questions to guide students effectively and unobtrusively,
and to enhance their problem-solving skills through imitation and practice.
The book is divided into four sections:
In the Classroom:
This section begins with a concise table that carries the reader through
the four phases of problem solving: 1) understanding the problem, 2) devising
a plan and recognizing the connection of parts of the problem, 3) carrying
out the plan, and 4) looking back: reexamining, discussing, and checking
the results in order to aid future problem solving. This section then
details these aspects of Polya's approach and walks the reader through
several examples.
How to Solve
It: An imaginary dialogue between a student and teacher illustrates
Polya's approach with respect to a particular mathematical problem.
Short Dictionary
Heuristic: The dictionary provides references for particular aspects
of problem solving, including such topics as the following: using analogies
to aid problem solving, introducing auxiliary elements to aid problem
solving, checking the result and deriving it differently, using the results
of earlier problems to solve new problems, decomposing and recombining
problems, thinking inductively, using notation, setting up equations,
varying problems, and recognizing signs of progress.
Problems,
Hints, Solutions: This section provides many sample problems to let
readers test their knowledge and understanding of the approach introduced
in this book. By encouraging the teacher/readers to participate in the
learning process from a student's perspective, Polya helps readers internalize
the approach and integrate it with their teaching skills.
Although the
first edition dates from 1945 and the author is writing to teachers of
mathematics, How to Solve It offers insights and practical solutions
for the difficult task of teaching students to solve problems in several
disciplines. If you find yourself solving problems for your students because
they can't do it themselves, or frustrated that you can't get them to
understand, try Polya's approach! |
Related Articles
Calculators are rarely first on list of things college students enjoy buying. And yet, without a little bit of research, some students can end up spending several hundred dollars on the devices throughout their college career.
While no guide can cover every calculator, course and instructor requirement, here you will find compiled a few basic guidelines for getting the right calculator during your time at Parkland College.
To get the straight scoop, Buster sat down with Omar Adawi, Associate Professor in mathematics and physics, and KeikoKircher, a part-time instructor who also pulls double duty, teaching physics and math as well.
These first questions were addressed to Kircher:
Buster Bytes: What calculators are allowed for PHY 141?
KeikoKircher: Any calculator is allowed, since our goal is not to test your algebra skills.
BB: Are any calculators not allowed for PHY 141?
KK: There is none.
BB: Do you use one calculator in particular during class? If so, would having that calculator help students follow along?
KK: I don't use any during lectures, but if I do use one during discussion time, I would probably be using the TI-83. It may help students to use the same calculator just because I'm able to help them with it, but not because the calculator itself does a better job.
BB: I'm not sure if you teach other Physics courses, but if you do, would any of your answers change for those courses?
KK: I have taught PHY 143 as well, and my answers would be the same with that course.
BB: I believe you also teach math? Are any particular calculators required for those courses?
KK: In the math class that I teach, pre-algebra, they are not allowed to use any calculator unless I explicitly tell them to use one. They are required to buy a calculator that can deal with basic functions such as square roots and trigonometric functions for when they deal with those. But the calculator doesn't have to be as fancy as theTI series.
Similar questions were asked of Adawi:
Buster Bytes: Professor Adawi, you teach MAT 129, what calculators are allowed for that course?
Omar Adawi: Depending on the instructor, the TI-83+, TI - 84+ or TI-89 is allowed.
BB: Are any calculators not allowed for MAT 129? Are these answers the same for all sections?
OA: Again it depends on the instructor. For example if the required calculator is a TI-83+ or TI-84+ then a TI-89 would not be permitted on quizzes and exams.
BB: Do you use one calculator in particular during class? If so, would having that calculator help students follow along?
OA: In MAT 129 I use the TI-89. The students may use this calculator to carry out explorations or sketch graphs, etc. The use of the calculator enhances the learning process of the class material.
BB: Do you know of any different regulations for other courses?
OA: There are different regulations, especially for MAT 128 and lower level courses, but for Calculus 3 and above as well.
BB: Do you teach courses in any other department? Are any particular calculators required for those courses?
OA: I teach PHY 142 in the summers. There is no particular calculator required for this class but I recommend the TI-89, since the engineering students will need this calculator for their future engineering courses.
Hopefully this information will help making your calculator purchases a little easier. To save money, you might want to look ahead and try to pick a calculator that will work for upcoming classes in your program, as well.
Another great way to save money is to shop around. Often, used calculators can be found for a discount from other students who have taken the course and no longer need their calculator. Look to the bulletin boards around campus for notifications about calculators for sale.
Used calculators can also be found on Craigslist, eBay and Amazon. Amazon even offers a free Prime membership to college students, which provides free two day shipping for many purchases.
Wherever you find your calculator, make sure you "do your homework," when it comes to shopping for the best price. And if you don't need it after the course, why not get a little money back and pass on the savings to another student who needs it? |
Interdisciplinary Uses of Graphing Calculators in Mathematics and Social Studies
Unformatted Document Text:
Graphing calculators 2
Professional organizations in mathematics and social studies are encouraging teachers
and students to use graphing calculator technology. The National Council of Teachers of Mathematics (NCTM 2000) advocates that problem solving, reasoning, communication, and interdisciplinary connectionsMethodology
We used an interpretive case study methodology to focus on how mathematics and socialThree economics and two mathematics 11
th
and 12
th
grades classrooms from a school inData Collection
In this study, five integrated lessons were used for instruction. As indicated earlier,An additional benefit of this collaborative inquiry was in the preparation of mathematicsDiscussion
A change in the teachers and most of the students in their relation to mathematics and
Authors: Okoka, Clara. and Lee, John.
Page 2 of 3
Graphing calculators 2
Professional organizations in mathematics and social studies are encouraging teachers
and students to use graphing calculator technology. The National Council of Teachers of Mathematics (NCTM 2000) advocates that problem solving, reasoning, communication, and interdisciplinaryconnections
Methodology
We used an interpretive case study methodology to focus on how mathematics and social
Three economics and two mathematics 11
th
and 12
th
grades classrooms from a school in
Data Collection
In this study, five integrated lessons were used for instruction. As indicated earlier,
An additional benefit of this collaborative inquiry was in the preparation of mathematics
Discussion
A change in the teachers and most of the students in their relation to mathematics and |
...
More About
This Book
and trig. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.
Calculus For Dummies is intended for three groups of readers:
Students taking their first calculus course – If you're enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.
Students who need to brush up on their calculus to prepare for other studies – If you've had elementary calculus, but it's been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies will give you a thorough, no-nonsense refresher course.
Adults of all ages who'd like a good introduction to the subject – Non-student readers will find the book's exposition clear and accessible. Calculus For Dummies takes calculus out of the ivory tower and brings it down to earth.
This is a user-friendly math book. Whenever possible, the author explains the calculusconcepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. Then, you'll see how the calculus concepts work in concrete examples. All explanations are in plain English, not math-speak. Calculus For Dummies covers the following topics and more:
Real-world examples of calculus
The two big ideas of calculus: differentiation and integration
Why calculus works
Pre-algebra and algebra review
Common functions and their graphs
Limits and continuity
Integration and approximating area
Sequences and series
Don't buy the misconception. Sure calculus is difficult – but it's manageable, doable. You made it through algebra, geometry, and trigonometry. Well, calculus just picks up where they leave off – it's simply the next step in a logical progression.
"My best day in Calc 101 at Southern Cal was the day I had to cut class to get
a root canal."
- Mary Johnson
"I keep having this recurring dream where my calculus professor is coming
after me with an axe."
- Tom Franklin, Colorado College sophomore
"Calculus is fun, and it's so easy. I don't get what all the fuss is about."
- Sam Einstein, Albert's great grandson
In this chapter, I answer the question "What is calculus?" in plain English,
and I give you real-world examples of how calculus is used. After reading
this and the following two short chapters, you will understand what calculus
is all about. But, here's a twist, why don't you start out on the wrong foot by
briefly checking out what calculus is not.
False, false, false! There's this mystique about calculus that it's this ridiculously
difficult, incredibly arcane subject that no one in their right mind would sign up
for unless it was a required course.
Don't buy into this misconception. Sure calculus is difficult - I'm not going to
lie to you - but it's manageable, doable. You made it through algebra, geometry,
and trigonometry. Well, calculus just picks up where they leave off - it's
simply the next step in a logical progression.
And calculus is not a dead language like Latin, spoken only by academics. It is
the language of engineers, scientists, and economists - okay, so it's a couple
steps removed from your everyday life and unlikely to come up at a cocktail
party. But the work of those engineers, scientists, and economists has a huge
impact on your day-to-day life - from your microwave oven, cell phone, TV,
and car to the medicines you take, the workings of the economy, and our
national defense. At this very moment, something within your reach or within
your view has been impacted by calculus.
So What Is Calculus Already?
Calculus is basically just very advanced algebra and geometry. In one sense,
it's not even a new subject - it takes the ordinary rules of algebra and geometry
and tweaks them so that they can be used on more complicated problems.
(The rub, of course, is that darn other sense in which it is a new and more difficult
subject.)
Look at Figure 1-1. On the left is a man pushing a crate up a straight incline.
On the right, the man is pushing the same crate up a curving incline. The
problem, in both cases, is to determine the amount of energy required to
push the crate to the top. You can do the problem on the left with regular
math. For the one on the right, you need calculus (assuming you don't know
the physics shortcuts).
For the straight incline, the man pushes with an unchanging force, and the
crate goes up the incline at an unchanging speed. With some simple physics
formulas and regular math (including algebra and trig), you can compute
how many calories of energy are required to push the crate up the incline.
Note that the amount of energy expended each second remains the same.
For the curving incline, on the other hand, things are constantly changing. The
steepness of the incline is changing - and not just in increments like it's one
steepness for the first 10 feet then a different steepness for the next 10 feet - it's constantly changing. And the man pushes with a constantly changing force - the
steeper the incline, the harder the push. As a result, the amount of energy
expended is also changing, not every second or every thousandth of a second,
but constantly changing from one moment to the next. That's what makes it a
calculus problem. By this time, it should come as no surprise to you that calculus
is described as "the mathematics of change." Calculus takes the regular
rules of math and applies them to fluid, evolving problems.
For the curving incline problem, the physics formulas remain the same, and
the algebra and trig you use stay the same. The difference is that - in contrast
to the straight incline problem, which you can sort of do in a single shot - you've
got to break up the curving incline problem into small chunks and do
each chunk separately. Figure 1-2 shows a small portion of the curving incline
blown up to several times its size.
When you zoom in far enough, the small length of the curving incline becomes
practically straight. Then, because it's straight, you can solve that small chunk
just like the straight incline problem. Each small chunk can be solved the same
way, and then you just add up all the chunks.
That's calculus in a nutshell. It takes a problem that can't be done with regular
math because things are constantly changing - the changing quantities
show up on a graph as curves - it zooms in on the curve till it becomes
straight, and then lets regular math finish off the problem.
What makes calculus such a fantastic achievement is that it actually zooms in infinitely. In fact, everything you do in calculus involves infinity in one way or
another, because if something is constantly changing, it's changing infinitely
often from each infinitesimal moment to the next.
Real-World Examples of Calculus
So, with regular math you can do the straight incline problem; with calculus
you can do the curving incline problem. Here are some more examples.
With regular math you can determine the length of a buried cable that runs
diagonally from one corner of a park to the other. With calculus you can
determine the length of a cable hung between two towers that has the shape
of a catenary (which is different, by the way, from a simple circular arc or a
parabola). Knowing the exact length is of obvious importance to a power
company planning hundreds of miles of new electric cable. See Figure 1-3.
You can calculate the area of the flat roof of a home with regular math. With
calculus you can compute the area of a complicated, nonspherical shape like
the dome of the Houston Astrodome. Architects designing such a building
need to know the dome's area to determine the cost of materials and to figure
the weight of the dome (with and without snow on it). The weight, of course,
is needed for planning the strength of the supporting structure. Check out
Figure 1-4.
With regular math and some simple physics, you can calculate by how
much a quarterback must lead his receiver to complete a pass. Note that
the receiver runs in a straight line and at a constant speed. But when NASA,
in 1975, calculated the necessary "lead" for aiming the Viking I at Mars, it
needed calculus because both the Earth and Mars travel on elliptical orbits
(of different shapes) and the speeds of both are constantly changing - not to
mention the fact that on its way to Mars, the spacecraft is affected by the
different and constantly changing gravitational pulls of the Earth, the moon,
Mars, and the sun. See Figure 1-5.
You see many real-world applications of calculus throughout this book. The
differentiation problems in Part IV all involve the steepness of a curve - like
the steepness of the curving incline in Figure 1-1. In Part V, you do integration
problems like the cable-length problem shown back in Figure 1-3. These
problems involve breaking up something into little sections, calculating each
section, and then adding up the sections to get the total. More about this in
ChapterIf you wonder:
Why study calculus?
What's it good for?
What's the bottom line of all this mumbo-jumbo?
then this book's for you.
Offering excellent bird's-eye views of key calculus concepts, Mr. Ryan then develops them in a clear, easy-to-understand manner that will give you one "Eureka!" moment after another.
I recommend this book as an overview for the curious, a review for the rusty, or as introductory material prior to tackling Calculus I.
It includes an excellent summary of the highlights of pre-calculus material
for review.
For more hands-on exercises, "Calculus Workbook for Dummies" is an excellent companion volume.
1 out of 1 people found this review helpful.
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Anonymous
Posted June 13, 2008
A Rosetta stone for Calculus
I love this book so much I could marry it. I am sending this guy a gift basket when I graduate. Every Calc student should buy this. My comments aren't hyperbole. I now understand calculus. It is easy, quick reading. There are no negatives to say.
1 out of 1 people found this review helpful.
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Anonymous
Posted February 26, 2006
A True Calculus Bible
I could always DO calculus, but I never really 'understood' it. The explanations in this book worked, where the many textbooks I have used failed. Hats off to the author for a most excellent approach to simplifying the 'whats and whys' of a challenging and confusing subject.
1 out of 1 people found this review helpful.
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Anonymous
Posted December 24, 2004
Connie B
I never thought that I could master anything higher than Algebra. Well, I did. Caluculus is for dummies after all. I love this book and will pass the good news on to fellow students. Thanks, C.B.
1 out of 1 people found this review helpful.
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Anonymous
Posted June 20, 2012
Makes ANYONE able to understand calculus!
I'm currently in Algerba 1 and I'm also in the lowest math class at school. I have an interest for calculus and I started to read this book. I was surprised at how easily I understood this book. The author presents calculus with an easy mindset instead of a difficult one which greatly helps the reader to come at it at an easy approach. I would highly recommend this to ANYONE.
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Anonymous
Posted November 27, 2011
Helps...
I'm taking a Calculus II course, and I was recently struggling with it. I bought this book because a friend of mine who already took the course said it helped them. They were right. It helps. Not as much as I would've liked, but the book definitely does help. It focuses more on Calc I stuff though.
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In Calculus for Dummies, Mark Ryan presents the fundamental concepts of calculus in a clear and concise manner.
After recently finishing a Master's in Evolutionary Ecology, I decided to pursue higher mathematics and engineering to contribute to the growing field of ecological engineering. I haven't taken Calculus in four years, and my first course is Multivariate Analysis. I grabbed Calculus for Dummies to brush up. I was extremely surprised by the text's approach. It provided a simplified conceptual overview, but left with me many relevant ways to view calculus in a simplified manner. This book will help remove the FEAR of taking a calculus course!
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Anonymous
Posted May 28, 2008
A good introduction:
This is a great entry-level book to calculus. I originally bought it to help out with my AP Calculus course in high school, but never read it ¿ had I, I might not have had to retake basic calculus (limits through antiderivatives) again in college. This is a great book that makes the fundamentals of first year calculus really stick. It is NOT a universal calculus panacea and several times Ryan notes that a particular topic is outside the scope of the book ¿ this does not mean it won't be on your test. This book - is- lacking with regards to the complexity of the examples given ¿ any calculus courses taken as a preparation for a math-based science (such as chemistry, engineering, or physics) -will- be harder, but with this book, you'll be able to concentrate on the methods and higher applications, rather than bogging yourself down trying to understand the more basic material.
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Anonymous
Posted May 9, 2008
Makes a complex topic understandable and even enjoyable!
Wow, it's been a long time since I've been in a math class, and to say that I am rusty is a gross understatement. So it was with great relief that I discovered that not only can I follow this book, I actually enjoyed it. I even laughed out loud a few times. Mark's humor made all the difference, and when the going got a little tough, the lively writing kept me engaged. Mark's experience as a teacher is evident throughout-- he knows where the pitfalls are and addresses them before the reader trips. I would describe myself as a slightly above-average math student, and calculus was not an intuitively easy topic for me. There were parts of the book that I had to read several times and then digest. Fortunately, I was being taught by a great teacher. I could well imagine learning calculus from a lessor instructor and getting totally lost. Soon my daughter will be taking calculus in school, and I'm going to make sure that she reads this book. Congratulations Mark on writing the definitive book for 'dummies,' scholars and everyone in between.
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Anonymous
Posted April 20, 2007
Good book
Good choice to help understand calculus
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Anonymous
Posted May 6, 2006
a terrific resource
The author does a great job of making a complex topic accessible to people (like me) that are equipped only with good basic math skills. He walks you through all of the greek symbols and translates them into plain english. Importantly, the material is not dumbed down, it is just presented in a way that is far more understandable than most text books.
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Anonymous
Posted November 12, 2004
Great book!
I used this book for calc 1 and 2 during college. It greatly helped my scores! It provides plain explanations and easy step-by-step walkthroughs.
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Anonymous
Posted October 17, 2004
Best Introduction to Calculus out there
Ryan uses everyday language and tons of the most common examples to help you learn. He also makes it as fun as it can be with jokes and everyday applications. To top it off, he covers every topic that you need to know for an introduction, and also where the ideas came from. Great book, you won't need anything else.
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Anonymous
Posted January 7, 2004
Excellent Introduction to Calculus Basics
I have been using 'Calculus for Dummies' to review my Calc I and Calc II basics. I am surprised and pleased with how well Ryan's descriptions and explanations have allowed me to better understand the underlying principles in Calculus. If you are serious about groundwork and maintenance of your mathematical education, you know that it is frequently beneficial to return to the basics for those fundamental concepts which can occasionally grow fuzzy with time. If you can find a text that treats the material in a new and entertaining way, the review can be enjoyable as well as instructive. I highly recommend Ryan's book. He limits complexity (and warns you when he is doing that) to keep the material accessible. For the ultra rigorous analysis, there are many college texts available. But if you are new to Calculus, or looking for a different and refreshing approach to the basics, you will find 'Calculus for Dummies' a wise investment. If you are taking Calculus in school and are having some problems understanding the material (and who hasn¿t?), this book will help you 'decode' some of the more difficult concepts. I am sure that it is destined to become a valuable catalyst text on many a struggling math student's desktop.
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After their carts collide in a hardware store, two teachers discover that they both bought the same items in different quantities. With limited information, this segment demonstrates how to use an equation to determine the cost of each item.
Students practice using algebraic expressions by recording data from a video segment in which two staircases ascend at different rates. They record the patterns in two-column tables, draw line graphs and write simple algebraic relations.
Using segments and web interactives from Get the Math, this lesson helps students see how Algebra I can be applied to the world of fashion, challenging them to use algebraic concepts and reasoning to modify garments and meet target price points.
Using segments and web interactives from Get the Math, this lesson helps students see how Algebra I can be applied in the music world, challenging them to use algebraic concepts and reasoning to calculate the tempos of different music samples.
Using video segments and web interactives from Get the Math, this lesson helps students see how Algebra I can be applied in the world of videogame design and challenges them to use algebraic concepts and reasoning to plot the linear paths of items in a videogame.
Students are introduced to algebraic expressions that use more than one variable and have multiple solutions. They figure out combinations of two items at different costs, with each combination adding up to 100.
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Major funding for Teachers' Domain was provided by the National Science Foundation. |
A collection of problems, some with a very practical application, others designed as purely theoretical puzzles, that will offer something of interest to all. Understand algebra, and all other branches of mathematics and arithmetic will suddenly open up in front of you.
Offers assistance in understanding algebra, which is vital to all higher level maths and therefore to a wide range of careers. Written by an experienced Maths teacher, who is also a parent of school age children, the book takes parents and students back to basics.
This text presents the complete proceedings of the Special Year and International Conference on Geometric Group Theory in Canberra, Australia, 1996. Topics covered include: geometric automorphism groups; geometric invariants of a group; and Artin groups and braid groups. |
Trigonometry With Infotrac
9780534403928
ISBN:
0534403921
Edition: 5 Pub Date: 2003 Publisher: Thomson Learning
Summary: This text provides students with a solid understanding of the definitions and principles of trigonometry and their application to problem solving. Identities are introduced early in Chapter 1. They are reviewed often and are then covered in more detail in Chapter 5. Also, exact values of the trigonometric functions are emphasized throughout the textbook. There are numerous calculator notes placed throughout the textTH EDITION, CD NOT INCLUDED. Intact & readable. PLEASE NOTE~ we rated this book USED~ACCEPTABLE due to likely defects such as highlighting, writing/markings, folds, creas [more]
5TH EDITION, CD NOT |
DJVUThis is a standard two-semester text for a first course in numerical analysis at the advanced undergraduate level, offering unique coverage of numerical approximation/interpolation, graphics, and parallel computing. A portion of the programs are written in Turbo Pascal. The remainder are pseudocode or generalized algorithms. Because other texts use FORTRAN or just pseudocode, the Turbo Pascal flavor of the Buchanan/Turner text sets it apart and makes it particularly appropriate for the typical undergraduate with Pascal programming skills and access to a personal computer. ...
NumericalMethodsnbsp; The book continues to maintain a student-friendly approach and numerical problem solving orientation.
Numerical continuation methods have provided important contributions toward the numerical solution of nonlinear systems of equations for many years. The methods may be used not only to compute solutions, which might otherwise be hard to obtain, but also to gain insight into qualitative properties of the solutions. Introduction to Numerical Continuation Methods, originally published in 1979, was the first book to provide easy access to the numerical aspects of predictor corrector continuation and piecewise linear continuation methods. Not only do these seemingly distinct methods share many common features and general principles, they can be numerically implemented in similar ways. Introduction to Numerical Continuation Methods also features the piecewise linear approximation of implicitly defined surfaces, the algorithms of which are frequently used in computer graphics, mesh generation, and the evaluation of surface integrals. ...
This
Book Description This new book from the authors of the classic book NumericalMethods addresses the increasingly important role of numericalmethods in science and engineering. More cohesive and comprehensive than any other modern textbook in the field, it combines traditional and well-developed topics with other material that is rarely found in numerical analysis texts, such as interval arithmetic, elementary functions, operator series, convergence acceleration, and continued fractions.
The various chapters within this volume include a wide variety of applications that extend far beyond this limited perception. As part of the Reliable Lab Solutions series, Essential Numerical Computer Methods brings together chapters from volumes 210, 240, 321, 383, 384, 454, and 467 of Methods in Enzymology. These chapters provide a general progression from basic numericalmethods to more specific biochemical and biomedical applications.
NumericalMethods in Engineering with Python is a text for engineering students and a reference for practicing engineers. The numerous examples and applications were chosen for their relevance to real world problems, and where numerical solutions are most efficient. ...
This introduction to numerical analysis shows how the mathematics of calculus and linear algebra are implemented in computer algorithms. It develops a deep understanding of why numericalmethods work and exactly what their limitations are.
Steven Chapra's second edition, Applied NumericalMethods with MATLAB for Engineers and Scientists, is written for engineers and scientists who want to learn numerical problem solving. This text focuses on problem-solving (applications) rather than theory, using MATLAB, and is intended for NumericalMethods users; hence theory is included only to inform key concepts. The second edition feature new material such as Numerical Differentiation and ODE's: Boundary-Value Problems. For those who require a more theoretical approach, see Chapra's best-selling NumericalMethods for Engineers, 5/e (2006), also by McGraw-Hill. ...
NumericalMethods in Engineering with Python, 2nd Edition is intended for engineering students and as a reference for practicing engineers interested in exploring Python. This new edition features 18 more exercises, more robust computer codes, and the addition of rational function interpolation, Ridder's method, and the downhill simplex method.
The contributions in this volume emphasize analysis of experimental data and analytical biochemistry, with examples taken from biochemistry. They serve to inform biomedical researchers of the modern data analysis methods that have developed concomitantly with computer hardware. ...
NumericalMethods in Engineering with MATLAB® is a text for engineering students and a reference for practicing engineers, especially those who wish to explore the power and efficiency of MATLAB®. Examples and applications were chosen for their relevance to real world problems, and where numerical solutions are most efficient. Numericalmethods are discussed thoroughly and illustrated with problems involving both hand computation and programming. MATLAB® mfiles accompany each method and are available on the book web site. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method. |
Mathematics
The number one book in the field, Basic Mathematics covers addition, subtraction, fractions, decimals, percent, and prepares readers for a first ...Show synopsisThe number one book in the field, Basic Mathematics covers addition, subtraction, fractions, decimals, percent, and prepares readers for a first course in algebra |
This webinar offers a quick and easy way to learn some of the fundamental concepts for using Maple. Learn the basic steps on how to compose, plot and solve various types of mathematical problems. This webinar will also demonstrate how to create professional looking documents using Maple, as well as the basic steps for using Maple packages.
The typical multivariate calculus course begins with a unit on vectors, lines and planes, material that serves as an introduction to the linear geometry of R³. This webinar will provide solutions for a number of typical, and not-so-typical problems in the "lines-and-planes" section of the calculus course. The problems will be solved with a suite of commands in the Student MultivariateCalculus package, commands specifically designed to handle manipulations of lines and planes in R³. In addition, these problems will also be solved in a syntax-free way via the Context Menu system because the relevant commands have been completely incorporated into that environment.
In this webinar, an introduction is made to modeling, simulation, and analysis with MapleSim by studying a full vehicle model equipped with electric power steering. The webinar will cover modeling and simulation aspects such as a 3D multibody steering mechanism, and multi-domain modeling by inclusion of electrical and 1D translational components.
This webinar, presented by Dr. Robert Lopez, Maple Fellow and Emeritus Professor from the Rose-Hulman Institute of Technology, will provide you with tips and techniques that will help you get started with Maple 17.
The Möbius Project is a revolutionary initiative that brings the power of Maple to even more people, in even more ways. This webinar will demonstrate: how to create Möbius Apps in Maple, how to share Möbius Apps with your colleagues and students using the MapleCloud, and how to grade Möbius Apps in Maple T.A. |
Algebra 2 covers factoring, rational exponents, quadratic equations, functions, imaginary and complex numbers, and exponential and logarithmic functions and equations. We would always endeavor to tie into the world around us, the subject matter in Algebra 2. The student and I would work through... |
Find a Marlow Heights, MD Algebra 2Topics include simplifying expressions, evaluating and solving equations and inequalities. Algebra 2 is build on algebraic and geometric concepts. It advances in algebraic skills such as systems of equations, advanced polynomials, imaginery and complex numbers, quadrat |
GeoGebra Workshop 2012
Thank you for your participation
The GeoGebra Workshop is a FREE professional development workshop for teachers working at elementary, secondary or terciary level. The workshop was held Wednesday May 23, 2012Keynote Speaker
Michael Todd Edwards Ph.D. is an associate professor at Miami University in Oxford, Ohio. He also serves as co-director of the GeoGebra Institute of Ohio. Dr. Edwards' scholarly interests center around the teaching and learning of mathematics with technology, specifically graphing calculators, computer algebra systems, and dynamic geometry software. He is dedicated to providing students with authentic mathematics-oriented experiences in the classroom using technology as a vehicle for fostering such experiences.
Speakers
Todd Edwards, Miami University, Oxford, Ohio Getting Started with GeoGebra - In addition to the keynote address, Dr. Edwards will present a session that provides a friendly introduction to GeoGebra for novices. No prerequisite skills are required.
Ryan Hedstrom, University of New England Department of Mathematical Sciences, Biddeford, Maine Using Probability Distributions in GeoGebra - In this presentation, we will explore how GeoGebra can be used to find probabilities from known probability distributions.
Doug Kuhlmann, Phillips Academy, Andover Massachusetts. GeoGebra in AP calculus - Several demos of GeoGebra applets that are used in teaching AP calculus (BC/AB). These applets are useful for both differential and integral calculus. Applets can be downloaded from Here are the materials associated with this presentation: Conference materials
James Quinlan, University of New England, Biddeford, Maine. This session will contain various GeoGebra skills centered around creating lessons/developing GeoGebra applets. In particular, we will look at making variable text, basic interface design, and conditional showing/hiding objects.
Mike May, Saint Louis University Mathematics Dept., St.Louis, Missouri
Using GeoGebra to create random drill problems - We will look at how to use the capabilities of GeoGebra to drill students with problems containing random values. (Advanced Session). Audience: Classroom Teachers.
James Factor, Alverno College, Mathematics Dept., Milwaukee, Wisconsin
The Geometric Construction and Animation of Bezier Curves - Students often have difficulty relating geometry constructs to algebra. This presentation will illustrate how to start with a geometric construction and derive the algebraic description of a curve. The GeoGebra software will be used in the construction of a geometric framework and the illustration of an animation that will show how the framework generates the curve. This process can be done for a polynomial curve of any degree. Dr. Factor's Conference materials
Pam Buffington with Peter Tierney-Fife, Gardiner, Maine
Taking Advantage of GeoGebra 4 - This session highlights new features in GeoGebra 4 and includes example classroom-ready created resources. Topics will include GeoGebra Script, graphing inequalities, the Function Inspector, and creating custom buttons with polygons and text. Created resource examples will focus on the topics of number sense, algebra, and geometry.
Pam Buffington with Peter Tierney-Fife, Gardiner, Maine
OER in Math Professional Development Project Resources - During the 2010-2011 school year Maine schools in RSU#11 and RSU#54 partnered with Education Development Center (EDC) to create grades 7-12 classroom-ready resources as part of the OER in Math Professional Development project. The session will focus on exploring project resources including formative assessments, GeoGebra applets, lesson plans, student activity handouts, and screencasts. All resources are free and available online, with most in formats that can be easily adapted by educators. Results of the project evaluation will be discussed briefly.
Map & Directions
Location
The Harold Alfond Center for Health Sciences, which opened in 1996, provides a significant focus for the University. Located at the center of campus, this three-story building houses numerous laboratories and lecture halls. It is a join-use building between the College of Osteopathic Medicine and the College of Arts and Sciences.
The Community
Biddeford is a small city with a population of about 23,000 and Saco with a population of about 17,000 sits across the river. Although originally settled a few hundred years ago, the cities really developed as major textile center in the 1800s; most of the original textile mills have since been converted to residential, professional, commercial and light industrial use. The section of town where the University is located is about four miles from the two downtowns were the Saco River meets the ocean in an area known as Hills Beach. This seaside area is commercially undeveloped and is primarily a summer cottage and residential area. Neighboring towns include Old Orchard Beach, Arundel, Kennebunk, Kennebunkport, and Wells.
Portland, the largest city in Maine, is a 25-minute drive north from Biddeford. This growing metro area is justifiably proud of its fine symphony orchestra, active theater groups, concert halls and other live music venues, museums and many other arts organizations. There is an abundance of one-of-a-kind restaurants - more per capita than almost any city in the country - and countless shops in the downtown area. Portland is home to two minor-league sports teams: the Portland Seadogs are affiliated with the Boston Red Sox and the Portland Pirates play hockey at the Cumberland County Civic Center.
For individual athletic pursuits, there are opportunities in southern Maine for canoe and kayaking, mountain and road biking, hiking and camping, Nordic and alpine skiing, snowshoeing, bird watching, and fishing and hunting. Somtimes just spending time on the beaches is enough to please visitors and residents alike. The campus is only an hour's drive from the mountains and minutes from numerous rivers.
Located off exit 32 of the Maine Turnpike (Route I-95), Biddeford is within easy reach of most major eastern cities. Portsmouth, New Hampshire, a 40-minute drive from campus, affords the curious an opportunity for exposure to a seaport steeped in North American history. In addition, the historical and cultural riches of Boston are a 90-minute drive from the campus. |
In terms of math lessons, algebra equations are often tricky, just to lose one step in the process due to a simple error will throw off the entire answer. Unlike some forms of mathematics, because there are so many steps in the process the student often finds it difficult to get the correct answers. Of course if students have not learned the steps from the beginning they will never success with their math lessons, algebra equations. Algebra is simply not a subject that can be learned if students do not start from the beginning and make sure they master each step along the way.
Many college teachers are just amazed and quite bewildered to find that their undergrad students have not learned the basics sufficiently to understand the more complex theorems. Many students fail their math lessons, algebra and other mathematical concepts in high school and manage to enter college and university without them, often because they are a mature student. Colleges and universities have had to incorporate high school preparatory classes; just to prepare these students sufficiently for the more advanced math lessons - algebra and then statistics and other forms of forms of higher math necessary for their college programs.
Therefore it is the onus of the math teachers to make sure that math lessons, algebra equations are mastered at these fundamental course levels in order for their students to proceed to the higher levels required of them. Many of these students do not understand the concepts of vectors and vector spaces and so earlier math lessons are required before Euclidean spaces and matrices can be taught. Understanding the underlying concepts from which the theorems are built upon is key to success with the algebraic equations and formulas.
Early on in the math lessons, especially at the college level, algebra concepts would have to include lessons on polynomials in order for students to grasp the properties of linear algebra. From this point the students can begin to understand that on complex vector spaces eigenvalues are certain to exist. At this point, the student can grasp the concept of the upper triangle matrix and how a linear operator on a real vector space has an invariant subspace.
Students are mastering their college level courses by this point and are studying orthonormal bases, the Gram-Schmidt procedure and adjoints. Other theorems useful in college math are the spectral theorem, and more. However, none of these college students will be even understood with other the proper high school courses and prerequisites. |
Math Principles for Food Service Occupations, 5th Edition
Math Principals for Food Service Occupations teaches readers that the understanding and application of mathematics is critical for all food service jobs, from entry level to executive chef or food service manager. All the mathematical problems and concepts presented are explained in a simplified, logical, step by step manner. It is a book that guides food service students and professionals in the use of mathematical skills to successfully perform their duties as a culinary professional or as a manager of a food service business. Now out in the 5th edition, this book is unique because it follows a logical step-by-step process to illustrate and demonstrate the importance of understanding and using math concepts to effectively make money in this demanding business. Part 1 trains the reader to use the calculator, while Part 2 reviews basic math fundamentals. Subsequent parts address math essentials in food preparation and math essentials in food service record keeping while the last part of the book concentrates on managerial math. New to this 5th edition, "Chef Sez", quotes from chefs, managers and presidents of companies, are used to show readers how applicable math skills are to food service professionals. "TIPS" (To Insure Perfect Solutions) are included to provide hints on how to make problem solving simple. Learning objectives and key words have also been expanded and added at the beginning of each chapter to identify key information, and case studies have been added to help readers understand why knowledge of math can solve problems in the food service industry. The content meets the required knowledge and competencies for business and math skills as required by the American Culinary Federation |
Concept of a set, operations on sets, Venn diagrams.De Morgan laws.Cartesian product, relation, equivalence relation. Representation of real numbers on a line.Complex numbers - basic properties, modulus, argument, cube roots of unity.Binary system of numbers.Conversion of a number in decimal system to binary system and vice-versa.Arithmetic, Geometric and Harmonic progressions.Quadratic equations with real coefficients.Solution of linear inequations of two variables by graphs.Permutation and Combination.Binomial theorem and its application.Logarithms and their applications.
2.Matrices and Determinants:
Types of matrices, operations on matrices Determinant of a matrix, basicproperties of determinant.Adjoint and inverse of a square matrix, Applications - Solution of a system of linear equations in two or three unknowns by Cramer's rule and by Matrix Method.
3.Trigonometry:
Angles and their measures in degrees and in radians.Trigonometrical ratios.Trigonometric identitiesSum and difference formulae.Multiple and Sub-multiple angles.Inverse trigonometric functions.Applications - Height and distance, properties of triangles.
4.Analytical Geometry of two and three dimensions:
Rectangular Cartesian Coordinate system.Distance formula.Equation of a line in various forms.Angle between two lines.Distance of a point from a line.Equation of a circle in standard and in general form.Standard forms of parabola, ellipse and hyperbola.Eccentricity and axis of a conic.
Point in a three dimensional space, distance between two points.Direction Cosines and direction ratios.Equation of a plane and a line in various forms.Angle between two lines and angle between two planes.Equation of a sphere.
5.Differential Calculus:
Concept of a real valued function - domain, range and graph of a function.Composite functions, one to one, onto and inverse functions.Notion of limit, Standard limits - examples.Continuity of functions - examples, algebraic operations on continuous functions.Derivative of a function at a point, geometrical and physical interpreatation of a derivative - applications.Derivatives of sum, product and quotient of functions, derivative of a function with respect of another function, derivative of a composite function.Second order derivatives.Increasing and decreasing functions.Application of derivatives in problems of maxima and minima.
6.Integral Calculus and Differential equations:
Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric, exponential and hyperbolic functions.Evaluation of definite integrals - determination of areas of plane regions bounded by curves - applications. Definition of order and degree of a differential equation, formation of a differential equation by examples.General and particular solution of a differential equation, solution of first order and first degree differential equations of various types - examples.Application in problems of growth and decay.
7. Vector Algebra :_
Vectors in two and three dimensions, magnitude and direction of a vector.Unit and null vectors, addition of vectors, scalar multiplication of vector, scalar product or dot product of two-vectors.Vector product and cross product of two vectors.Applications-work done by a force and moment of a force, and in geometrical problems.
Part 'A' - ENGLISH (Maximum Marks 200). The |
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Math Courses
Course:
Adv. Algebra I A,B/Algebra I A,B
Category:
P G W
Credits:
10
Grade Level:
9-12
Prerequisite:
Mastery of Pre-Algebra B or equivalent
Overview:
This is a two semester course for the first year of algebra. The first semester emphasizes the language of algebra, operating with rational numbers, inequalities, monomials, and polynomials. The second semester emphasizes functions and graphs, lines and slopes, systems of open sentences, radicals, quadratics, and factoring.
Course Essentials:
• Arithmetic Properties (CA Standard 1.0, 2.0) o Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable: o Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. • B.Functions (CA Standard 16.0, 17.0, 18.0) o 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. o 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. o 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. • Polynomials (CA Standard 10.0, 12.0) o 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques. o 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. • D.Factoring (CA Standard 11.0) o 11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. • E.Rational Expressions (CA Standard 13.0) o 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
• F.Equations (CA Standard 3.0, 4.0, 8.0, 9.0)/Inequalities (CA Standard 3.0, 4.0) o Students solve equations and inequalities involving absolute values. o Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12. o 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. o 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. • G.Quadratics Equations (CA Standard 140, 19.0, 20.0, 21.0, 22.0, 23.0) o 14.0 Students solve a quadratic equation by factoring or completing the square. o 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. o 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. o 21.0 Students graph quadratic functions and know that their roots are the x- intercepts. o 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. o 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. • H.Graphs (CA Standard 6.0, 7.0) o Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). o Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. • I.Axioms (CA Standard 24.0, 25.0) o 24.0 Students use and know simple aspects of a logical argument: 25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: • K. Word Problems (CA Standard 5.0, 15.0) o 5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. o 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
Course:
Adv. Algebra/Trigonometry A,B
Category:
P
Credits:
10
Grade Level:
10-12
Prerequisite:
Mastery of Adv. Geometry (grade of "B" or better)
Overview:
This two semester, college preparatory course examines linear, circular, trigonometric, and logarithmic functions; matrices, vectors, and linear systems; and trigonometric formulas, graphs, inverses, and their applications. The second semester examines polar coordinates; sequences and series with introduction of limits; graphs of lines and conics; probability and descriptive statistics.
Course:
Adv. Geometry A,B/Geometry A,B (Basic)
Category:
P G W
Credits:
10
Grade Level:
9-12
Prerequisite:
Mastery of Algebra I
Overview:
This two semester course develops methods of logical thinking areas used to develop a collection of useful statements about plane figures and relationships between them. All basic geometric content and many applications are presented--points, lines, distances, angles, and other figures. The second half develops methods of logical thinking used to examine statements about plane figures and relationships between them. All basic geometric content and many applications are presented--points, lines, distances, angles, and other figures.
Course Essentials:
1. Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2. Students write geometric proofs, including proofs by contradiction. 3. Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. 4. Students prove basic theorems involving congruence and similarity. 5. Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 6. Students know and are able to use the triangle inequality theorem. 7. Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. 8. Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. 9. Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders. 10. Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. 11. Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. 12. Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. 13. Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. 14. Students prove the Pythagorean theorem. 15. Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. 16. Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. 17. Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. 18. Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan( x ) = sin( x )/cos( x ), (sin( x )) 2 + (cos( x )) 2 = 1. 19. Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. 20. Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles. 21. Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. 22. Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
Course:
Algebra I W (P)/Algebra I W
Category:
P G W
Credits:
5
Grade Level:
9-12
Prerequisite:
Mastery of Pre-Algebra B or equivalent
Overview:
This is the first semester of a two-year course in Algebra. The first semester emphasizes the language of algebra, operating with rational numbers and inequalities.
This two semester course studies operations of powers, exponents, radicals, quadratics, and polynomials. The second semester studies exponents, exponential and logarithmic functions, probability, statistics, and trigonometric functions and identities.
Course Essentials:
1. Equations and Inequalities: Students solve linear equations, linear inequalities and absolute value equations and inequalities. 2. Linear Equations and Functions: Students graph linear and absolute value functions. They graph linear inequalities and write equations of lines. 3. Systems of Linear Equations and Inequalities: Students solve systems of equations in two and three variables by using methods of graphing, substitution, and elimination. They graph and solve systems of linear inequalities. 4. Quadratic Functions: Students graph quadratic functions and solve quadratic equations by factoring, completing the square, and using the quadratic formula. Students add, subtract, multiply and divide complex numbers. They can graph complex numbers. 5. Polynomials and Polynomial Functions: Students add, subtract, multiply, and divide polynomials. Students graph and solve polynomial functions. 6. Powers, Roots, and Radicals: Students operate with rational exponents and exponential functions. They solve radical equations. Students add, subtract, multiply and divide functions. They find inverses and compositions of two functions.
1. Develop skills necessary for success in math. 2. Add two or more numbers, regrouping as necessary, using 4- to 6-digit numbers. 3. Subtract a 4- or 5-digit number from a 5- or 6-digit number, with regrouping. 4. Multiply a 3- or 4-digit number by a 3-digit number, using numbers with one or more zeros. 5. Divide a 2- to 4-digit number by a 1- or 2-digit number and write the answer as a decimal. 6.Change fractions and mixed numbers to lowest terms. 7. Rename a whole number or mixed number as an equivalent mixed number. 8. Add and subtract Identify the question and insufficient information in word problems. 16. Identify distractors (extra facts) in problems. 17. Develop a strategy by determining the pattern or organization of word problems. 18. Use the strategy of estimation to test a problem. 19. Use the strategy of working backward to solve a problem. 20. Use bar, line and circle graphs and tables to gain information to problem solve. 21. Organize data into a table, diagram, or graph. 22. Apply the seven steps of problem-solving. 23. Identify standard units of measurement. 24. Select appropriate metric unit to measure weight and distance; convert, add, subtract, multiply and divide metric and non-metric units. 24. Find perimeters, areas and volumes of geometric forms. 25. Analyze, simplify and solve algebraic expressions; use formulas to solve problems. 26. Be able to read and interpret schedules, charts, and maps. 27. Be able to add and subtract negative numbers on a number line. 28. Be able to compare negative numbers using a number line. 29. Solve word problems using a calculator.
This two semester, college preparatory course examines derivatives, the chain rule, derivatives of trigonometric functions, applications of derivatives (concavity, points of inflection, maxima and minima) definite and indefinite integrals, application of definite integrals, transcendental functions, and methods of integration. The second semester examines conic sections and other plane curves, parametric equations for conics, hyperbolic functions, inverse hyperbolic functions, polar equations of conic sections and other curves, integrals in polar coordinates, infinite sequences and infinite series, power series and taylor polynomials, vectors, vector functions and motion, and differential equations--first order, second order and higher order equations.
Course Essentials:
1. Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity: 2. Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function. 3. Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem. 4. Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability: 5. Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions. 6. Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth. 7. Students compute derivatives of higher orders. 8. Students know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule. 9. Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing. 10. Students know Newton's method for approximating the zeros of a function. 11. Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts. 12. Students use differentiation to solve related rate problems in a variety of pure and applied contexts. 13. Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals. 14. Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals. 15. Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives. 16. Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work. 17. Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate. 18. Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals. 19. Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square. 20. Students compute the integrals of trigonometric functions by using the techniques noted above. 21. Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators or computers or both to approximate integrals numerically. 22. Students understand improper integrals as limits of definite integrals. 23. Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges. 24. Students understand and can compute the radius (interval) of the convergence of power series. 25. Students differentiate and integrate the terms of a power series in order to form new series from known ones. 26. Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term. 27. Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.
Course:
Integrated Math I A,B
Category:
P
Credits:
10
Grade Level:
9-12
Prerequisite:
Mastery of Pre-Algebra B or equivalent
Overview:
This college preparatory course examines integrated mathematics, which includes concepts from algebra, geometry, logic, probability, and statistics.
Course:
Integrated Math II A,B
Category:
P
Credits:
10
Grade Level:
9-12
Prerequisite:
Integrated Math I
Overview:
This college preparatory course examines integrated mathematics, which includes concepts from algebra, geometry, logic, probability, and statistics.
Course:
Integrated Math III A,B
Category:
P
Credits:
10
Grade Level:
9-12
Prerequisite:
Integrated Math II
Overview:
This college preparatory course examines integrated mathematics, which includes concepts from algebra, geometry, logic, probability, and statistics.
Course:
Intermediate Algebra A,B
Category:
G W
Credits:
5
Grade Level:
10-12
Prerequisite:
Mastery of Algebra I or Geometry
Overview:
(Not recommended for college bound Math/Science majors.) (Can not be used as a prerequisite toward fourth year of math.) This two semester course studies operations of powers, exponents, radicals, quadratics, and polynomials. The second half studies exponents, exponential and logarithmic functions, probability, statistics, and trigonometric functions and identities.
Course Essentials:
1. Extent knowledge of exponents and: Simplify expressions with negative integers. Express large and small numbers in scientific notation. Perform computations involving scientific notation. Write expressions written in either radical or exponent form. 2. Recognize and apply the basic properties of logarithms. Recognize the exponential and logarithmic functions are inverses. Evaluate logarithmic expressions. Solve verbal problems using logarithms. Solve equations using the product, quotient, and power properties of logarithms. Express logarithms as the sum or difference of simpler logarithmic expressions. Use logarithms to compute powers and roots. 3. Achieve and understanding of sequence and series, including: Arithmetic Factorial notation Geometric Special sequences Binomial expansion General Term of sequence 4. Use probability mathematics. Calculate combinations Calculate permutations Find the probability and odds of success and failure 5. Use basic statistical methods. Use table and graphs to present data. Calculate - mean, mode, median, standard deviation Use prediction equations. 6. Strengthen past learned skills acquired in the first course of algebra, and: Follow order of operations. Evaluate expressions. Apply the addition, subtraction, multiplication and division properties of equality to solve equations. Translate and solve verbal expressions. Evaluate expressions with absolute value and solve equations with absolute value. Compare numbers and expressions using inequality symbols. Solve inequalities using inequality properties. 7. Calculate and graph using the coordinate plane. Find the value of a function for a given element of the domain. Graph linear equations. Find the slope of the line passing through a given pair of points. Find the slope, y-intercept, and x-intercept from a given linear equation. 8. Solve systems of linear equations using: Graphs Substitution Elimination Three Variables Calculate for graphs of equations which parallel, perpendicular or neither. 9. Operate with polynomials, including: Simplification Factorization Long Division of Polynomials Addition Subtraction Multiplication Division 10. Operate with radical expressions and: Find a given root of numbers and algebraic expressions. Simplify and multiply radicals. Add and subtract radicals; multiply radicals using FOIL Divide radicals; name conjugates; rationalize the denominator; simplify; approximate value to three decimals. 11. Solve quadratic equations by: Factoring. Representing word problems with pictures, then writing and solving quadratic equations. Completing the square. Using the quadratic formula. 12. Operate with rational polynomial expressions to: Simplify and multiply expressions. Find monomial, binomial, and trinomial LCD's. Add and subtract expressions. Solve rational equations. Use rational expressions to solve problems.
Course:
Math Analysis A, B
Category:
P
Credits:
10
Grade Level:
11-12
Prerequisite:
Mastery of Algebra II B (grade of "B" or better)
Overview:
This two semester, college preparatory course examines linear and quadratic functions, polynomial functions, inequalities in one and two variables, exponents and logarithms, trigonometric functions, equations and applications including sine curves, cosine curves, and trigonometric identities. The second semester examines triangle trigonometry: solving right triangles, law of sines, law of cosines, trigonometric additions formulas, matrices, combinatorics, probability including the binomial probability theorem, probability of combinations, and descriptive statistics.
Course Essentials:
1. Find an equation of a line given the geometric properties of the slope. 2. Model real-world situations by means of linear functions. 3. Solve and graph quadratic equations when solutions are both real and complex. 4. Use synthetic division to apply the remainder and factor theorems. 5. Find maximum and minimums of polynomial functions. 6. Solve polynomial equations. 7. Solve applied problems using linear programming. 8. Use exponential and logarithmic functions: use the natural logarithm. 8. Solve exponential equations. 9. Find the arc length and area of a sector of a circle and solve problems of apparent size. 10. Evaluate the six trigonometric functions and solve trigonometric equations. 11. Evaluate the inverse trigonometric functions. 12. Find the equations of different sine and cosine curves in applications. 13. Use trigonometric functions to model periodic behavior. 14. Prove trigonometric identities. 15. Use a graphical calculator to study linear, polynomial and trigonometric functions. 16. Use trigonometry to find unknown sides or angles of a right triangle. 17. Find the area of a triangle given the length of two sides and the measure of the included angle. 18. Use the law of sines and cosines. 19. Use the trigonometric addition formulas. 20. Use identities to solve trigonometric equations. 21. Find the sum, difference, scalar multiples, and products of matrices. 22. Find the inverse of a 2x2 matrix and solve linear systems using matrices. 23. Solve communications network problems using matrices. 24. Solve problems involving permutations and combinations. 25. Solve counting problems that involve permutations with repetition and circular permutations. 26. Use the binomial theorem and Pascal's triangle. 27. Find a sample space of an experiment and the probability of an event or either two events two events. 28. Find the probability of events occurring together and to determine whether two events are independent 29. Use the binomial probability theorem. 30. Use combinations to solve probability problems. 31. Calculate the mean, median, mode, variance and standard deviation of a set of data. 32. Use tables, graphs, box-and -whisker plots for data analysis. 33. Analyze the normal distribution.
Course:
Mathematics 7
Category:
Jr. H.S.
Credits:
10
Grade Level:
7
Prerequisite:
Required
Overview:
This course enables the student to improve fundamental skills in everyday arithmetic, including basic operations of decimals and percents and solving problems with geometric shapes.
Course Essentials:
Number Sense 1. Students know the properties of, and compute with, rational numbers expressed in a variety of forms: 2. Students use exponents, powers, and roots and use exponents in working with fractions: Algebra and Functions 1. Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs: 2. Students interpret and evaluate expressions involving integer powers and simple roots: 3. Students graph and interpret linear and some nonlinear functions: 4. Students solve simple linear equations and inequalities over the rational numbers: Measurement and Geometry 1. Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems: 2. Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale: 3. Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures: Statistics, Data Analysis, and Probability 1. Students collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set by hand and through the use of an electronic spreadsheet software program: Mathematical Reasoning 1. Students make decisions about how to approach problems: 2. Students use strategies, skills, and concepts in finding solutions: 3. Students determine a solution is complete and move beyond a particular problem by generalizing to other situations.
Course:
Mathematics 8
Category:
Jr. H.S.
Credits:
10
Grade Level:
8
Prerequisite:
Required
Overview:
This course enables the student to practice and improve fundamental skills in everyday arithmetic, including addition, subtraction, multiplication and division of fractions and decimals.
Course Essentials:
1. Add two or more numbers, regrouping as necessary, using 4-to 6- digit numbers. 2. Subtract a 4- or 5- digit number from a 5- or 6- digit number, with regrouping. 3. Multiply a 3- or 4- digit number by a 3-digit number, using numbers with one or more zeros. 4. Divide a 2- to 4- digit number by a 1- or 2- digit number and write the answer as a decimal. 5. Change fractions and mixed numbers to lowest terms. 6. Rename a whole number or mixed number as an equivalent mixed number. 7. Add mixed numbers with unlike denominators. 8. Subtract Apply whole numbers, fractions, decimals, proportions and percents to solve consumer, technical, and business problems.
Course:
Pre-Algebra A, B
Category:
G W
Credits:
10
Grade Level:
7-12
Prerequisite:
Director Approval
Overview:
This two semester course enables the student to practice and improve fundamental skills prerequisite to those encountered in formal algebra. Course content includes percents, measurements, conversions, formulas, and pre-algebra tasks and applications.
Course Essentials:
1. Determine if proportions are true; write and solve proportions with the unknown in each of four positions. 2. Convert fractions, decimals, and percents. 3. Find a percentage of a given number. 4. Find what percent one number is of another. 5. Find a number if a percentage of the number is known. 6. Graph, add, subtract, multiply and divide integers. 7. Rewrite numbers with positive and negative exponents; simplify expressions with and without parentheses; evaluate and write numbers using scientific notation. 8. Evaluate and simplify variable expressions. 9. Translate verbal expressions into mathematical expressions using given variables or by assigning variable names. 10. Solve equations of the forms; x + a = b; ax = b; ax + b = c. 11. Solve equations with unknowns on both sides. 12. Solve equations containing parentheses. 13. Solve a formula for one variable, given the values of the other variables. 14. Translate verbal sentences into mathematical sentences and solve. 15. Use equations to solve consumer, technical and business problems.
1. Use synthetic division to apply the remainder and factor theorems. 2. Find maximums and minimums of polynomial functions by graphing. 3. Solve polynomial equations. 4. Use exponential and logarithmic functions; use the natural logarithm. 5. Solve exponential equations. 6. Find the arc length and the area of a sector of a circle and solve problems of apparent size. 7. Evaluate the six trigonometric functions and solve trigonometric equations. 8. Evaluate the inverse trigonometric functions. 9. Find the equations of different sine and cosine curves in applications. 10. Use the trigonometric functions to model periodic behavior. 11. Prove trigonometric functions. 12. Use trigonometry to find unknown sides or angles of a right triangle. 13. Find the area of a triangle given the length of two sides and the measure of the included angle. 14. Use the law of sines and cosines. 15. Use the trigonometric additions formulas. 16. Use identities to solve trigonometric equations. 17. Use a graphical calculator to study linear, polynomial and trigonometric functions. 18. Find equations of circles and find the coordinates of any points where circles and lines meet. 19. Find equations of ellipses, hyperboles and paraboles. 20. Solve systems of second-degree equations. 21. Graph polar equations. 22. Write complex numbers in polar form and find products in polar form. 23. Use De Moivre's theorem to find powers of complex numbers. 24. Find roots of complex numbers. 25. Use vector and parametric equations to describe motion in the plane. 26. Apply dot and cross product. 27. Extend vectors to three dimensions and apply them. 28. Find the formula for the nth term of an arithmetic and geometric sequence. 29. Use sequences recursively to solve problems. 30. Find the sum of a finite arithmetic and geometric series. 31. Find the limit of an infinite sequence: determine when limit does not exist. 32. Find the sum of an infinite geometric series. 33. Use identities to solve trigonometric equations. 34. Use mathematical induction to prove that a statement is true. 35. Determine whether a function is continuous. 36. Use limits to graph rational functions. 37. Use the power series of a given function to find an infinite series for a functional value or a related function. 38. Find derivatives of functions. 39. Sketch the graphs of functions using derivatives. 40. Solve extreme value problems using derivatives. 41. Find instantaneous velocities and accelerations. 42. Use a graphic calculator to study functions.
Course:
Probability and Statistics A,B
Category:
P
Credits:
10
Grade Level:
11-12
Prerequisite:
Mastery of Algebra II
Overview:
This two semester, college preparatory course provides a solid background in descriptive statistics, probability, random variables, and discrete and continuous distributions.
Course Essentials:
1. Define "independent events" and demonstrate the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces 2. Define "conditional probability" and solve for probabilities in finite ample space. 3. Solve for probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin toss using "Discrete Random Variables" 4. Use standards (normal, binomial, and exponential) to solve events in problems in which the distribution belongs to those families. 5. Determine the mean and the standard deviation of a normally distributed random variable 6. Define mean, median, and mode of a distribution of data and compute each in particular situations. 7. Compute the variance and standard deviation of a distribution of data 8. Using frequency tables, histograms, standard line and bar-graphs, stem and leaf displays, scatterplots and box-and-whisker plots, organize and describe distributions of data.
Course:
Statistics
Category:
P
Credits:
5
Grade Level:
11-12
Prerequisite:
Mastery of Algebra II
Overview:
This one semester, college preparatory course examines all standard topics in statistics through two-way analysis of variance. Concentration is on the basic concepts with secondary emphasis on their application.
This one semester, college preparatory course examines trigonometric functions, inverse trigonometric functions, analytic trigonometry, graphing of functions, and additional topics such as Law of Sines and Cosines.
Course Essentials:
1. Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians. 2. Students know the definition of sine and cosine as y- and x- coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions. 3. Students know the identity cos 2 (x) + sin 2 (x) = 1: 4. 5. Students know the definitions of the tangent and cotangent functions and can graph them. 6. Students know the definitions of the secant and cosecant functions and can graph them. 7. Students know that the tangent of the angle that a line makes with the x- axis is equal to the slope of the line. 8. Students know the definitions of the inverse trigonometric functions and can graph the functions. 9. Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. 10. Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/ or simplify other trigonometric identities. 11. Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/ or simplify other trigonometric identities. 12. Students use trigonometry to determine unknown sides or angles in right triangles. 13. Students know the law of sines and the law of cosines and apply those laws to solve problems. 14. Students determine the area of a triangle, given one angle and the two adjacent sides. 15. Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa. 16. Students represent equations given in rectangular coordinates in terms of polar coordinates. 17. Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form. 18. Students know DeMoivre's theorem and can give n th roots of a complex number given in polar form. 19. Students are adept at using trigonometry in a variety of applications and word problems. |
Determinant
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.
Matrices
In mathematics, a matrix is a rectangular table of elements, which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters.
Matrices are described by the field of matrix theory. Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra.
The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (written m × n) and m and n are called its dimensions. |
Home > Education > Mathematics and Statistics for the New Zealand Curriculum
Welcome to the Mathematics and Statistics for the New Zealand Curriculum website.
Here you will find a new series of textbooks, workbooks, CD-ROMs and supporting resources for Years 9 to 13 in the New Zealand Curriculum. Use the following links to access resources and learn about the New Zealand books and authors.
Click on the covers below for further information including pricing and contents. |
This revised introduction to the basic methods, theory and applications of elementary differential equations employs a two part organization. Part I includes all the basic material found in a one semester introductory course in ordinary differential equations. Part II introduces students to certain specialized and more advanced methods, as well as providing a systematic introduction to fundamental theory. |
I'm a student educator. This blog is a record of my development as a person and as a professional. If you are looking for specific kinds of posts, click on the appropriate category below.
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Pre-Calculus, Weeks 01 and 02
Remember, when I said I was refreshing on my Pre-calculus? I was expecting it to be a quick refresher, but it turns out that there are many concepts from pre-calculus that I never learned properly the first time. As I go through and relearn these pre-calculus skills, I realise that they are the exact same skills that screwed me over in my first calculus classes.
Simplifying Radical Forms
Sounds so easy, until you get something like this,
and you have to figure out how to turn this into its simplest form. Just so you know, the correct answer is
Algebraic Symbol Manipulation
This one got me twice. The first time, and then the second time when I was reviewing it two days later, to see if I still knew how to do it.
I had to play around with the variables for half an hour before I realised you just have to get the x's on the same side, and then factor the x out.
Solving Equations in Radical Form
This one is a skill you need to succeed with integrals. Most equations in radical form are easy. It's when you throw in weird shit like fractional exponents that you have to really think "out of the box".
The trick here is to turn the equation into a classical quadratic equation,
then make a substitution.
Now you can solve the equation like normal.
u now is equal to either -3 or 1. Substitute y^(1/3) back in… and you have.
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2 thoughts on "Pre-Calculus, Weeks 01 and 02"
Haha, yea, I'm also having trouble getting excited about math. But, I'm going back to school for engineering, so I have to make this stuff my bread and butter. How about yourself? How are your studies coming? |
Intended Outcomes for the course
Use quadratic, rational and radical models in academic and non-academic environments.
Recognize connections between graphical and algebraic representations in academic and non-academic settings. Interpret graphs in academic and non-academic contexts.
Be successful in future coursework that requires the use of algebraic concepts and an understanding of functions.
Outcome Assessment Strategies
Assessment shall include:
1. The following topics must be assessed in a closed-book, no-note, no-calculator setting:
a.finding the equation of the linear function given two ordered pairs stated using function notation
b.simplifying rational expressions
c.solving rational equations
d.solving radical equations
e.graphing quadratic functions
f.determining the domain of radical and rational functions
g.evaluating algebraic expressions that include function notation
2.At least two proctored closed-book, no-note examinations, one of which is the comprehensive final. These exams must consist primarily of free response questions although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate.
3.Assessment must include evaluation of the student's ability to arrive at correct and appropriate conclusions using proper mathematical procedures and proper mathematical notation. Additionally, each student must be assessed on their ability to use appropriate organizational strategies and their ability to write conclusions appropriate to the problem.
2.3.3.Represent the domain in both interval and set notation, where appropriate
2.3.4.Apply unions and intersections ("and" and "or") when finding and stating the domain of functions
2.3.5.Understand how the context of a function used as a model can limit the domain
2.4. Range
2.4.1.Understand the definition of range (set of all possible outputs)
2.4.2.Determine the range of functions represented graphically, numerically and verbally
2.4.3.Represent the range in interval and set notation, where appropriate
2.5. Function notation
2.5.1.Evaluate functions with given inputs using function notation where functions are represented graphically, algebraically, numerically and verbally (e.g. evaluate )
2.5.2.Algebraically simplify and distinguish between different examples such as , , and
2.5.3.Interpret in the appropriate context e.g. interpret where models a real-world function
2.5.4.Solve function equations where functions are represented graphically, algebraically, numerically and verbally (i.e. solve for and solve for where and should include but not be limited to linear functions, quadratic functions, and absolute value functions)
2.5.5.Solve function inequalities algebraically (i.e. , , and where and are linear functions and and where is an absolute value function)
2.5.6.Solve function inequalities graphically (i.e. ,, and where and should include but not be limited to linear functions, and for quadratic and absolute value functions)
2.6. Graphs of functions
2.6.1.Use the language of graphs and understand how to present answers to questions based on the graph (i.e. read the value of an intersection to solve an equation and understand that is a number not a point)
2.6.2.Determine function values, solve equations and inequalities, and find domain and range given a graph
5.6.1.Solve distance, rate and time problems involving rational terms using well defined variables and stating conclusions in complete sentences including appropriate units
5.6.2.Solve problems involving work rates using well defined variables and stating conclusions in complete sentences including appropriate units
Addendum
·Functions should be studied symbolically, graphically, numerically and verbally.
·As much as possible, instructors should present functions that model real-world problems and relationships to address the content outlined on this CCOG.
·Function notation is emphasized and should be used whenever it is appropriate in the course.
·Students should be required to use proper mathematical language and notation. This includes using equal signs appropriately, labeling and scaling the axes of graphs appropriately, using correct units throughout the problem solving process, conveying answers in complete sentences when appropriate, and in general, using the required symbols correctly.
·Students should understand the fundamental differences between expressions and equations including their definitions and proper notations.
·All mathematical work should be organized so that it is clear and obvious what techniques the student employed to find his answer. Showing scratch work in the middle of a problem is not acceptable.
·Since technology is used throughout the course, there is a required calculator packet for students that gives directions for several graphing calculators. The students should understand the limitations of calculator—i.e. when the calculator gives misleading information. Examples of the calculator's limitations include the following: when finding horizontal intercepts, the calculator sometimes gives something like y = 3E-13; the calculator rounds to 12 or fewer decimal places; some calculators appear to show vertical asymptotes on the graphs of rational functions; it appears that the graph of touches the x axis; the calculator does not show holes on rational function graphs; the calculator cannot handle very large numbers, e.g. etc.
·For dividing rational expressions as in 5.3.3 and 5.3.3.1, focus on examples where the letters represent real numbers and linear polynomials. E.g. , , and .
·Exploration of difficult rational exponents, as in 4.5, should be limited. Basic understanding is essential and a deep understanding takes more than one course to develop. Examples should be limited to one or two variables, keeping things as simple as possible while covering all possibilities. E.g. , , , .
·As much as possible, instructors should present functions that model real-world problems and relationships to address the content outlined on this CCOG.
·In 3.3.1, when solving applications of quadratic equations, a complex solution should be interpreted as the graph never reaching a particular real world y-value.
·For simplifying complex rational expressions as in 5.3.6.1, a major emphasis shall be placed on cases where , , , , , and (as above) represent real numbers, linear polynomials in one variable. For example, or would be good examples. |
MATH 225: Mathematics as a Decision Making Tool
The focus of this course is to develop the quantitative skills and reasoning ability necessary to help students read critically and make decisions in our technical information society. A project tying this course to the student's own interest is a course requirement. Major topics include collecting and describing data, inferential statistics and probability, geometric similarity, geometric growth, symmetry and patterns. 3 hrs. lecture/wk.... more »
Credits:0
Overall Rating:0 Stars
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Thanks, enjoy the course! Come back and let us know how you like it by writing a review. |
Edgenuity Geometry is a two-semester, hands-on and lecture-based course featuring an introduction to geometry, including reasoning and proof and basic constructions. Triangle relationships (similarity and congruency) and...
Edgenuity English-Language Arts Grade 9 is a year-long course that provides rigorous training in the foundations of English Language Arts skills and strategies. It also expands on and applies traditional concepts to...
Edgenuity Algebra is a two-semester course that provides in-depth coverage of writing, solving and graphing a variety of equations and inequalities, as well as linear systems. Functions are a central theme of the course...
Edgenuity English-Language Arts Grade 9 (common Core) provides rigorous training in the foundations of English Language Arts skills and strategies. In addition, it expands on and applies traditional concepts to modern,...
Edgenuity English-Language Arts, Grade 10 (Common Core provides rigorous training in the foundations of English Language Arts skills and strategies. In addition, it expands on and applies traditional concepts to modern,...
The goal of this course in American literature is to explore the pursuit of the American Dream through American literature. The emphasis is on instruction in critical reading, and textual analysis demonstrated through...
AP English Literature and Composition is designed to be a college/university-level course, thus the "AP" designation on a transcript. This course will provide students with the intellectual challenges and workload...
The purpose of English I is to develop effective skills in writing, reading, and analyzing that can be applied broadly to literary genre (i.e. poetry, short story, novel, and drama) and across disciplines (i.e. science,...
Algebra I A: This course covers such key concepts as variables, function patterns, and graphs. Students learn operations with rational numbers and properties of rational numbers. Students
solve linear equations and... |
Introduction to Linear Functions
This lesson is designed to introduce students to the idea of functions composed of two operations, with specific attention to linear functions and their representations as rules and data tables, including the mathematical notions of independent and dependent variables. |
Algebra success for allBasic concepts and properties of algebra are introduced early to prepare stud ... more »ents for equation solving. Abundant exercises graded by difficulty level address a wide range of student abilities. The Basic Algebra Planning Guide assures that even the at-risk student can acquire course content.Multiple representations of conceptsConcepts and skills are introduced algebraically, graphically, numerically, and verbally-often in the same lesson to help students make the connection and to address diverse learning styles.Focused on developing algebra concepts and skillsKey algebraic concepts are introduced early and opportunities to develop conceptual understanding appear throughout the text, including in Activity Labs. Frequent and varied skill practice ensures student proficiency and success. « less
"Note: This seller does not offer expedited shipping. Ships same or next day! Almost perfect! isbn|0133500403 Prentice Hall Mathematics: algebra 1 (c. )2011 (JoV) wq " -- all american textbooks @ Michigan, United States Teachers Source is dedicated to providing the best textbook buying experience possible. The books and CD's we sell are purchased brand new and shipped out the same way. We ship the same day if ordered before 3pm PST with free shipping confirmation. If you have any questions please feel free to send us an email and we will get back to you the same day. Thank you for your business. . . " -- teacherssource @ California, United States |
Interdisciplinary Uses of Graphing Calculators in Mathematics and Social Studies
Unformatted Document Text:
Graphing calculators 2
Professional organizations in mathematics and social studies are encouraging teachers
and students to use graphing calculator technology. The National Council of Teachers of Mathematics (NCTM 2000) advocates that problem solving, reasoning, communication, and interdisciplinary connectionsMethodology
We used an interpretive case study methodology to focus on how mathematics and socialThree economics and two mathematics 11
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In this study, five integrated lessons were used for instruction. As indicated earlier,An additional benefit of this collaborative inquiry was in the preparation of mathematicsDiscussion
A change in the teachers and most of the students in their relation to mathematics and
Authors: Okoka, Clara. and Lee, John.
Page 2 of 3
Graphing calculators 2
Professional organizations in mathematics and social studies are encouraging teachers
and students to use graphing calculator technology. The National Council of Teachers of Mathematics (NCTM 2000) advocates that problem solving, reasoning, communication, and interdisciplinaryconnections
Methodology
We used an interpretive case study methodology to focus on how mathematics and social
Three economics and two mathematics 11
th
and 12
th
grades classrooms from a school in
Data Collection
In this study, five integrated lessons were used for instruction. As indicated earlier,
An additional benefit of this collaborative inquiry was in the preparation of mathematics
Discussion
A change in the teachers and most of the students in their relation to mathematics and |
Contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and of Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner's inequality, and many, many more. The inequalities are used to obtain properties of function spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. This textbook complements and fills... |
REQUIREMENT: All students who enroll in this course must have taken the Making Math Real: Overview course within the last 3 years. This is the one and only Making Math Real course for learning and applying the 9 Lines symbol imaging mental organizer for developing automaticity with the multiplication facts. The 9 Lines is the most crucial Making Math Real unifying system for connecting elementary content to higher-level mathematics through algebra and calculus. This 2-day course provides the most comprehensive instructional coverage for acquiring the specific language, sequencing, assessment points, applied practice, and daily maintenance for correctly and successfully applying the 9 Lines.
The 9 Lines mental organizer is a key component within the structure of mathematics that spans elementary through algebraic development. In addition to symbol imaging development, The 9 Lines strategy goes far beyond the multiplication facts. It is a crucial mental organizer designed to image and connect lowest terms, equivalent fractions and greatest common factor.
Topics include learning the precise structure and language for imaging the 9 Lines, daily maintenance and re-imaging, special cases for imaging, multiple ways to interact with the 9 Lines, condensed versions for re-imaging, whole number factoring strategies and applications, lowest terms, equivalent fractions and greatest common factor, and developing synthesis through high-interest games the 2-day 9 Lines Intensive course: $349 for tuition and reader, paid to Making Math Real Institute; Additional fee for optional UC Extension academic units, paid to UC Regents on the first day of class. |
9th Grade Math
Thu, 27 Oct 2011 14:06:02 +0000en-UShourly1 Into 9th Grade Math
27 Oct 2011 13:47:38 +0000admin school math is the next step in a child's life where they are taken out of the basic principles of mathematics and ushered into a world of greater depth, learning skills that they will carry with them into adulthood and hopefully exercise their critical thinking skills in a way that will help them in their career field of choice. Starting with the first year of high school, 9th grade math builds upon the basics and also introduces new components of practical math application, such as consumer math and finances.
9th Grade Math
An example of a typical 9th grade math course will include the following :
Consumer Mathematics
Applied Business Math
Pre-Algebra
Algebra 1
Geometry
Some students may excel beyond the required courses in their first year and go on to study Algebra 2, Advanced Algebra, and perhaps even venture into calculus. Every student is different and now is the time for them to truly understand and put into practice all of the equations they were taught up to transitioning from elementary education, to high school.
An important element of approaching 9th grade math, is to consider the student and what interests they have, and the goals they have for themselves in the future. If they have plans to attend a college, and what degree they wish to achieve can affect the course material they pursue while working toward their high school diploma. When looking at 9th grade math the entire four years of high school should also come into focus and contribute to what and how the student studies.
Along with learning new math expressions students will begin to learn how to use a variety of problem solving strategies. They will learn how to determine what information is needed while problem solving, and how to use multiple representation to explain situations ( numerically, algebraically, graphically, etc.) They will begin to understand how to reason, prove their reasoning mathematically, and then communicate their reasoning to others.
It is important for teachers to take the time to connect with their students on this subject. Young people who are struggling in grasping basic concepts, or did not fully learn foundational principles in elementary and middle school will have a much more difficult time transitioning into higher math. Algebra in particular can become a source of frustration without basic understanding. Parents should also be aware of where their child is at, and what they should be working on. Having teachers and parents who are willing to be involved can help a student feel more comfortable with asking questions and less inhibited in the math world.
9th grade truly is a transition from child to young adult. Diving into the world of higher math is one step closer to equipping the next generation with the tools they need to succeed. With proper support from teachers, parents, and a school that facilitates an atmosphere of learning through asking questions and hands on application, 9th grade math is one more brick in the foundation of a child's education that they can build upon for the rest of their lives.
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]]> Mathematics Tests in the Classroom
27 Oct 2011 13:44:07 +0000admin mathematics tests are the new frontier in education. Although schools have been utilizing online resources for years, there are more options in online mathematics and mathematics testing then ever before. Like anything, there are pros and cons to using online resources for any subject. Although it seams that the benefits of using it in the classroom outweigh the negatives as the percentage of schools implementing new programs utilizing it are on the rise.
Traditional textbooks in schools are usually passed down from class to class and suffer from huge amounts of wear. In some cases schools can't afford to replace textbooks every year or even every 5 years. Once textbooks are purchased it is difficult to replace them with up to date curriculum. As society and culture become more progressive the way mathematics is taught needs to follow the progression. Online learning can accommodate that by providing up to date resources to schools at a fraction of the cost of purchasing new materials. Using online mathematics tests makes it easier to include experts from around the world, and students from other schools. Opening up a students world to include information from teachers and students outside of their school is one more opportunity for them to grow, learning things in ways they may not otherwise have had access too. In addition the amount of new material is almost limitless when the academic world is connected online.
Specifically online mathematics tests can be a benefit for students who need a little extra help. In a large classroom setting it is true that there is only so much time that can be spent one on one with the students. With math testing online the students can have access to online tutoring . They won't get a specific answer to the problem they are working on, but will have the ability to go over similar problems showing them the formula they may have forgotten, or not quite grasped during their regular classroom time. Outside of the tutoring program it is also beneficial for the accountability it creates in not allowing the student to correspond outside of the program being used . With most programs their work is recorded along with the time it took them to complete it.
Generations of students are now growing up with computers and the internet as large parts of their daily lives. This familiarity with technology can help students feel more comfortable in their approach to taking tests. Sitting down to a computer feels natural to them, and when using online mathematics tests the task can feel less dauntingly formal.
On the more traditional side, are hard copies of textbooks, paper and sharpened pencils along with real faces and voices. Even with all of the benefits of online testing they should not replace the hands on experience of traditional testing. Part of moving forward in academia is not leaving behind traditional methods and ideas, but rather fusing the two together to create something new where each of the ideas benefit from the other. This is real progressive thinking in education.
]]> Help Fractions – Where to Begin Helping with Fractions
27 Oct 2011 13:43:03 +0000admin should be a key building block in a students mathematical foundation. It is an area of math that they will use in their daily lives as adults. It can sometimes be overlooked as a principle concept to understand, but if a child is struggling with fractions it can disrupt their continuing studies is math. When teaching fractions to 9th grade students it shouldn't be assumed that everyone in the class has retained all of the information that they learned during their elementary education. It is advisable anytime a new subject is going to be learned or expanded upon that a brief overview of the basics is gone over in order to refresh their memories. With proper review of fractions many struggles and frustrations can be avoided.
Fractions can be difficult because the student has to change the way he or she looks at numbers. With most other equations they are dealing with the numbers being whole numbers, rather then the whole being "fractioned" into different parts. An illustration of this is if you had two children each holding slices from an apple. Child one is holding 3 slices and child two is holding 5 slices. Automatically the student will view the apple in it's slices and think only of child 2 having more apples then child one, rather then seeing the apple slices as making a whole. Being able to help a child grasp onto the concept of each whole being made up of several parts and then solving mathematical equations using those parts will aid them as they begin to study fractions at a higher level.
All people have differing learning styles but it is fairly universal that using visual aids can help when learning about new things. Presenting fractions as story problems, while also having the problem written out and even having physical objects present that can represent the problem are all ways to help them understand fractions. Another key component is vocabulary. Make sure that the student remembers all of the proper terms to the parts of the fraction. Remembering terms like numerator and denominator and which numbers they represent go long way in understanding the different steps in working with fractions.
In the 9th grade students will begin to learn how to use fractions in algebraic equations. Many students find working with algebra to be intimidating, but with good illustration and instruction the concept can be grasped even by a struggling student. Helping with fractions can be as simple as going over some steps that the student may be missing, or may have forgotten. Some students can become frustrated when they forget a step in solving the equation, such as checking to see if they have properly simplified the problem. Simplifying fractions is a key step in solving them, especially when it comes to algebraic equations.
In conclusion the outline for beginning to help with fractions is :
Review
Visualize aid
Go over vocabulary
Reinforce steps in the equation
And perhaps the most important part of helping with fractions is exercising patience, using encouragement and not allowing the student to give up. A discouraged student will most likely spend the rest of their lives believing they can't do it. Good tutoring in math can instill a deep sense of academic satisfaction giving them the confidence to continue learning.
]]> Importance of Math Practice For Kids
27 Oct 2011 13:41:40 +0000admin makes perfect" is an age old idea that is as true today as the first time it was stated. As in all other areas of life practicing in math is quintessential in not only perfecting their skills, but also in retaining the information that they have learned. The problem is coming up with new ways to keep a child interested and motivating them to practice.
In this article two different methods for math practice with kids using the internet will be outlined. With the many online resources for studying math, it's an excellent way to make practice time for your kids a reality. Not all parents have the opportunity to sit down with their children one on one while they go over their math lessons. These programs make it not only easy but fun for kids to fine tune their math skills.
An excellent program for math practice for kids can be found at . There are many programs that are similar that are web based, this one offers many benefits. Allowing a student who is older to work with programs like this one can be a simple way to practice. The program is free and consists of online tests with immediate scoring and assessments as well as access to an online tutor. The site also features a " Math to Graduate" program, which helps students study all of the different areas of math they will need in order to graduate with higher scores. It can accommodate students at all different levels offering over 4000 help and remedial pages. It also includes a section that focuses on finances and business math, which take the math skills they already posses and puts them into real life application. Programs like this are an excellent tool for students who are old enough to study on their own.
Besides the traditional problem solving format, taking math practice for kids and turning it into a challenging game is another wonderful way to get kids excited about using their math skills. Math games are not just for elementary aged kids. There are many programs out their that offer interesting and challenging math games for high school ages. A great example is found at and are kid safe websites that offer math games for children k-12 covering subjects from basic counting all the way up to trigonometry and games using integers. Using math in this type of setting is a great way for a student who may be getting bored with their current form of study and get them excited about learning new things and practicing the skills they already have. Adding a competitive edge can be a driving force to get better and better.
Without math practice for kids the information that children are learning during their school years won't survive into adulthood. Much if it will be lost along the way as they go through their life. It's important to reinforce the things they are learning with creativity. Instilling a commitment to practicing in children is a lifelong lesson they will learn in addition to their math lessons which is just as important in their successful development.
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]]> Answers For Pre-Algebra
27 Oct 2011 13:40:29 +0000admin the 9th grade there are many different items that need to be considered for math answers for pre-algebra students. Here the complexities will increase as the student continues to advance in learning. These topics can range from linear, quadratic, polynomial, trigonometric, exponential, logarithmic, and rational functions, each necessary for the student to understand.
The functions, processes and equations will begin to require more effort on the part of the student. Teachers and facilitators will need to be more prepared and able to help the student more effectively understand how they arrive at math answers for pre- algebra scenarios. Making sure that they understand what goes into the solution ensures the student and educator that they understand what the point of the process actually is! Here the student needs to have a good grasp and be able to pose and prove a variety of theorems that they have been focusing on for a while. These answers need to be designed in a way that not only arrives at the correct answer but is also designed to allow the student to be able to use the data in a meaningful way and understand the method behind the proverbial "madness".
Students will also begin deriving answers that use matrices for representing data. as well as follow equations that require a mastery of problems using a variety of polynomials (with first degree) using the four operations and the first degree. These are more advanced than the previous information that has been required and understandably requires more time.
Students will also need to make sure that they can use inverse operations to solve variables in a formula. As the students will embark into deeper topics in future classes and throughout college they will benefit greatly from the way the questions and explanatory answers are set up. Many educators like the ability to review the topics in the same order with the answers that are provided in the text.
Math answers for pre algebra often are provided in the text book itself. This allows the student to review selected answers and see if they are completing and navigating each area correctly. If they are not arriving at the correct conclusions during study they are better able to determine what areas they may need improvement on. Many text books will provide answers and ensure that they are not duplicated in test areas so that the student is not able to simply memorize the answers but needs to have a clear understanding of the solution and how they arrived at that conclusion.
Math answers for pre-algebra takes more time for students to complete, requiring the designer to take into consideration that the textbooks will need to be set up to allow for the student to complete the assignments in a reasonable time frame. Students in high school have a lot of requirements and often have a relatively small amount of tolerance for redundancy in educational material.
Math can be fun if the the right work goes into not only the curriculum but also the math answers for pre-algebra as well and can be a tremendous part of a well designed textbook. |
A valuable text for introductory course work in computer science for mathematicians, scientists and engineers. This book demonstrates that Mathematica is a powerful tool in the study of algorithms, allowing the behavior of each algorithm to be studied separately. Examples from mathematics, all types of science, and engineering are included, as well as computer science topics. This book is also useful for Mathematica users at all levels. |
Synopsis
Need some serious help solving equations? Totally frustrated by polynomials, parabolas and that dreaded little x? THE MATH DUDE IS HERE TO HELP!Jason Marshall, popular podcast host known to his fans as The Math Dude, understands that algebra can cause agony. But he's determined to show you that you can solve those confusing, scream-inducing math problems--and it won't be as hard as you think! Jason kicks things off with a basic-training boot camp to help you review the essential math you'll need to truly "get" algebra. The basics covered, you'll be ready to tackle the concepts that make up the core of algebra. You'll get step-by-step instructions and tutorials to help you finally understand the problems that stump you the most, including loads of tips on:
Working with fractions, decimals, exponents, radicals, functions, polynomials and more
Solving all kinds of equations, from basic linear problems to the quadratic formula and beyond
Using graphs and understanding why they make solving complex algebra problems easier
Learning algebra doesn't have to be a form of torture, and with The Math Dude's Quick and Dirty Guide to Algebra, it won't be. Packed with tons of fun features including "secret agent math-libs," and "math brain games," and full of quick and dirty tips that get right to the point, this book will have even the biggest math-o-phobes basking in a-ha moments and truly understanding algebra in a way that will stick for years (and tests) to come. Whether you're a student who needs help passing algebra class, a parent who wants to help their child meet that goal, or somebody who wants to brush up on their algebra skills for a new job or maybe even just for fun, look no further. Sit back, relax, and let this guide take you on a trip through the world of algebra |
Courses in
Mathematics
and Statistics
Honours
timetable
MT2001 MATHEMATICS
This is a core second level mathematics module which must be taken by anyone planning to pursue Honours level material in Mathematics, Statistics or Physics.
Aims
To extend the knowledge and skills gained by students in the module MT1002 in preparation for Honours study.
In particular
- to enhance their skills and understanding of calculus to include functions of several variables;
- to develop an appreciation of basic ideas in Linear Algebra.
Objectives
By the end of the course a student should be able to demonstrate:
- how to determine the Taylor series of a function and estimate the error associated with a Taylor polynomial;
- an understanding of the concepts of continuity and differentiability of functions of several variables; partial differentiation and an appreciation of the difference between partial and ordinary differentiation;
- how to determine the local behaviour of surfaces at stationary points;
- how to determine integrals over areas and volumes (and when necessary use different co-ordinate systems including cylindrical and spherical polars);
- an ability to work with the fundamental concepts of vector space theory;
- an ability to manipulate matrices in order to find inverses and determinants;
- how to determine the eigenvalues and eigenvectors of a square matrix (and diagonalisation when appropriate);
- how to obtain a Fourier series representation of simple function and interpret its properties. |
Everyday Math Demystified: A Self-teaching Guide
Includes what you need to know about mathematics, including arithmetic, ratios, and proportions, working with money, the International System of Units, perimeter and area, graphs, stock returns, square roots, rates of change, and much more.
Description of this Book
Say goodbye to dry presentations, gruelling formulas, and abstract theories that would put Einstein to sleep, now there's an easier way to master the disciplines you really need to know. Everyday Math Demystified has everything you need to know about essential mathematics, including arithmetic, ratios, and proportions, working with money, the International System of Units, perimeter and area, graphs, stock returns, square roots, rates of change, and much more.
Awards & Reviews
Author's Bio professional handbooks. He is the author of the TAB Encyclopedia of Electronics for Technicians and Hobbyists, Teach Yourself Electricity and Electronics, and The Illustrated Dictionary of Electronics. Booklist named his McGraw-Hill Encyclopedia of Personal Computing a Best Reference of 1996. |
books.google.com - This... to Solve Word Problems in Algebra
How to Solve Word Problems in Algebra:
This and college students take algebra annually, and almost all are stumped by word problems-- Features Solution methods that are easy to learn and remember, plus a self-test-- Stands alone or can be used with any standard Algebra I or II textbook
From inside the book
Review: How to Solve Word Problems in Algebra: A Solved Problem Approach
User Review - Ron Moreland - Goodreads
This is an excellent resource for any student in any math class. Typically there is a great deal of stress when student's are facing any type of word problem. This book gives detailed step-by-step solutions to the type of problems students will encounter in an Algebra class!Read full review |
Product Description
Singapore Math's Primary Math, U.S. Edition series features the Concrete> Pictorial> Abstract approach. Students begin by learning through concrete and pictorial means before moving into abstract thought and development, which encourages an active thinking process, communication of mathematical ideas and problem solving. Lessons are designed for a mix of teacher instruction and independent work, and students are encouraged to discuss ideas and explore additional problem-solving methods.
This set of textbooks and workbooks is designed specifically for U.S. students. Names, terms in examples, measurement, spellings, currency and other such elements have been changed to reflect American names and stylistic preferences. Review included. 104 pgs, non-consumable and non-reproducible. Paperback.
Product Reviews
Singapore Math: Primary Math Textbook 3A US Edition
4.4
5
5
5
Higher grade levels lack Instruction
I grew up in Japan and Korea as a kid and love their education systems. As a homeschool parent, I am always looking for affordable curriculum. We are in middle school math now and have had to switch from Singapore Math not because of aging out, but because the books lack the instructions for how to do the harder math it assigns. We use the Textbooks, Workbooks, and Tests books as supplements now, and are using Lifepac math now, but we are still looking for any other math curriculum, our goal for University is Engineering so we want to be prepared.
May 2, 2013
Wonderful Product for Math-Oriented Student
We have used several other math programs before coming to Singapore. This program is just what my daughter needs. It's fast-moving, with little review. It has challenging word problems that keep her thinking about number and how to manipulate them to get the answer.
She found most of the other programs dreadfully boring, but is excited every day when we pull out her math book.
March 16, 2011
Excellent resource. We would give all the Singapore Math books 5 stars! The books are visually clear and easy to understand. I recommend taking the placement test to make accurate book selection. We did and ended up starting in a lower book that I thought we would, but quickly gained speed. Now our child is faster than a calculator and loves math!
June 25, 2010
I've been homeschooling my son for three years (this is our fourth) and for the first time, my son loves doing his math work! Singapore gets 5 stars from me, and my son!
November 10, 2009 |
is a comprehensive course that provides an in-depth exploration of geometric concepts. Through a "Discovery-Confirmation-Practice" based exploration of geometric concepts, students are challenged to work toward a mastery of computational skills, to deepen their conceptual understanding of key ideas and solution strategies, and to extend their knowledge in a variety of problem-solving applications. Course topics include reasoning, proof, and the creation of a sound mathematical argumentWithin each Geometry lesson, students are supplied with a post-study "Checkup" activity, providing them the opportunity to hone their computational skills in a low-stakes, 10-question problem set before moving on to a formal assessment. Additionally, many Geometry lessons include interactive-tool-based exercises and/or math explorations to further connect lesson concepts to a variety of real-world contexts.
To further assist students for whom language presents a barrier to learning, this course includes audio resources in both Spanish and English.
The content is based on the National Council of Teachers of Mathematics (NCTM |
The mathematics program at Fresno Pacific University is designed to prepare students to solve problems, communicate their understanding and appreciate the beauty of mathematics and its role in human history. |
Mathematical Modeling and Simulation
This concise and clear introduction to the topic requires only basic knowledge of calculus and linear algebra - all other concepts and ideas are developed in the course of the book. Lucidly written so as to appeal to undergraduates and practitioners alike, it enables readers to set up simple mathematical models on their own and to interpret their results and those of others critically. To achieve this, many examples have been chosen from various fields, such as biology, ecology, economics, medicine, agricultural, chemical, electrical, mechanical and process engineering, which are subsequently discussed in detail.
Based on the author`s modeling and simulation experience in science and engineering and as a consultant, the book answers such basic questions as: What is a mathematical model? What types of models do exist? Which model is appropriate for a particular problem? What are simulation, parameter estimation, and validation?
The book relies exclusively upon open-source software which is available to everybody free of charge. The entire book software - including 3D CFD and structural mechanics simulation software - can be used based on a free CAELinux-Live-DVD that is available in the Internet (works on most machines and operating systems).
Customer Reviews:
Nice book, but very expensive
By cincosauces - July 7, 2010
This is a good an interesting book. I particularly like the author's focus on the basic concepts and philosophy of modelling as well as the use of open source software as Maxima. The book is clearly intended for beginners, like students at early university courses. I have only two negative comments on this book. One is that there is no chapter or section on dimensional analysis and scaling. The second, and more important is the high price, which is a hinder to the supposed intended audience.
A one-of-a-kind introduction to the theory and application of modeling and simulation techniques in the realm of international studies. Modeling and Simulation for Analyzing Global Events provides an ...
This book provides an introduction to the theory and applications of modeling and simulation with a multidisciplinary perspective, and the authors offer a concise look at the key concepts that make ...
This book constitutes the refereed post-proceedings of the third Asian Simulation Conference, AsiaSim 2004, held in Jeju Island, Korea in October 2004. The 78 revised full papers presented together ... |
ProblemsMagic Problems Creator for Mathematics has a powerful Wizard to help you Create your own custom problems collections. Magic Problems Creator for Mathematics has a powerful Wizard to help you Create your own custom problems collections.ALGEBRA: - Matrices, determinants - Matricial equations - Study of system of equations- Easy and fast to use.- User's manual in...
Just like children, some computer programs dont like to cleanup after themselves. Just like children, some computer programs dont like to cleanup after themselves. They leave their toys laying around for Windows to come along and trip over, often crashing your computer. PC Repair can find and eliminate these problems before...
This is a program that take the fear and mystery out of solving algebraic word problems. There are 70 problems in the program, and each method is fully explained step by step, so you never are lost and wondering "how did he do that"! The program contains a solution guide with commentary, as needed, to tutor you to success. The solution guide is easily printed out for reference and for learning.
An easy to use program to aide students in learning to identify key words that identify mathematical operations and work through mathematical word problems. Word problems are considered important because they take math into the real world. Unfortunately solving word problems is one of the areas where students have the most difficulties in elementary math. Master Math Word Problems helps students develop word problem skills through practice. It...
Calculate geometry problems with this tool. Calculate geometry problems with this tool. Geometry Solver 3D will solve analytic geometry problems easily. It will provide tools for calculations in 3D as well as graphic OpenGL demonstrations. Targeted platforms are Linux and Windows both x86...
This program contains everything you ever wanted to know about vectors. This program contains everything you ever wanted to know about vectors. It graphically demonstrates how vectors are added.It can be used to provide students with vector problems which they can solve graphically or mathmatically. It contains...
Tools for Solving Mathematics problems. Tools for Solving Mathematics problems. Mathematics Tools is a tools that help people in solving Mathematical problems such as: Solving quadratic equation and cubic equationSolving System of equations (2 or 3 unknowns)Working in the Base Number...5Rating Tmj Problems teaches you How to quickly find out if you have an airway block in your nose, and 2 exercises to quickly start curing it. 5Rating Tmj Problems teaches you How to quickly find out if you have an airway block in your nose, and 2 exercises to quickly start curing it. (If your sinuses are blocked, this is extremely important). Never again will you... Suffer from horrible... |
An Introduction to the History of Algebra: Solving Equations from Mesopotamian Times to the Renaissance (Mathematical World)
Book Description: This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever applicable, to link these developments to the extension of the number system. Various examples of problems, with their typical solution methods, are analyzed, and sometimes translated completely. Indeed, it is another aim of this book to ease the reader's access to modern editions of old mathematical texts, or even to the original texts; to this end, some of the problems discussed in the text have been reproduced in the appendices in their original language (Greek, Latin, Arabic, Hebrew, French, German, Provençal, and Italian) with explicative notes |
ImproveyourThinking SkillsinMaths is written based on the latest Mathematics syllabus issued by the Ministry of Education. The questions in this book are designed to stimulate creative thinking skills solving problem sums.
ImproveyourThinkingSkillsinMaths is written based on the latest Mathematics syllabus issued by the Ministry of Education. The questions in this book are designed to stimulate creative thinking skills in solving problem sums.
ImproveyourThinking SkillsinMaths is written based on the latest Mathematics syllabus issued by the Ministry of Education. The questions in this book are designed to stimulate creative thinking skills problem sums.
Topical Maths Normal (Academic) is a series of books written for students to build a strong foundation in Maths. Each book provides a comprehensive coverage of the Secondary Maths syllabus, and serves as a good revision for students before their examinations.
Topical Maths Normal (Academic) is a series of book written for students to build a strong foundation in Maths. Each book provides a comprehensive coverage of the Secondary Maths syllabus, and serves as a good revision for students before their examinations. |
Intended for students who plan to continue in the calculus sequence, this course involves the study of basic functions: polynomial, rational, exponential, logarithmic, and trigonometric. Topics include a review of the real number system, equations and inequalities, graphing techniques, and applications of functions. Includes problem-solving laboratory sessions. Permission of instructor is required. This course does not count toward the major or minor in mathematics. (Offered annually)
A study of selected topics dealing with the nature of
mathematics, this course has an emphasis on its origins and a focus on mathematics as a creative
endeavor. This course does not normally count toward the major or minor in mathematics. (Offered each semester)
This course offers a standard introduction to the concepts and techniques of the differential calculus of functions of one variable. A problem-solving lab is included as an integral part of the course. This course does not count towards the major in mathematics. (Offered each semester)
This course is a continuation of the topics covered in MATH 130 with an emphasis on integral calculus, sequences, and series. A problem-solving lab is an integral part of the course. Prerequisite: MATH 130 or permission of the instructor. (Offered each semester)
This course emphasizes the process of mathematical reasoning, discovery, and argument. It aims to acquaint students with the nature of mathematics as a creative endeavor, demonstrates the methods and structure of mathematical proof, and focuses on the development of problem-solving skills. Specific topics covered vary from year to year. MATH 135 is required for the major and minor in mathematics. Prerequisite: MATH 131 or permission of the instructor. (Offered each semester)
This course is an introduction to the concepts and methods of linear algebra. Among the most important topics are general vector spaces and their subspaces, linear independence, spanning and basis sets, solution space for systems of linear equations, linear transformations and their matrix representations, and inner products. It is designed to develop an appreciation for the process of mathematical abstraction and the creation of a mathematical theory. Prerequisite: MATH 131, and MATH 135 strongly suggested, or permission of the instructor. Required for the major in mathematics. (Offered
annually)
A continuation of linear algebra with an emphasis on applications. Among the important topics are eigenvalues and eigenvectors, diagonalization, and linear programming theory. The course explores how the concepts of linear algebra are applied in various areas, such as, graph theory, game theory, differential equations, Markov chains, and least squares approximation. Prerequisite: MATH 204. (Offered every third year)
This course offers an introduction to the theory, solution techniques, and applications of ordinary differential equations. Models illustrating applications in the physical and social sciences are investigated. The mathematical theory of linear differential equations is explored in depth. Prerequisites: MATH 232 and MATH 204 or permission of the instructor. (Offered annually)
This course couples reason and imagination to consider a number of theoretic problems, some solved and some unsolved. Topics include divisibility, primes, congruences, number theoretic functions, primitive roots, quadratic residues, and quadratic reciprocity, with additional topics selected from perfect numbers, Fermat's Theorem, sums of squares, and Fibonacci numbers. Prerequisites: MATH 131 and MATH 204 or permission of the instructor. (Offered every third year)
A graph is an ordered pair (V, E) where V is a set of elements called
vertices and E is a set of unordered pairs of elements of V called
edges. This simple definition can be used to model many ideas and
applications. While many of the earliest records of graph theory
relate to the studies of strategies of games such as chess,
mathematicians realized that graph theory is powerful well beyond the
realm of recreational activity. In class, we will begin by exploring
the basic structures of graphs including connectivity, subgraphs,
isomorphisms and trees. Then we will investigate some of the major
results in areas of graph theory such as traversability, coloring and
planarity. Course projects may also research other areas such as
independence and domination. (Offered occasionally)
This course offers a careful treatment of the definitions and major theorems regarding limits, continuity, differentiability, integrability, sequences, and series for functions of a single variable. Prerequisites: MATH 135 and MATH 204. (Offered annually)
This course begins with a generalization of the notions of
limit, continuity, and differentiability (developed in MATH 331), and extends them to the
two-dimensional setting. Next, the Fundamental Theorem of Calculus is extended to line integrals
and then to Green's Theorem. The course culminates with a brief introduction to analysis in the
complex plane. Prerequisites: MATH 232 and MATH 331. (Offered occasionally)
This is an introductory course in probability with an emphasis on the development of the student's ability to solve problems and build models. Topics include discrete and continuous probability, random variables, density functions, distributions, the Law of Large Numbers, and the Central Limit Theorem. Prerequisite: MATH 232 or permission of instructor. (Offered alternate years)
This is a course in the basic mathematical theory of statistics.
It includes the theory of estimation, hypothesis testing, and linear models, and, if time permits, a
brief introduction to one or more further topics in statistics (e.g., nonparametric statistics,
decision theory, experimental design). In conjunction with an investigation of the mathematical
theory, attention is paid to the intuitive understanding of the use and limitations of statistical
procedures in applied problems. Students are encouraged to investigate a topic of their own
choosing in statistics. Prerequisite: MATH 350. (Offered alternate years)
Typical reading: Larsen and Marx, Mathematical Statistics and Its Applications
Drawing on linear algebra and differential equations, this course investigates a variety of mathematical models from the biological and social sciences. In the course of studying these models, such mathematical topics as difference equations, eigenvalues, dynamic systems, and stability are developed. This course emphasizes the involvement of students through the construction and investigation of models on their own. Prerequisites: MATH 204 and MATH 237 or permission of the instructor. (Offered every third year)
An introduction to the axiomatic method as illustrated by
neutral, Euclidean, and non-Euclidean geometries. Careful attention is given to proofs and
definitions. The historical aspects of the rise of non-Euclidean geometry are explored. This course
is highly recommended for students interested in secondary-school teaching. Prerequisite: MATH
331 or MATH 375. (Offered every third year)
Typical reading: Greenberg, Euclidean and Non-Euclidean Geometries: History and
Development
Each time this course is offered, it covers a topic in mathematics that is not usually offered as a regular course. This course may be repeated for grade or credit. Recent topics include combinatorics, graph theory, and wavelets. Prerequisite: MATH 135 and MATH 204 or permission of instructor.
(Offered alternate years)
This course studies abstract algebraic systems such as groups, examples of which are abundant throughout mathematics. It attempts to understand the process of mathematical abstraction, the formulation of algebraic axiom systems, and the development of an abstract theory from these axiom systems. An important objective of the course is mastery of the reasoning characteristic of abstract mathematics. Prerequisites: MATH 135 and MATH 204 or permission of the instructor. (Offered annually)
First order logic is developed as a basis for understanding the nature
of mathematical proofs and constructions and to gain skills in dealing with formal languages.
Topics covered include propositional and sentential logic, logical proofs, and models of theories.
Examples are drawn mainly from mathematics, but the ability to deal with abstract concepts and
their formalizations is beneficial. Prerequisite: MATH 204, PHIL 240, or permission of
instructor. (Offered every third year)
This course covers the fundamentals of point set topology, starting from axioms
that define a topological space. Topics typically include: topological equivalence, continuity,
connectedness, compactness, metric spaces, product spaces, and separation axioms. Some topics
from algebraic topology, such as the fundamental group, might also be introduced. Prerequisite
MATH 331 or permission of the instructor. (Offered occasionally)
This course presents a careful study of various concepts of analysis. Such
topics as convergence and continuity are briefly examined, first on the real line and then in more
general metric spaces. Other topological properties of metric spaces are studied. An examination of
different types of integrals concludes the course. Prerequisite: MATH 331 or permission of
instructor. (Offered occasionally)
An introduction to the theory of functions of a
complex variable. Topics include the geometry of the complex plane, analytic functions, series
expansions, complex integration, and residue theory. When time allows, harmonic fuctions and
boundary-value problems are discussed. Prerequisite: MATH 331 or permission of instructor.
(Offered every third year) |
Hi, my high school classes have just begun and I am shocked at the amount of algebra age problem homework we get. My basics are still not clear and a big assignment is due within few days. I am really worried and can't think of anything. Can someone guide me?
Can you please be more descriptive as to what sort of assistance you are expecting to get. Do you want to learn the fundamentals and solve your assignments on your own or do you want a software that would provide you a line by line answer for your math problems?
Welcome aboard dear. This subject is very interesting, but you need to know your basics and techniques first. Algebrator has guided me a lot in my course. Do give it a try and it will work for you too.
Algebrator is the program that I have used through several math classes - Algebra 2, Algebra 2 and Pre Algebra. It is a truly a great piece of math software. I remember of going through problems with quadratic formula, trigonometry and evaluating formulas. I would simply type in a problem from the workbook, click on Solve – and step by step solution to my math homework. I highly recommend the program. |
M150 Math Grade 10 - Geometry
$40.00
This is a study of form,
position, and magnitude. As the student progresses in this study, he or
she will come to a better understanding of the connective thread that
ties all parts of this subject together. A knowledge of the vocabulary
terms are explained so that the student can have an understanding of
them in a geometric sense. |
This carefully crafted learning resource helps students develop their problem-solving skills while reinforcing their understanding with detailed explanations, worked-out examples, and practice problems. Students will also find listings of key ideas to master. Each section of the main text has a corresponding section in the Study Guide. [via]
This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, so that when students finish the course, they have a solid foundation in the principles of mathematical thinking. This comprehensive, evenly paced book provides complete coverage of the function concept and integrates substantial graphing calculator materials that help students develop insight into mathematical ideas. The authors' attention to detail and clarity, as in James Stewart's market-leading Calculus text, is what makes this text the market leader.
This book is a response to those instructors who feel that calculus textbooks are too big. In writing the book james stewart asked himself: what is essential for a three-semester calculus course for scientists and engineers? stewart's essential calculus offers a concise approach to teaching calculus that focuses on major concepts and supports those concepts with precise definitions, patient explanations, and carefully graded problems. Essential calculus is only 850 pages-two-thirds the size of stewart's other calculus texts (calculus, fifth edition and calculus, early transcendentals, fifth edition)-and yet it contains almost all of the same topics. The author achieved this relative brevity mainly by condensing the exposition and by putting some of the features on the website, Despite the reduced size of the book, there is still a modern flavor: conceptual understanding and technology are not neglected, though they are not as prominent as in stewart's other books. Essential calculus has been written with the same attention to detail, eye for innovation, and meticulous accuracy that have made stewart's textbooks the best-selling calculus texts in the world [via]
This book is a response to those instructors who feel that calculus textbooks are too big. In writing the book james stewart asked himself:what is essential for a three-semester calculus course for scientists and engineers? stewart's essential calculus: early transcendentals offers a concise approach to teaching calculus, focusing on major concepts and supporting those with precise definitions, patient explanations, and carefully graded problems. Essential calculus: early transcendentals is only 850 pages-two-thirds the size of stewart's other calculus texts (calculus, fifth edition and calculus, early transcendentals, fifth edition)-yet it contains almost all of the same topics. The author achieved this relative brevity mainly by condensing the exposition and by putting some of the features on the website Despite the reduced size of the book, there is still a modern flavor: conceptual understanding and technology are not neglected, though they are not as prominent as in stewart's other books. Essential calculus: early transcendentals has been written with the same attention to detail, eye for innovation, and meticulous accuracy that have made stewart's textbooks the best-selling calculus texts in the world |
Welcome
Welcome to CPM Educational Program,
an educational non-profit organization dedicated to improving grades 6-12 mathematics instruction. CPM offers professional development and curriculum materials. We invite you to learn more about the CPM mathematics program by clicking the "Learn about CPM" link at left. The other sections offer support materials for teachers, parents and students.
Headlines
CPM to offer Integrated I-III
CPM will develop an Integrated series based on the Appendix A pathway. Click here for full details.
Middle School acceleration pathways now available
The timelines, pacing guides and tables of contents for acceleration to take algebra in 8th grade are now available here.
CPM offers regional conferences for Summer 2013
CPM releases Common Core series, Core Connections
Click on "Learn About CPM" at left to see tables of contents and sample chapters.
For an eBook preview, click here.
News
New curriculum and technical support available
Three CPM mentor teachers are now available to help parents, students and teachers who have questions about the CPM program. This support is primarily for questions about using the program, its technology and the website. To use CPM's support service, go to to see the available services. Follow the prompts to get help. Note that this service is not a "homework helpline." That support is at
CPM now offers a new series of textbooks to meet the grade 6-8 and high school CCSS content and practice standards: Core Connections, Courses 1 - 3 and Core Connections Algebra 1 & 2 and Geometry. Learn how this series as well as the original Connections series of CPM textbooks are fully aligned with the CCSS Content and Mathematical Practice Standards. CPM can also provide professional development centered around embedding the eight CCSS Mathematical Practices into your current lessons and current textbook from any publisher. Start moving on the path to the CCSS today!
Parent e-book licenses are now available!
Parents may purchase a one-year e-book license of their student's book for $10 by calling CPM and using a credit card. See order form for a complete list of available one-year licenses. Contact CPM at (209) 745-2055.
Sample Problem
Core Connections Geometry : 3-35.
GEORGE WASHINGTON'S NOSE
On her way to visit Horace Mann University, Casey stopped by Mount Rushmore in South Dakota. The park ranger gave a talk that described the history of the monument and provided some interesting facts. Casey could
not believe that the carving of George Washington's face is 60 feet tall from his chin to the top of his head!
However, when a tourist asked about the length of Washington's nose, the ranger was stumped! Casey came to her rescue by measuring, calculating and getting an answer. How did Casey get an answer?
Your Task: Figure out the length of George Washington's nose on the monument. Work with your team to come up with a strategy. Show all measurements and calculations on your paper with clear labels so anyone could understand your work.
DISCUSSION POINTS
What is this question asking you to find?
How can you use similarity to solve this problem?
Is there something in this room that you can use to compare to the monument?
What parts do you need to compare?
Do you have any math tools that can help you gather information? |
0130114170
9780130114174
Mathematics for Technical and Vocational Students: This self-paced instruction guide serves as mathematics reference to understanding and solving problems in technical and trade vocations. It contains over 4000 exercises and 1300 word problems that illustrate principles in practical situations encountered in the labor market. This book only requires a prior "knowledge" of number awareness in order to master topics such as percentages, ratio and proportion, practical algebra, metric measurements, and geometrical constructions. For anyone interested in a technical or vocational career and looking for a chance to apply useful mathematical principles to practical problems. «Show less
Mathematics for Technical and Vocational Students: This self-paced instruction guide serves as mathematics reference to understanding and solving problems in technical and trade vocations. It contains over 4000 exercises and 1300 word problems that illustrate principles in practical situations... Show more»
Rent Mathematics for Technical and Vocational Students 2nd Edition today, or search our site for other Spangler |
Essential Mathematics Testmaker Plus! VCE CD-ROM
Summary:
Essential Mathematics Testmaker Plus! VCE contains 50 sets of
questions covering the course requirements for Mathematics Methods
Units 1-4, Advanced General Mathematics and Specialist Mathematics.
Each set covers a specific topic and contains 44 questions. The sets
can be used to generate tests, worksheets or assignments on single or
mixed topics. The generated documents can be printed or placed on the
school's network where they can be completed on-screen and
automatically marked. A detailed results editor, which automatically
collates the on-screen versions results, also allows the teacher to
add results from other sources. The results editor includes
differentiated levels of entry for the teacher and coordinator.
Progressive assessment and a powerful electronic book mark for every
teacher facilitates efficient VCE record keeping. The CD-ROM is
compatible with both PCs and Macs and also includes a network
licence |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
Winter Intersession 2006-2007 MAT 092 MARK BOOTHNAME:Solve each problem and block your answer. Solve using the five-step problem-solving process. 1) The sum of twice a number and 18 less than the number is the same as the differencebetween -10 and
Radical Supplement for Math 92Objective Simplify and perform operations on numerical square-roots. Square Roots In prior chapters, you have squared numbers such as 22=4. Now we are going to reverse this operation and that is called finding a numbers
Positivity of IntersectionsApril 2, 2008Abstract Notes of a lecture by Dusa McDuff on positivity of intersections of J-holomorphic curves in dimension 4.There are 3 places where one needs information on local structure of Jholomorphic curves: 1. |
Five Hundred Mathematical Challenges
Edward J. Barbeau, Murray S. Klamkin, William O.J. Moser
The organization of the book makes it a superb pedagogical instrument... Throughout the book are interspersed fables concerning mathematicians and occasional "bons mots." They are wonderful...The book is a paperback, done in a large elegantly printed format. I suggest you try it out on some of your talented undergraduate students. — The Mathematical Intelligencer
The book is an excellent source of problems for high school or college teachers who wish to challenge mathematically oriented students. The problems cover a wide range of topics, including geometry, algebra, number theory, trigonometry, probability and combinatorics...I recommend this book highly for mathematics teachers as a source of nontrivial precalculus problems. — AAAS, Science Books and Films
This book contains 500 problems that range over a wide spectrum of areas of high school mathematics and levels of difficulty. Some are simple mathematical puzzlers while others are serious problems at the Olympiad level. Students of all levels of interest and ability will be entertained and taught by the book. For many problems, more than one solution is supplied so that students can see how different approaches can be taken to a problem and compare the elegance and efficiency of different tools that might be applied.
Teachers at both the college and secondary levels will find the book useful, both for encouraging their students and for their own pleasure. Some of the problems can be used to provide a little spice in the regular curriculum by demonstrating the power of very basic techniques.
This collection provides a solid base for students who wish to enter competitions at the Olympiad level. They can begin with easy problems and progress to more demanding ones. A special mathematical tool chest summarizes the results and techniques needed by competition-level students. |
1.Evaluate monomial and polynomial expressions given a value or values for the variable(s)
2.Multiply and divide monomial expressions with a common base, using the properties of exponents
3.Add, subtract, and multiply monomials and polynomials
4.Divide a polynomial by a monomial
5.Find values of variables for which an algebraic fraction is undefined
6.Multiplying Binomials – FOIL
B.Exponents
1.Rules of Exponents
2.Zero Exponent
3.Negative Exponent
4.Scientific Notation
a. Converting to and from scientific notation
b. Products and quotients using scientific notation
IV.Factoring
A.Factors
1.GCF
2.Trinomials with a leading coefficient of one (after GCF is factored out)
3.Trinomials with a leading coefficient other than 1
4.Difference of Perfect Squares
B.Algebraic Fractions
1. Finding a value(s) for which an algebraic expression is undefined
2. Simplify fractions with polynomials in the numerator anddenominator
3.Add and subtract fractional expressions with monomial or like binomial denominators
4.Multiply and divide algebraic fractions, expressing the result in simplest form
Note: Item 3 will be taught but whether or not it will be tested will be at the discretion of the individual teacher. This item is not listed in the NYS Integrated Algebra curriculum, however we feel by teaching both types of factoring problems together, our students will have a better understanding of the material.
2. Solve algebraic proportions in one variable which result in a quadratic equation.
3. Understand the relationship between the roots of a quadratic equation and the factors of a quadratic expression.
Note:The introduction of the quadratic formula in this unit of study is optional.
Note:The individual teacher may chose to test multiple times within this unit.Whenever possible, the instructor should stress to students the importance of being able to evaluate all expressions, identify and distinguish between the different types of equations, and know the appropriate/best method for solving the linear or quadratic equation. |
This module introduces science students to the basic principles and good practices in the collection, recording and evaluation of data, and to the use of modern libraries and information resources. It introduces a range of basic mathematical skills that are essential in a wide range of scientific endeavour. The concept of calculus is introduced. |
ATLAST Computer Exercises for Linear Algebra
The ATLAST collection of computer exercises represents the best creative
efforts of the more than 350 faculty members who participated in the thirteen
ATLAST workshops offered between 1992 and 1995. Workshop participants designed
computer exercises and projects suitable for use in undergraduate linear
algebra courses. From the entire ATLAST database of materials, the editors
have selected a comprehensive set of exercises covering all aspects of
the first course in linear algebra. Each chapter is divided into two sections.
The first section consists of shorter exercises and the second section
consists of longer projects.
The computer exercises are all based on MATLAB.
A unique feature of MATLAB is that it is the only major mathematical software
package that is based almost entirely on matrices. A collection of MATLAB
routines (M-files) has been developed to accompany this book. Many of these
routines are designed to give visual illustrations of important linear
algebra concepts such as coordinate systems, linear transformations, and
eigenvalues. Other M-files illustrate visual applications such as using
linear transformations for computer animations or using matrix factorizations
for digital imaging. Still other M-files can be used to generate special
structured matrices. Students are then challenged to discover properties
of the special matrices.
The entire collection of ATLAST M-files can be obtained from the ATLAST
Web page:
These files are required for many of the exercises and projects in the
book. The collection of files will be updated and expanded a few times
a year; so check this web page periodically for the latest versions. Of
the sixty M-files currently in the collection some of the more interesting
files to preview are:
Cogame - a utility for visualizing linear
combinations and coordinate systems in the plane
Eigshow - a utility for visualizing eigenvalues
and eigenvectors of 2x2 matrices
Pyr - a utility for studying rotations in
three space by examining the pitch, yaw, and roll of an airplane
Svdshow - a utility for visualizing the
singular value decomposition of a 2x2 matrix
The ATLAST book has computer lab projects for each of these utilities.
To take full advantage of the ATLAST M-files they should be used in conjunction
with the exercises and projects in the ATLAST book.
The book is available as an inexpensive paperback manual that can be
used in conjunction with any of the standard linear algebra textbooks.
Anticipated list price should be about $12 or $13 depending on the markup
at your bookstore. This price may go up by a few dollars in Fall 1998.
Super Bargain:
Prentice-Hall is offering a 50% discount if the manual is adopted along
with one of the Prentice-Hall linear algebra texts.
Super Super Bargain:
The manual is available for free if adopted for Spring 1998 in conjunction
with
Linear Algebra with Applications,
5th ed., by
Steven J. Leon.
To receive the two book package at the bargain rate order both using
the single
ISBN 0-13-907858-4
The package may be ordered with this ISBN after Spring 1998, however, there
will be a
nominal charge of a few dollars for the ATLAST book added to the package
price.
Unlike other manuals of computer exercises, the ATLAST collection is
a massive collaboration representing a wide variety of views. |
Mathematics and Politics
Registration Fee: $325 by May 5, $450 after
The field of Mathematics and Politics seeks to answer questions
arising in political science from a mathematical perspective. For
example:
Who should have won the 2000 presidential election, and—more
generally—what are the best voting procedures to use when there are
three or more candidates?
What fraction of power does the president have in the US Federal
System, and—more generally— how does one measure political influence in
legislative systems?
How can marital assets be divided fairly, and how is this related
to the resolution of international disputes?
Many questions like these can be answered with little or no
background in either mathematics or political science. The
accessibility of the material, therefore, makes it a perfect backdrop
to introduce mathematical reasoning to undergraduates in the humanities
and social sciences, giving them the opportunity to investigate active
areas of mathematics research and refine their critical thinking skills
in a setting that is largely noncomputational.
This workshop will introduce several areas of mathematics and
politics, including escalation, conflict, voting systems (including
Arrow's Impossibility Theorem and the Gibbard-Satterthwaite
Manipulability Theorem), power, fair division of divisible and
indivisible goods, and apportionment. Participants will develop the
tools needed to introduce such topics to their students, by either
enriching courses with new material or by developing a new course
devoted entirely to the subject in as early as the fall of 2008. There
is also an abundance of material here that mathematics majors will find
interesting and challenging as well. Participants will be provided with
some reading materials and related questions to think about before the
workshop. |
This page is updated frequently. Hit "reload" to make sure you are
viewing the most recent version and not an old cache.
Remember: Learning mathematics is like constructing a building; it
first
requires a strong foundation. We believe that completing all
of the
problems listed below provides a
foundation
for understanding the material in that section. Only the problems
in bold red should be turned
in. We strongly
suggest working other problems in each section to reinforce these
foundations
and prepare you for upcoming material.
Section
Prep Problems
Study Problems
0.1
9/14
3,5,7,11,12,18,29,32,33,37,42,49,60,65,68,75,81,85,89,93
0.2
9/8
0,16,17
9/14
3,7,11,18,22,23,27,29,32,33,37,42,49,51,65,69,70
0.3
9/11
0,6
9/14
1,2,4,12,13,15,18,21,23,35,46,52,58,66,67,72,90
1.1
9/14
0,2
9/18
3,5,8,12,15,17,21,22,23,24,25,27,29,31,33,35,37,41
1.2
9/14
0
9/18
4,5,6,7,9,10,11,13,15,22,25,26,28,29,30,31,33,35,38,39
1.3
9/15
0,2,6
9/18
8,10,11,13,16,17,21,22,25,29,31,32,36,38,39,41,42
1.4
9/19
0,7
9/22
9,12,14,17,21,22,25,26,28,31,34,40,44,45,47,49,51
1.5
9/21
0
9/25
2,6,8,10,11,12,25,26,31,32,36,38,41,42,50,58,62,68,71,72,75,100
1.6
9/25
0,4
9/28
1,7,10,14,15,20,31,33,34,41,44,48,58,62,63,68,70,71,72,75,77,76
1.7
9/26
0,7,8
9/28
6,10,12,14,16,19,21,24,26,29,30,31,34,36,37,42
2.1
9/29
0,4
10/2
13,14,15,23abefgh,24bcdefg,32,34
2.2
10/2
0,6
10/6
3,10,12,13,14,15,16,21,22,40,41,52,53,54
2.4
10/5
0
10/6
16,32,34,36,39,43,47
2.5
10/5
0,13
10/9
14,16,18,20,21,29,32,35,37,39,41,42
2.6
10/6
0,2
10/9
5,10,13,14,20,22,27,36,37,42,44,48,49,52,55,61
2.7
10/6
0
10/10
9,11,17,18,20,22,23,25,27,37,39,46,48,51
3.1
10/12
0
10/16
3.2
10/13
0,3
10/19
6,7,9,11,15,16,18,19,22,28,30,31,33,35,36
3.3
10/16
0
10/19
5,13,14,15,17,18,23
3.4
10/17
0
5,6,7,12,36,37,39,41,42,43,49,51,54,55,56,57,58,67,69
3.5
0
3.6
0
3.7
0,2
3.8
0
4.1
0
4.2
0
4.3
0
4.4
0
4.5
0
5.1
0
5.2
0
5.3
0
5.4
0
6.1
0
6.2
0
6.3
0
6.4
0
7.1
0
7.2
0
All assignments are from Integrated Calculus by Taalman, (Houghton Mifflin, Boston
& New York, 2005). |
Successful application of mathematical principles to solve a range of challenging problems. Clear integration of knowledge, understanding and skills from different areas. Comprehensive responses containing all necessary detail.
B
Broad knowledge and understanding, although some responses lacked detail or contained minor errors.
Broad knowledge and understanding, although some responses lacked detail or contained minor errors.
Successful application of mathematical principles to solve a variety of problems. Some integration of knowledge, understanding and skills from different areas. Some responses lacked necessary detail or contained minor errors. |
Historical Modules For The Teaching And Leaning Of Mathematics - 05 edition
Marketplace Sellers for Historical Modules For The Teaching And Leaning Of Mathematics
Just for you, we curate a growing list of independent booksellers, giving you even MORE choices when shopping for your textbooks.Keep in mind: Marketplace orders do NOT qualify for free shipping. More about the Marketplace
Brand new. We distribute directly for the publisher. These eleven historical modules are collections of lesson materials designed to demonstrate the use of the history of mathematics in the teaching ...show moreof mathematics. They have been written by teams of college and high school teachers and have been field tested in a variety of situations. The materials can fit many different types of objectives and can be used in a variety of mathematics classes, from pre-algebra through calculus. The eleven modules are: Archimedes; Combinatorics; Exponentials and Logarithms; Functions; Geometric Proof; Lengths, Areas, and Volumes; Linear Equations; Negative Numbers; Polynomials; and Statistics and Trigonometry |
Algebra Having trouble doing your Math homework? This program can help you master basic skills like reducing, factorising, simplifying and solving equations. Step by step explanations teach you how to solve problems concerning fractions,binomials, trinomials etc. ... Algebra has three different functionalities: you start learning each type of problem by seeing the program solve it step by step. ... Algebra will compare your answer to its own. ... Each type of problem has three levels to help you to start with easy problems and slowly building up your skills. - 895Kb
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Math Function Mania Math Function Mania is a fun multimedia game that teaches functions, algebra and problem solving skills. Functions are very important in math! By mastering them, you will greatly increase your math skills. This game teaches you by the "hands on" method - you will discover how functions work by... Topics covered include equations, algebra, problem solving, critical thinking, polynomials, factoring, remainders, number bases, and prime numbers. ... In the Combat! ... game, you play head to head against another player. - 877Kb
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EMMentor_Light Interactive multilingual mathematics software for training problem-solving skills offers more than 500 of math problems, a variety of appropriate techniques to solve problems and a unique system of performance analysis with methodical feedback. The software allows students at all skill levels to... Covered subject areas are arithmetic, pre-algebra, algebra, trigonometry and hyperbolic trigonometry. ... Included are basic and advanced math topics, such as solutions of linear, quadratic, biquadratic, reciprocal, cubic and fractional algebraic identities, equations and inequalities, solutions... - 3Mb
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EMSolution Algebra This bilingual program offers 70690 of fully explained step by step solutions of algebra problems together with test authoring tools. Problems of 11 levels of complexity vary from basic to advanced: linear, quadratic, biquadratic, reciprocal, cubic, high degree and complex fractional expressions... Special options help teachers create quizzes, tests, variant tests, exams and homeworks of varied complexity and develop problem solving lesson plans. ... A number of variant tests are ready for students to review and reinforce their skills. - 5Mb
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EMSolutionLight Free problem-solving mathematics software allows you to work through more than 500 math problems with guided solutions, and encourages to learn through in-depth understanding of each solution step and repetition rather than through rote memorization. ... Covered subject areas are arithmetic, pre-algebra, algebra, trigonometry and hyperbolic trigonometry. ... The software supports a number of interface styles. ... This includes basic math and advanced topics, such as solutions of linear, quadratic, biquadratic, reciprocal, cubic and fractional algebraic identities, equations and... - 3Mb
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MathAid College Algebra Interactive,... The demo version contains selected lessons from the full version, fully functional, all features included. ... Topics covered: rectangular coordinate system, functions and graphs, linear and absolute value functions, quadratic functions, polynomial and rational functions, exponential and logarithmic... - 658Kb
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EMMentor Algebra short Interactive multilingual mathematics software for training problem-solving skills offers 70439 algebraic problems, a variety of appropriate techniques to solve problems and a unique system of performance analysis with methodical feedback. Included are linear, quadratic, biquadratic, reciprocal,... Dozens of tests are ready for learners to review and reinforce their math skills. ... The software supports bilingual interface and a number of interface styles. ... Test preparation options facilitate development of printable math tests and automate preparation of 12 test variants around each... - 5Mb
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Mathematics, Physics, Chemistry, Biology, and Astronomy for All
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An Active Learning Approach.
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Data Mining and Optimization for Decision Making
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0764196758
9780764196751. The manual also provides a detailed 10-chapter review covering topics for both exams. The enclosed CD-ROM presents two additional practice tests, one in Calculus AB, and the other in Calculus BC. Tests on the CD-ROM come with solutions explained and automatic scoring of the multiple-choice questions. The authors also offer an overview of the AP Calculus exams, which includes advice to students on making best use of their graphing calculators. «Show less.... Show more»
Rent Barron's AP Calculus with CD-ROM 10th Edition today, or search our site for other Hockett Calculus textbooks. All of our textbooks come with a 21 day Satisfaction Guarantee, 14 days for eTextbooks.
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TABLE OF CONTENTS
Introduction
The Courses
Topics That May Be Tested on the Calculus AB Exam
Topics That May Be Tested on the Calculus BC Exam
The Examinations
The Graphing Calculator: Using Your Graphing Calculator on the AP Exam
Grading the Examinations
The CLEP Calculus Examination
This Review Book
DIAGNOSTIC TESTS
Calculus AB
Calculus BC
TOPICAL REVIEW AND PRACTICE
Functions
Definitions
Special Functions
Polynomial and Other Rational Functions
Trigonometric Functions
Exponential and Logarithmic Functions
Parametrically Defined Functions
Practice Exercises
Limits and Continuity
Definitions and Examples
Asymptotes
Theorems on Limits
Limit of a Quotient of Polynomials
Other Basic Limits
Continuity
Practice Exercises
Differentiation
Definition of Derivative
Formulas
The Chain Rule; the Derivative of a Composite Function
Differentiability and Continuity
Estimating a Derivative
Numerically
Graphically
Derivatives of Parametrically Defined Functions
Implicit Differentiation
Derivative of the Inverse of a Function
The Mean Value Theorem
Indeterminate Forms and L'Hopital's Rule
Recognizing a Given Limit as a Derivative
Practice Exercises
Applications of Differential Calculus
Slope; Critical Points
Tangents and Normals
Increasing and Decreasing Functions
Functions with Continuous Derivatives
Functions Whose Derivatives Have Discontinuities
Maximum, Minimum, and Inflection Points: Definitions
Maximum, Minimum, and Inflection Points: Curve Sketching
Functions That Are Everywhere Differentiable
Functions Whose Derivatives May Not Exist Everywhere
Global Maximum or Minimum
Differentiable Functions
Functions That Are Not Everywhere Differentiable
Further Aids in Sketching
Optimization: Problems Involving Maxima and Minima
Relating a Function and Its Derivatives Graphically
Motion Along a Line
Motion Along a Curve: Velocity and Acceleration Vectors
Tangent-Line Approximations
Related Rates
Slope of a Polar Curve
Practice Exercises
Antidifferentiation
Antiderivatives
Basic Formulas
Integration by Partial Fractions
Integration by Parts
Applications of Antiderivatives; Differential Equations
Practice Exercises
Definite Integrals
Fundamental Theorem of Calculus (FTC); Definition of Definite Integral
Properties of Definite Integrals
Integrals Involving Parametrically Defined Functions
Definition of Definite Integral as the Limit of a Sum: The Fundamental Theorem Again
Approximations of the Definite Integral; Riemann Sums
Using Rectangles
Using Trapezoids
Comparing Approximating Sums
Graphing a Function from Its Derivative; Another Look
Interpreting In x as an Area
Average Value
Practice Exercises
Applications of Integration to Geometry
Area
Area Between Curves
Using Symmetry
Volume
Solids with Known Cross Sections
Solids of Revolution
Arc Length
Improper Integrals
Practice Exercises
Further Applications of Integration
Motion Along a Straight Line
Motion Along a Plane Curve
Other Applications of Riemann Sums
FTC: Definite Integral of a Rate Is Net Change
Practice Exercises
Differential Equations
Basic Definitions
Slope Fields
Euler's Method
Solving First-Order Differential Equations Analytically
Exponential Growth and Decay
Exponential Growth: dy/dt = ky
Restricted Growth: dy/dt = k(A-y)
Logistic Growth: dy/dt = ky(A-y)
Practice Exercises
Sequences and Series
Sequences of Real Numbers
Infinite Series
Definitions
Theorems About Convergence or Divergence of Infinite Series
Tests for Convergence of Infinite Series
Tests for Convergence of Nonnegative Series
Alternating Series and Absolute Convergence
Power Series
Definitions; Convergence
Functions Defined by Power Series
Finding a Power Series for a Function: Taylor and Maclaurin Series
Approximating Functions with Taylor and Maclaurin Polynomials
Taylor's Formula with Remainder; Lagrange Error Bound
Computations with Power Series
Power Series over Complex Numbers
Practice Exercises
Miscellaneous Multiple-Choice Practice Questions
Miscellaneous Free-Response Practice Exercises
AB PRACTICE EXAMINATIONS
AB One
AB Two
AB Three
BC PRACTICE EXAMINATIONS
BC One
BC Two
BC Three
Appendix: Formulas and Theorems for Reference
Index
677 |
Adoption of the "Mathematical Modelling and Problem Solving" practice to support problem solving in key 1st year modules ("Electronic Materials and Devices" and "Fundamentals of Electrical Engineering") will help students engage much better with the theoretical concepts involved, which will lay a strong foundation for more advanced modules in these areas in the 2nd & 3rd years of study. Competence in mathematical modelling and problem solving will also improve the students' ability to cope with other theoretical modules in their degree programmes.
Project Aims
Students commencing the School's degree programmes generally have high mathematics grades (at least B), but offer various different types of qualifications (A levels, BTEC, European Baccalaureate, International Baccalaureate and other international equivalents). Despite the inclusion of a 20 credit mathematics module in our first year curriculum, in which most students perform very well, many students struggle when required to apply mathematical skills to solve electronics problems in our other 1st year modules.
Project Objectives
The learning materials will be adapted for use within the abovementioned 1st year modules, and will
comprise self-study resources which have both a formative and summative assessment function.
The learning materials will be transferred into a web-based format, which allows exercises to be modified automatically, and provides instant feedback to students on their performance and the source of errors. The academic leaders of each module will be responsible for specifying the problem sets, and an experienced postgraduate student will implement the web-based formatting. To reduce development costs, we will use an existing web interface such as Pearson "MyMathLab", which we already successfully use for core engineering mathematics teaching. |
Course Overview
Calculus I with Review Part 2 (MATH 131Q) is a continuation of Calculus I with Review Part 1 (MATH 130): an introduction to differential and integral calculus. Background material in algebra and trigonometry is reviewed as needed. It is designed for students studying science, mathematics, computer science, and those planning on certain types of graduate work. Others are welcome. The two-semester sequence MATH 130 – MATH 131Q is equivalent to Calculus I (MATH 141Q). A course in calculus emphasizes skills, theory, and applications, and it opens doors to higher mathematics, science, and technology. You may not enter MATH 131Q without taking MATH 130 at Stetson.
Texts and Calculator
In addition to the texts and calculator used in MATH 130 you will need the trade book A Tour of the Calculus by David Berlinski.
Grading
Your grade will be based on 3 quizzes, 3 tests, and a final exam, 1000 points total. The grading scale is A: 900 - 1000 points, B: 800 - 899, etc., with +'s and -'s in the top and bottom 20 points.
Policies and Due DatesTests and Quizzes must be taken during the scheduled time unless you have a valid excuse cleared with me ahead of time. Make-ups must take place by class time on Friday of the same week. Grade penalties will be imposed for infringements. Tests and quizzes are on Tuesdays; please see the syllabus. The final exam is Wednesday 12/8, 8-10 am.
Homework should consume about 8 hours per week outside of class. Homework is in three parts: review, current exercises, anticipate — past, present, future. All three are important for successfully mastering the material. Step 1: review recent work, catch up on problems you could not do previously, review your notes to familiarize yourself with examples done in class. Step 2: problems are grouped on the syllabus by type. Within each group, do problems until you have mastered the technique. You need not do all the problems the first night, but should do most problems before the test. Step 3: read the section for the next class. This prepares the ground for planting new ideas, helping you make the most of class time. Homework is not collected: I trust you to keep current and to ask timely questions.
Stewart contains answers to odd-numbered problems in the Appendix. Those marked with a graph symbol usually require graphical or calculator answers. Safier puts all answers with the problems. Questions from Berlinski will appear on most quizzes and tests – see the syllabus for specific assignments and timing. Your knowledge and thoughtfulness about the contents will be noted in discussions (Re-)Read the lecture Class, College, and Life online, and frequently review the Study Tips. For Stewart, use the |
My Advice to a New Math 175 Student:
Go to every single class!!!! Missing one class will put you
behind. Sleeping an extra hour is not worth the hours you will have to
put in to catch up. However, just being in class isn't enough. Pay
attention and follow along. Don't be afraid to ask questions even if you
think you are the only one that doesn't know the answer. You are in the
class for your benefit, not your classmates. Also, make sure you attempt
all of the assignments and do them the day that you talked about that
particular subject. Don't tell yourself you'll do it some other time because
you will only fall behind and all of the assigned sections will start piling
up on you. If you get stuck on a problem, just ask him to go over
it in class the next day. It may take awhile to do the assigned problems
and you probably won't feel like doing them, but push through it anyway because
it pays off in the end. It's a great feeling to encounter a problem on
an exam that you actually know how to do.
In addition to all of the above, there is always extra
help. I was only able to attend one Tuesday night review session but it is
very beneficial if you have any questions on assigned problems that weren't covered
in class. What helped me the most though was utilizing his office hours
to ask him questions. Going over a problem one- on-one made me a lot more
confident that I knew what I was doing because I was forced to listen. He
was always available during his office hour times and you could also make
an appointment if you needed to. Using these outside resources gets you
more involved in the class and your chance of succeeding increases.
One more thing...be prepared to put a lot of time and effort
into this class. It isn't easy, but it is possible to do well. It's
also kind of amazing how all of the application problems that you encounter
can actually be used in real life. It's even more rewarding when you actually
know how to solve those problems! |
MATH 500: Fundamentals of Mathematics
This course provides students with a thorough foundation in the topics of whole numbers, fractions, decimals, ratios and proportions, percents, geometric figures and measurement. (Offered in lab and lecture formats.) Lecture: 3 hours
Credits:0
Overall Rating:0 Stars
N/A
Thanks, enjoy the course! Come back and let us know how you like it by writing a review. |
Algebra 2 is a mathematics course designed for students who have completed Algebra 1 and includes the study of linear, quadratic, square root, rational, exponential, and logarithmic functions, equations, and inequalities. Real-world problem solving situations will also be covered. This course is for Toltech T-STEM Academy sophomores and is rigorous in its study of algebraic theory and situations; studying outside of class time is an absolute necessity.
vWarm Ups are a daily grade due at the beginning of each class period and cannot be made up. If you miss a warm up due to an absence, an alternative assignment will be given.
vClass work is due before the end of the period on the day it is assigned, no exceptions. You may attend tutoring to complete class work not finished in class. If you miss class work due to an absence, you must attend tutoring to make up the assignment.
vHomework is the due the class period after it is assigned and is graded on completion. If it is not completed, it will be accepted one (Pre-AP) to three (non Pre-AP) class periods after it is assigned. If homework is not turned in within this time frame, it's a zero. The highest grade a late homework assignment can receive is an 80.
ATTENDENCE:
Mathematics can be very challenging; therefore, instructional time is very important. Students who are absent miss vital pieces of information, fall behind completing coursework, and risk credit denial.If a student is absent, it is his/her responsibility to collect missing assignment(s) and notes.These can be obtained from the website, a classmate, or by making an appointment with the teacher.For each day a student has an excused absence, one additional class day will be given to complete missing assignment(s).If a student is absent on the day of a test, the test can be made-up by making an appointment with the teacher. It is the student'sresponsibility to make appointments and turn in all missing assignments and tests completed and in a timely manner.
CLASS ORGANIZATION:
Notebooks/Binders are required for this class.The notebook should be a 1.5 to 3 inch 3-ring binder and should consist of items in the following order: Course Syllabus, Classroom Policies, List of Assignments, and then dividers. We will discuss notebook organization in detail in class. Notebooks will be graded two or three times each nine-weeks.
SCHOLASTIC INTEGRITY:
Scholastic dishonesty shall include, but not be limited to: cheating on tests, copying of homework, and the discussion of tests with other classes before all classes have completed the test.The consequence of scholastic dishonesty will come in the form of a zero (0) for that assignment, test, or project. |
Algebra, 1st Edition
ISBN10: 0-495-38798-3
ISBN13: 978-0-495-38798-5
AUTHORS: Kaufmann/Schwitters
INTERMEDIATE ALGEBRA'S simple, three-step problem-solving approach—learn a skill, use the skill to solve equations, and then use the equation to solve application problems—keeps you focused on building skills and reinforcing them through practice. This straightforward approach, in an easy-to-read format, has helped many students grasp and apply fundamental problem-solving skills. The carefully structured pedagogy includes learning objectives, detailed examples to help you see how concepts are used and applied, practice exercises, and helpful end-of-chapter reviews. Problems and examples reference a broad array of topics as well as career areas such as electronics, mechanics, and health, showing you that mathematics is part of everyday life |
Track Description: Herb Gross talks about a specific type of Differential Equations, namely those that are linear, 2nd order, homogeneous and with constant coefficients. He gives examples of the three types of possible general solutions and then shows why they ARE the solutions |
MAA Review
[Reviewed by Tom Schulte, on 01/12/2011]
This slim collection of varied visual "proofs" (a term, it can be argued, loosely applied here) is entertaining and enlightening. I personally find such representations engaging and stimulating aids to that "aha!" moment when symbolic argument seems not to clarify. Since such pictures can be found sprinkled in the pages of many mathematics periodicals and even in the occasional textbook, others obviously feel the same way. A collection of a gross of them in a single volume is really a delight to peruse.
The proofs are arranged by topic into six main chapters: Geometry & Algebra, Trigonometry, Calculus & Analytic Geometry, Inequalities, Integer Sums, Sequences & Series, and a final chapter of Miscellaneous. Geometry & Algebra leads off with the classic subject for this approach: the Pythagorean Theorem. This includes one of my favorites, the one by President James Garfield. Having several to compare can be particularly illustrative (literally) of how proof-like reasoning can be done in an image. I would wager that any student that can apply the Pythagorean to find a missing side of a right triangle can also, with little or no help, figure out how most of these Pythagorean proofs "work". Of course, this particular theorem lends itself very well to such an approach, which makes it very apt initial material
The Inequalities chapter presents various approaches to the five basic means and slightly more arcane inequalities, such as the Cauchy-Schwarz Inequality and those of Napier, Bernoulli, and Aristarchus. Up to this point an ambitious high school student or college student in first semester calculus would make fine progress.
This leads me to my only complaint here. This book is part of the MAA's Classroom Resource Materials and as a teacher I find it an excellent resource to enliven the odd lecture. However, I also like to think of the independent mathematics enthusiast with this book in hand and in that situation I feel an opportunity was missed to add a few words of explanation here and there. For example, one could point out the geometric connection to topics such as inner product spaces and norms, or include a sentence or two on combinatorial notation, or describe the simple choreography that results in a cycloid.
Of course, very few pages here could really benefit from such added verbiage and a diligent reader will still enjoy the book even if a few of the proofs require further investigation. The collection does have a good explanation for the confection-inspired optical illusion of The Problem of Calissons. Along with non-attacking queens and characteristic polynomials, this is in the final miscellany section. Whether one is seeking to ornament undergraduate lectures or simply to exercise one's mathematically-inclined mind, this book is a worthwhile purchase.
Tom Schulte surprises the occasional drowsy undergrad with visual proofs at Oakland Community College in Michigan.
BLL* — The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries. |
The LIFEPAC Math (Algebra I) complete set contains all 10 student workbooks for a full year of study plus the comprehensive Teacher's Guide.
Topics covered include:
Variables and Numbers
Solving Equations
Problem Analysis and Solution
Polynomials
Algebraic Factors
Algebraic Fractions
Radical Expressions
Graphing
Systems
Quadratic Equations and Review Lifepac Math, Grade 9 (Algebra I), Complete Set
Review 1 for Lifepac Math, Grade 9 (Algebra I), Complete Set
Overall Rating:
5out of5
Lifepac 9 Math
Date:June 11, 2012
AnnaExcellent as a primary math book, and the teacher's guide has all the ideas you would need for anything "extra", based on the student's needs. Combining the Grade 9 with the Grade 10 Lifepac Math is also an excellent idea, depending on your classes motivation and progress.
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Review 2 for Lifepac Math, Grade 9 (Algebra I), Complete Set
Overall Rating:
5out of5
Date:June 17, 2010
Karen Breland
This is my first time to use the Lifepac curriculum and I must say I am very pleased with it. The lessons are very easy to understand. I have ordered this for a refresher for my daughter over the summer. I will most likely be using this curriculum for all of her subjects in the fall.
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+1point
1of1voted this as helpful.
Review 3 for Lifepac Math, Grade 9 (Algebra I), Complete Set
Overall Rating:
4out of5
Date:November 17, 2008
James Clark
It is a great curriculum but it would be better served by having more self testing areas more often.
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+1point
1of1voted this as helpful.
Review 4 for Lifepac Math, Grade 9 (Algebra I), Complete Set
Overall Rating:
5out of5
Date:September 27, 2007
Laura Savin
Very detail material. Make sure your student comprehends the pre-algebra first before doing this. In the beginning it's easy, as you go further it gets harder but has good examples to go by and explanations.
Share this review:
+1point
1of1voted this as helpful.
Review 5 for Lifepac Math, Grade 9 (Algebra I), Complete Set
Overall Rating:
4out of5
Date:September 14, 2007
Mimi Robinson
After having tried ABeka and Saxon I felt this gave more opportunity for comprehension than the others. It also is set up to work at a at your own pace style. Love It! |
Mathematical assessment and instruction of diverse students will include number and operations, algebra, geometry, measurement, data analysis, and probability. Instructional approaches will include inquiry and direct instruction, and will emphasize systematic presentation that moves from concrete to abstract and utilizes multi-sensory. |
Contents on this page
Staffing
Examiner: Bill McCann Moderator: John Williams-Mozley
Rationale
Students entering tertiary study require an opportunity to master and become confident with mathematics. This course will attempt to provide students with mathematical competencies and improve capabilities for tertiary studies within a broad spectrum of undergraduate degrees. Hence, the course will equip students with a broad mathematical background to assist with further tertiary studies.
Synopsis
Using concepts of self-paced instruction, the course aims to give the student a carefully sequenced series of topics, which will provide the foundation for mathematics that will be encountered in tertiary studies detailed above. The self-paced structure allows students to work at their own pace developing confidence with effective and efficient mathematical problem solving skills. In addition, content of selective modules is a culturally relevant way to represent and portray an Indigenous perspective of how mathematical forms are used within their communities, families, language, culture and traditional practices. Hence, Indigenous students will feel comfortable and will relate to the appropriate use of mathematical language and understanding of mathematical processes and concepts from an Indigenous and non-Indigenous view. As a result, this will ensure that students become successful and maintain interest within the arena of mathematics.
Objectives
On successful completion of this course, students should be able to:
combine knowledge of place value with formal notation to represent order and manipulate numbers, decimals and fractions of any size and use scientific notation;
display data to show frequency and spread and interpret and critique data, making adjustments and inferences where appropriate.
Topics
Description
Weighting(%)
1.
Mathematics: Indigenous and non-Indigenous: Past and Present
10.00
2.
Managing Mathematics
18.00
3.
Comparing Numbers
18.00
4.
Introduction to Personal Finance
18.00
5.
Measurement: Application in Everyday Life
18.00
6.
Dealing with Data - Statistics in Our Life
1845.00
Private Study
75.00
Assessment details
Description
Marks out of
Wtg (%)
Due Date
Notes
ASSIGNMENT 1
40
10
16 Mar 2012
ASSIGNMENT 2
40
16
06 Apr 2012
ASSIGNMENT 3
48
16
04 May 2012
1.5 HOUR TEST
60
35
21 May 2012
(see note 1)
ASSIGNMENT 4
40
23
08 Jun 2012
NOTES
Res School 2. Students will be advised of the examination date after Residential School timetable has been finalised
Important assessment information
Attendance requirements:
This course requires attendance at a residential school. It aggregare of the weighted marks obtained for each of the summative assessment items in the course.
Examination information:
There are no examinations for this course.
Examination period when Deferred/Supplementary examinations will be held:
As there are no examinations in this course, there will be no deferred or supplemtary examinations successfully |
Additional Mathematics for OCR
Val Hanrahan
Summary: Teach with confidence, knowing your students will be fully prepared for their exams, with this detailed textbook that is closely tailored to the specification and has been endorsed by OCR.
Endorsed by OCR for use with the OCR Additional Mathematics specification. This level 3 qualification in Key Stage 4 enables students to study higher level mathematics without having to embark on their AS modules.
- Accessible and concise, written by experienced authors to guide and encourage your higher level students towards success
- Includes an introduction to each topic followed by worked examples with commentaries
- Provides plenty of practice with hundreds of questions
- Ideal for students considering maths at AS/A level, accelerating their progress and aiding their future choices
This book is endorsed by OCR for use with Additional Maths (FSMQ) and covers the exact requirements of the specification
It is perfect for Higher Tier students taking their GCSE early as they can gain a recognised qualification without having to embark on their AS modules
It is also ideal for students who intend to follow AS/A level Further Maths by giving them a 'head start' on their course
It gives students a 'taster' of AS/A level Maths and aids their choice of AS/A level modules
Table of Contents: 1 Algebra: review Linear expressions Solving linear equations Changing the subject of an equation Quadratic expressions Solving a quadratic equation that factorises Completing the square Simultaneous equations 2 Algebra: techniques Linear inequalities Solving quadratic inequalities Manipulating algebraic fractions Solving equations involving fractions Simplifying expressions containing square roots 3 Algebra: polynomials Operations with polynomials The Factor Theorem The Remainder Theorem 4 Algebra: applications The binomial expansion The binomial distribution 5 Co-ordinate geometry I Co-ordinates The gradient of a line Parallel and perpendicular lines The distance between two points Midpoint of a line joining two points The equation of a straight line Drawing a line given its equation Finding the equation of a line Intersection of two lines The circle 6 Co-ordinate geometry II, applications Inequalities Using inequalities for problem solving 7 Trigonometry I Using trigonometry in right angled triangles Trigonometrical functions for angles of any size Solving trigonometrical equations Identities involving sinθ, cosθ, and tanθ The area of a triangle The Sine and Cosine rules 8 Trigonometry II – Applications Applications of the Sine and Cosine rules in 2-D Height and distance problems in 3-D 3-D problems involving solids 9 Calculus I - Differentiation Finding the gradient of a curve Differentiating by using standard results Tangents and normals Stationary points Curve sketching 10 Calculus II - Integration Reversing differentiation Using integration to find areas 11 Calculus III - Applications to kinematics Variable acceleration problems The formulae for constant acceleration
About the Author(s): Val Hanrahan has been Head of Maths at an Independent school. She has also written several Pure Maths A/AS texts. Roger Porkess has written and edited books for GCSE, A/AS level and Key Stage 3.
Reviews:
...it contains all the information you'll need for the course as well as a HUGE number of questions for practice and makes it all very interesting. |
Last Updated:
Section:
Mathematics in Education and Industry (MEI) was founded in the early 1960s and is an independent UK curriculum development body. It is also an independent charity, with any income generated through MEI's work is used to support mathematics education.
MEI has kindly shared the following resources with TES to support the use of ICT in A/AS Level mathematics. These are practical activities that look at how we can use such dynamic software as Autograph, Geogebra and the TI-Nspire graphical calculator to enhance the study of post-16 topics including co-ordinate geometry and calculus. They move away from a traditional textbook approach by challenging students to investigate, make predictions and explain results.
These activities are ideal for use in a computer room, or for students to investigate at home before more formal study in the classroom. In each case, the pdf file contains full instructions and tricky challenges, and the attached files can be used to demonstrate and consolidate understanding back in the classroom on the interactive whiteboard/projector |
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