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Calculus: Single and Multivariable 9780470089149 ISBN: 0470089148 Edition: 5 Pub Date: 2008 Publisher: Wiley, John & Sons, Incorporated Summary: Calculus is the leading resource among the 'reform' projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. This edition uses all strands of the 'rule of four' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. Ships From:Murray, KYShipping:Standard, Expedited, Second Day, Next DayComments: 5th Edition. Pre-loved books for the budget-conscious consumer. With more than 50 years' experie... [more]
Elementary Statistics With MultimediaAddison-Wesley is proud to celebrate the Tenth Edition of Elementary Statistics, now with a Multimedia Study Guide on CD-ROM included with every new copy of the book. The Multimedia Study Guide contains nine hours of review exercise videos presented by the author, animated flowcharts that walk students through complex concepts and procedures, and selected solutions from the Student's Solutions manual.This text is highly regarded because of its engaging and understandable introduction to statistics. The author's commitment to providing student-f... MOREriendly guidance through the material and giving students opportunities to apply their newly learned skills in a real-world context has made Elementary Statistics the #1 best-seller in the market.Students learning from Elementary Statistics should have completed an elementary algebra course. Although formulas and formal procedures can be found throughout the text, the emphasis is on development of statistical literacy and critical thinking. Datasets and other resources (where applicable) for this book are available here .
Annual Foundation Schools (AFS) Objectives of AFS Basic knowledge in algebra, analysis and topology forms the core of all advanced instructional schools organized in this programme. The objective of the Annual Foundation Schools, to be offered in Winter and Summer every year, is twofold: To bring up students with diverse backgrounds to a common level. To identify those who are fit for further training. Any student who wishes to attend the advanced instructional schools is strongly encouarged to enroll in the Annual Foundation Schools first. Format of AFS The topics listed in the syllabi will be quickly covered in the lectures. There will be intensive problem sessions in the afternoons. The objective is not just to cover the syllabus prescribed, but to inculcate the habit of problem solving. However, the participants will be asked to study all the topics in the syllabus at home since the syllabi of these schools will be assumed in all the advanced instructional schools devoted to individual subjects. Participants in AFS These schools will admit 40 students in their first and second years of Ph. D. programme, a few students of M. Sc. (II Year), 5 university lecturers and college teachers who lack the knowledge of basic topics covered in these schools. A participant who has attended AFS-I and II will never be allowed to attend these again. The Programme Director will invite an eminent mathematician to deliver a series of lectures for one week called Unity of Mathematics Lectures. These lectures will be at the level of the courses being taught and they will be devoted to topics which involve several diverse areas of mathematics.
study was performed using a convenience sample of 90 students at a northeastern community college to determine gender differences of math anxiety and its effect on math avoidance. Four sections of an introductory English class were given aDuring the last decade, new technologies created a deluge of potential drug targets. Sifting through thousands of potential drug targets is a major industry bottleneck. Pharmaceutical companies can save billions of dollars by identifying most... Each year thousands of students are tracked into mathematics classes. In these particular classes, students may struggle or find their mathematics skills less academically able than their classmates and give up on the tasks that are introduced to
* Allows the reader to quickly and easily grasp the math, fundamentals, and general concepts involved in astronomy. * Covers techniques for using telescopes, the challenges of amateur astrophotography, and the special problems of observing the sky at ''invisible wavelengths''. * Unlike most books on the topic, it presents general concepts first and... more... Electricity is a strong topic for McGraw-Hill, as seen with our tremendous sales figures with the TAB list. The books sell well across nearly every channel, and are extremely well received internationally. Electricity Demystified is a self-teaching guide intended for anybody who wants to get familiar with the basic concepts of electricity, be it for... more...morePart of the ''Demystified'' series, this title teaches complex subjects in an easy-to-absorb manner and is designed for users without formal training, unlimited time, or genius IQs. It helps users understand circle and triangle models; inverses of circular functions; graphs of functions; coordinate conversions; angles and distances; and more.Part of the ''Demystified'' series, this book covers various key aspects of trigonometry: how angles are measured; the relationship between angles and distances; coordinate systems; calculating distance based on parallax; reading maps and charts; latitude and longitude; and more. more...
An introduction to variables. The number-line is labeled and the different types of numbers are defined. Students manipulate simple equations, and practice constructing equations based on real world applications
Product Details See What's Inside Product Description By Frances Curcio, Theresa Gurl, Alice Artzt, Alan Sultan Connect the Process of Problem Solving with the Content of the Common Core Mathematics educators have long recognized the importance of helping students to develop problem-solving skills. More recently, they have searched for the best ways to provide their students with the knowledge encompassed in the Common Core State Standards (CCSS). This volume is one in a series from NCTM that equips classroom teachers with targeted, highly effective problems for achieving both goals at once. The 44 problems and tasks for students in this book are organized into the major areas of the high school Common Core: algebra, functions, geometry, statistics and probability, and number and quantity. Examples of modeling, the other main CCSS area, are incorporated throughout. Every domain that is required of all mathematics students is represented. For each task, teachers will find a rich, engaging problem or set of problems to use as a lesson starting point. An accompanying discussion ties these tasks to the specific Common Core domains and clusters they help to explore. Follow-up sections highlight the relevant CCSS Standards for Mathematical Practice that students will engage in as they work on these problems. This book provides high school mathematics teachers with dozens of problems they can use as is, adapt for their classrooms, or be inspired by while creating related problems on other topics. For every mathematics educator, the books in this series will help to illuminate a crucial link between problem solving and the Common Core State Standards$36.95 Customers Who Bought This Also Bought... This book focuses on essential knowledge for teachers about proof and the process of proving. It is organized around five big ideas, supported by multiple smaller, interconnected ideas—essential understandingsThe Center for the Study of Mathematics Curriculum (CSMC) leaders developed this volume to further the goal of teachers having opportunities to interact across grades in ways that help both teachers and their students see connections in schooling as they progress through the grades. Each section of this volume contains three companion chapters appropriate to the three grade bands—K–5, 6–8, and 9–12—focusing on important curriculum issues related to understanding and implementing the CCSSM. How do you help your students demonstrate mathematical proficiency toward the learning expectations of the Common Core State Standards (CCSS)? This teacher guide illustrates how to sustain successful implementation of the CCSS for mathematics for high school. Discover what students should learn and how they should learn it, including deep support for the Mathematical Modeling conceptual category of the CCSS. Comprehensive and research-affirmed analysis tools and strategies will help you and your collaborative team develop and assess student demonstrations of deep conceptual understanding and procedural fluency. You'll also learn how fundamental shifts in collaboration, instruction, curriculum, assessment, and intervention can increase college and career readiness in every one of your students. Extensive tools to implement a successful and coherent formative assessment and RTI response are included. This practical, useful book introduces tested tools and concepts for creating equitable collaborative learning environments that supports all students and develops confidence in their mathematical ability. By connecting the CCSSM to previous standards and practices, the book serves as a valuable guide for teachers and administrators in implementing the CCSSM to make mathematics education the best and most effective for all students. Nearly a decade ago, NCTM published Administrator's Guide: How to Support and Improve Mathematics Education in Your School. This updated Administrator's Guide now positions school and district leaders to make sense of the past decade's many recommendations, with special emphasis on the Common Core State Standards for Mathematics. The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
Bainbridge, PA SAT MathUnion and intersection are commonly used terms, especially referring to Venn Diagrams. The topics covered in a discrete math course vary depending on where the course is taken. Since discrete numbers are those that can be counted, topics in discrete math revolve around these numbers the tradition of Mrs. Korsgen, my sixth grade teacher, I still keep a list of words and definitions I learn each day and encourage my students to do likewise. From flash cards with questions on one side and answers on another to silly rhymes containing class material, I can help the student recognize and adapt the study and learning style that will work for him or her
Murphy's Office Hours Fall 2003 (PHSC 1115) Fall 2003 Activities and Sample Solutions: Watch this space for solutions to current tests and other work in the class (PDF files). The solutions provided here are from students in the class. These are not only correct solutions, but are meant to be examples of exemplary student work. For a limited time only, you may print out the quizzes from here and turn them in at the beginning of class on the day they are due. Do not get used to this because you will soon be required to use WebCT for these quizzes. Calc 3 resources §11.1 and §14.1: link to animation that shows the drawing of a cycloid (from Visual Calculus by Lawrence S. Husch at the University of Tennessee, Knoxville) §11.1 and §11.5: Java applets (by TJ Murphy at the University of Oklahoma) the Parametric Curves Applet behaves much like a graphing calculator but will simultaneously display the graphs of t versus x(t), t versus y(t), and x(t) versus y(t), on separate sets of axes. The user has additional options to (1) show the progression of points being plotted as t increases; (2) trace simultaneously along all three curves; and (3) trace simultaneously along all three curves with tangent lines. the Polar Curves and Area Applet behaves much like a graphing calculator but adds two features: the user has options to (1) plot points on the user-specified polar curve given by r(t) beginning at a user-specified value of t; and (2) fill in the area bounded by a user-specified polar curve given by r(t) beginning at a user-specified value of t.
HOME Welcome to MathFortress.com where you can "Fortify Your Math Knowledge". Here you will find unique Great Quality Videos and other resources for various mathematical disciplines all for FREE! You will need Adobe Flash Player to see the videos. This website is best viewed using Chrome or Firefox browsers. You can download any documents on this site for your own personal use either studying with them or incorporating them into your lesson plan if you are a teacher. Let everyone know about MathFortress.com If you spot an error or typo on some of these resources or you would like to see resources on a specific topic feel free to email me: [email protected] [May 12, 2013] Added Implicit Solutions (Level 1) in the DE section.This video introduces the basic concepts associated with solutions of ordinary differential equations. This video goes over implicit solutions of differential equations. The concept of a formal solution is also presented. [April 28, 2013] Added Solutions (Level 3) in the DE section.This video introduces the basic concepts associated with solutions of ordinary differential equations. This video goes over 3 slightly more challenging examples illustrating how to verify solutions to differential equations. In addition this video also covers how to determine an appropriate interval of definition. [April 21, 2013] Added Solutions (Level 2) in the DE section.This video introduces the basic concepts associated with solutions of ordinary differential equations. This video goes over 3 examples illustrating how to verify solutions to differential equations. In addition this video also covers how to determine an appropriate interval of definition. [April 1, 2013] Added Definitions and Terminology (Level 4) in the DE section.This video introduces the basic definitions and terminology of differential equations. This video goes over 8 examples covering how to classify Partial Differential Equations (PDE) by order and linearity. [March 24, 2013] Added Definitions and Terminology (Level 3) in the DE section.This video introduces the basic definitions and terminology of differential equations. This video goes over 6 examples covering how to classify ordinary differential equations by order and linearity. [March 17, 2013] Added Definitions and Terminology (Level 2) in the DE section.This video introduces the basic definitions and terminology of differential equations. This video goes over 4 basic examples covering how to classify ordinary differential equations by order and linearity. [March 10, 2013] Added Fractions (Part 5) in the GRE section.This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of fractions. Topics covered: Comparing Fractions, Comparing more then two Fractions, fractions with irrational Numbers, combination of operations with fractions. [March 3, 2013] Added Fractions (Part 4) in the GRE section.This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of fractions. Topic covered: division of fractions, complex fractions, and mixed numbers. [February 24, 2013] Added Fractions (Part 3) in the GRE section.This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of fractions. Topic covered: multiplication of fractions. [February 17, 2013] Added Fractions (Part 2) in the GRE section.This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of fractions. Topics covered include: addition and subtraction of fractions. [February 10, 2013] Added Fractions (Part 1) in the GRE section.This video is a review of basic arithmetic for the purpose of solving problems on the quantitative reasoning section of the GRE revised General Test. This video covers the basics of fractions. Topics covered include: The definition of a fraction and properties of fractions. [February 3, 2013] Added Introduction to Geometry (Level 7) in the Geometry section.This video continues introducing the basic building blocks for the successful study of geometry. This video culminates this video series by going over 9 examples, that cover naming and set problems. [January 27, 2013] Added Introduction to Geometry (Level 6) in the Geometry section.This video continues introducing the basic building blocks for the successful study of geometry. This video goes over 7 examples involving the union and intersection of geometric figures. [January 20, 2013] Added Introduction to Geometry (Level 5) in the Geometry section.This video continues introducing the basic building blocks for the successful study of geometry. This video reviews the basic concepts associated with sets. In addition, this video goes over 6 examples involving the union and intersection of geometric figures. [January 5, 2013] Added Introduction to Geometry (Level 4) in the Geometry section.This video continues introducing the basic building blocks for the successful study of geometry. This video goes over 3 slightly more challenging examples covering the appropriate way to denote and name angles formed by points on a geometric figure. [December 15, 2012] Added Introduction to Geometry (Level 3) in the Geometry section. This video continues introducing the basic building blocks for the successful study of geometry. This video goes over 5 examples covering the appropriate way to denote and name angles formed by points on a geometric figure.
More About This Textbook Overview This book is a compilation of all the formulae that a mariner is commonly called upon to use but the exact workings of which he has perhaps forgotten. For each subject category, the author states the basic parameters in narrative form, often including a figure, graph, chart, diagram, or table, and then provides accompanying equations and their amplifications. Although some formulae that are simpler in format are propounded in other texts, many of those formulae lead to confusion in that "special rules" must be applied to them in order to obtain a correct answer. However, the rules applied to the formulae in this book work for all problems. In a great circle sailing situation, for example, the fact of whether the vertex is ahead of you or behind you does not matter—if you apply the rule(s) given in this book, you will get the correct answer. Another important feature of the book is its devotion of over ten pages of material to the international system of units (S.I
Heya guys! Is someone here know about mathematic precalculas? I have this set of questions regarding it that I just can't understand. Our class was asked to answer it and understand how we came up with the answer. Our Algebra teacher will select random people to solve it as well as show solutions to class so I need thorough explanation regarding mathematic precalculas. I tried answering some of the questions but I think I got it completely incorrect. Please assist me because it's a bit urgent and the deadline is quite close already and I haven't yet figured out how to solve this. How about giving some more information of what exactly is your problem with mathematic precalculas? This would aid in finding out ways to search for an answer. Finding a coach these days quickly enough and that too at a charge that you can pay for can be a maddening task. On the other hand, these days there are programs that are to be had to assist you with your math problems. All you require to do is to go for the most suited one. With just a click the correct answer pops up. Not only this, it hand-holds you to arriving at the answer. This way you also get to find out how to get at the correct answer. I totally agree, Algebrator is awesome! I am much better in algebra now, and I have the best grades in the class! It helped me even with the most confusing math problems, like those on leading coefficient or adding exponents. I really think you should try it . Algebrator is a easy to use product and is surely worth a try. You will also find quite a few interesting stuff there. I use it as reference software for my math problems and can say that it has made learning math more fun.
Math 102 From Courses Contents Course overview As with any course on differential calculus, the central character in this course is the derivative. The course starts by building up to the limit definition of the derivative and procedes through analytical, graphical and numerical approaches to build students' understanding of several types of functions and their derivatives. Next, we cover optimization, with applications to biological systems as well as principles of data fitting. A section on growth, decay and periodic phenomena precedes an introduction to differential equations and their use in modeling of biological systems. We finish the term with an introduction to probability and statistics and their application to the life sciences. One big difference between this course and a more traditional calculus course is the inclusion of examples and applications from the life sciences in place of the more traditional emphasis on physics. These examples and applications come from a wide range of fields including biochemistry, cell biology, ecology, genetics, population biology and evolution. Another difference between this course and a more traditional calculus course is the inclusion of the section on probability and statistics. Although these topics are important in many areas of science, they are rarely addressed in a first year mathematics course. Given the importance of these topics in the life sciences, both through their role in experiment design and data analysis and their importance in characterizing biological diversity, and the fact that many students taking this class will not see these topics in any later mathematics courses, they have been included here to give students some early exposure.
Pearson Debuts Interactive NovaNET Geometry Pearson has launched a new online geometry course for its NovaNET 15.0 service targeted toward students in grades 6 through 12 and adult education. Person's NovaNET is an online, standards-based courseware system designed for middle- and high-school students. Aligned to the 2007 Prentice Hall Geometry textbook, the new NovaNET Geometry course includes 77 multimedia lessons and includes instructional strategies for each. Additional features include: Interactive practices; Feedback and remediation; Ongoing, formative and summative assessments for each lesson; and Support for special needs students, including struggling readers. According to Pearson, the previous geometry course remains available, but the new version is designed for split-semester geometry schedules divided into Geometry A and B
#1049919Function keys in a scientific calculator concepts of education and science advancement by johnkwan
Preface Preface This book is a result of feedback from many readers of the book Engineering with Mathcad: Using Mathcad to Create and Organize your Engineering Calculations. The goal of Engineering with Mathcad was to get readers using Mathcad's tools as quickly as possible. This was accomplished by providing a step-by-step approach that enabled easy learning. As a result of reader feedback, Essential Mathcad makes it even easier to learn Mathcad. We added a new Chapter 1 that quickly introduces many useful Mathcad concepts. By the end of Chapter 1 you should be able to create and edit Mathcad expressions, use the Mathcad toolbars to access important features, understand the difference between the various equal signs, understand math and text regions, know how to create a user-defined function, attach and display units, create arrays, understand the difference between literal subscripts and array subscripts, use range variables, and plot an X-Y graph. Readers felt that the discussion of Mathcad settings and templates in Part 1 slowed down their learning of Mathcad. As a result of this feedback, the chapters Mathcad Settings, Customizing Mathcad, and Templates have been moved to Part IV. These chapters will have more meaning after readers have a greater understanding of Mathcad. Most of the material from Engineering with Mathcad is included in this book, but it has been rearranged in order to allow quicker access to Mathcad's tools. Readers asked for more applied examples of using Mathcad from various disciplines. Essential Mathcad provides many additional examples from fields such as: Chemistry, water resources, hydrology, engineering mechanics, sanitary engineering, and taxes. These examples help illustrate the concepts covered in each chapter. A challenge with any book is to hit a balance between too little material and too much material. Based on feedback from Engineering with Mathcad, I feel that we have achieved a good balance in Essential Mathcad. Some have said that the first edition did not cover enough advanced topics for their math, physics or advanced engineering courses. Others asked for coverage of some essential engineering topics. On the other hand, some said that the book was too long and covered too much material. Essential Mathcad is an attempt to achieve an even better balance. By adding the new Chapter 1, An Introduction to Mathcad, and rearranging other chapters, I think we have helped make learning Mathcad even easier. By adding discussion of some requested topics, I think we have satisfied the desires of many readers who wanted discussion of more topics. This book cannot and does not include a discussion of all the many Mathcad functions and features. It does attempt to focus on the functions and features that will be most useful to a majority of the readers. Book Overview This book uses an analogy of teaching you how to build a house. If you were to learn how to build a house, the final goal would be the completed house. Learning how to use the tools would be a necessary step, but the tools are just a means to help you complete the house. It is the same with this book. The ultimate goal is to teach you how to apply Mathcad to build comprehensive project calculations. In order to begin building, you need to learn a little about the tools. You also need to have a toolbox where you can put the tools. When building a house, there are simple hand tools and more powerful power tools. It is the same with Mathcad. We will learn to use the simple tools before learning about the power tools. After learning about the tools, we learn to build. This book is divided into four parts: Part I—Building Your Mathcad Toolbox. This is where you build your Mathcad toolbox—your basic understanding of Mathcad. It teaches the basics of the Mathcad program. The chapters in this part create a solid foundation upon which to build. Part II—Hand Tools for Your Mathcad Toolbox. The chapters in this part will focus on simple features to get you comfortable with Mathcad. Part III—Power Tools for you Mathcad Toolbox. This part addresses more complex and powerful Mathcad features. Part IV—Creating and Organizing Your Project Calculations with Mathcad. This is where you start using the tools in your toolbox to build something—project calculations. This part discusses embedding other programs into Mathcad. It also discusses how to assemble calculations from multiple Mathcad files, and files from other programs. Additional Resources This book is written as a supplement to the Mathcad Help and the Mathcad User's Guide. It adds insights not contained in these resources. You should become familiar with the use of both of these resources prior to beginning an earnest study of this book. To access Mathcad Help, click Mathcad Help from the Help menu, or press the F1 key. The Mathcad User's Guide is a PDF file located in the Mathcad program directory in the "doc" folder. In addition to the Mathcad Help and the Mathcad User's Guide, the Mathcad Tutorials provide an excellent resource to help learn Mathcad. The Mathcad Tutorials are accessed by clicking Tutorials from the Help menu. Take the opportunity to review some of the topics covered by the tutorials. This book (if sold in North America) includes a CD containing the full, non-expiring version of Mathcad v.14. The software is intended for educational use only. The book along with CD provides a complete introduction to learning and using Mathcad. A companion website is provided along with the text and includes links to additional exercises and applications, errata, and other updates related to the book. Please visit Teminology There are a few terms we need to discuss in order to communicate effectively. The terms, "click," "clicking" or "select" will mean to click with the left mouse button. The terms "expression" and "equation" are sometimes used interchangeably. "The term "equation" is a subset of the term "expression." When we use the term "equation," it generally means some type of algebraic math equation that is being defined on the right side of the definition symbol ":=". The term "expression" is broader. It usually means anything located to the right of the definition symbol. It can mean "equation" or it can mean a Mathcad program, a user-defined function, a matrix or vector or any number of other Mathcad elements.
Search Course Communities: Course Communities Lesson 28: Radical Equations Course Topic(s): Developmental Math \| Radicals The lesson begins with an emphasis on isolating the radical expression in a radical equation and then highlights the importance of checking for extraneous solutions that may be generated when the equation is solved by applying even powers. Equations containing two radical expressions and then presented, followed by coverage of taking the (n)th root of (a^n).
problem in the text for chapters 1-20.Striking a balance between concepts, modeling, and skills, Calculus: Single & Multivariable, 4th Edition is a highly acclaimed book that arms readers with an accessible introduction to calculus. It builds on the strengths from previous editions, presenting key concepts graphically, numerically, symbolically, and verbally. Guided by this innovative Rule of Four approach, the fourth edition examines new topics while providing readers with a strong conceptual understanding of the material. «Show less ... Show more» Rent Student Solutions Manual to accompany Calculus: Single and Multivariable, 4th Edition 4th Edition today, or search our site for other Hughes-H
Function & Statistics for Math & Physics This source has three sections on Function. They are 'Function Basics and Simple Graphing', 'Kinds of Functions (Periodic, Odd, Even, Etc.)', and 'Function Squashing'. These three sections provide an overview of the different ways in which Functions can be calculated and used. To view these three sections click on the link below they are the second, third, and fourth link in order. This website is part of the Think Quest Education Foundation an international online educational community. This link contains online flash cards that can be used to study 'Derivatives of Functions'. You can find both shuffled and un-shuffled flashcards by looking at the middle of the page where it says Flash Cards. The 'Derivatives of Functions' flashcards are the second sets listed. This website is maintained by Russell Blyth. Professor Blythe is an Associate Professor in the Department of Mathematics and Computer Science at Saint Louis University. Scroll down the page to Section 1 which covers 'Algebra Review' and 'Functions and Their Limits' and click on any of the following links in the middle of the page, 'Lecture Notes', 'Practice Problems', or 'Solutions'. Section 2 that covers 'Differentiation of Functions' and click on any of the following links in the middle of the page, 'Lecture Notes', 'Practice Problems', or 'Solutions'. This site was designed by the University of Washington in order to prepare graduate students for advanced statistics courses. The website was maintained by the University of Washington Center for Statistics and Social Services. Scroll down the page to the header 'CLASS NOTES AND TUTORIALS'. Directly below that are links which cover several mathematical topics. This website is maintained by Professor Paul Dawkins a math professor at Lamar University. This website is maintained by a Professor Kirkman of College of Saint Benedict and Saint John's University. If you click on the different links on the page it provides you with information on what type of information the statistics test can calculate and real life examples for each test.
Learning Outcomes: On successful completion of this module, students should be able to: · Distinguish between and use various kinds of numbers: rational, real and complex; · Manipulate and simplify algebraic expressions; · Work with functions and graphs; · Recall and use the general properties of linear, quadratic, trigonometric, exponential and logarithm functions, and solve problems involving these functions; · Solve problems using counting techniques; · Solve problems in geometry and trigonometry, using vectors as necessary; · Use differentiation to solve extremal problems; · Solve problems involving sequences and series. Assessment: Total Marks 300: End of Year Written Examination 240 marks (120 marks for each of Section A and Section B); Continuous Assessment 60 marks (30 marks each for the continuous assessment associated with Section A and Section B. Format of continuous assessment to be in-class test(s) and/or homework assignment(s): students will be given written notification of the format and breakdown of marks for continuous assessment at the first lecture% Students must obtain not less than 40% neither in the combined mark for Section A (continuous assessment and end of year written examination) nor in the combined mark for Section B (continuous assessment and end of year written examination). For students who do not satisfy this requirement, the lower of the two marks, scaled relative to the total marks available for the module, will be returned. End of Year Written Examination Profile: 1 x 3 hr(s) paper(s). Requirements for Supplemental Examination: 1 x 3 hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward. Module Objective: To provide an introduction to techniques and applications of differential calculus. Module Content: Limits, continuity and derivatives of functions of one variable. Applications. Learning Outcomes: On successful completion of this module, students should be able to: · Solve inequalities involving real numbers and the modulus function, obtain upper and lower estimates for expressions; · Reason with both the intuitive idea and the formal definition of the limit of a real function f of one variable, compute limits using a variety of techniques, apply the concept of divergence towards infinity; · Reason with both the intuitive idea and the formal definition of continuity of a real function f, determine if a given function is continuous and if a discontinuity is removable, apply the Extremal Value Theorem and the Intermediate Value Theorem; · Reason with both the intuitive idea of the derivative of a real function in one variable and its formal definition, calculate the derivative of a wide variety of functions, derive properties of the function from properties of the derivative via the Mean Value Theorem; · Find local extrema of a function f by investigating its first and second derivative, solve extremal problems in one variable. Assessment: Total Marks 100: End of Year Written Examination 75 marks; Continuous Assessment 25 marks (in-class test(s) and/or homework assignment(s),students introduction to Integral Calculus and ordinary differential equations. Module Content: Techniques and applications of integration of functions of one variable; solution of ordinary differential equations. Learning Outcomes: On successful completion of this module, students should be able to: · Reduce simple indefinite integrals to standard form and evaluate them by means of integration by substitution, integration by parts, partial fractions, completion of the square; · Apply the Fundamental Theorem of Calculus to evaluate definite integrals; · Apply methods of integration to evaluate plane areas, volumes of rotation and arc length; · Derive simple properties of the natural logarithm and exponential from properties of definite integrals; · Apply the trapezoidal rule and Simpson's rule to find approximate values of definite integrals; · Use differential equations to set up mathematical models of simple growth and decay problems related to physical, sociological and biological phenomena; · Recognize and solve the following differntial equations: equations of type variables separable, the logistic equation and first-order linear equations Solve systems of linear equations by elimination. · Carry out matrix arithmetic, and invert matrices. · Find determinants. Use them to decide on solvability and invertibility. · Determine if vectors belong to subspaces, and in particular to spans. Use this to find bases of subspaces. · Find eigenvalues, and bases of eigenvectors for eigenspaces. · Diagonalize symmetric matrices by orthogonal matrices. · Give examples of matrices with or without various properties such as invertibility, othogonality, symmetry, nature of eigenvalues, and determine whether matrices have these. distance between two vertices in a weighted graph, and to produce examples of Euler and Hamilton cycles Detail the basic concepts and theorems of the integer number system including Proof by Induction, primes and the Division Algorithm; · Derive elementary properties of rational numbers and Gauss' Theorem; · Apply techniques which have been developed in the lecture to solve problems; · Explain the basic concepts and main theorems of the theory of congruences; · Solve problems concerning the Chinese Remainder Theorem, Fermat's Theorem and Euler's Theorem; · Apply the basic concepts to basic coding theory; · Develop the RSA algorithm Prove elementary theorems of planar and solid Euclidean geometry. · Explain the concepts of axiom and proof. · Carry out intuitive geometric reasoning Correctly use logical implications, negations, equivalences, in proving simple mathematical statements. · Perform operations with sets and display their results in Venn diagrams. · Discriminate when a relation is reflexive, symmetric, or transitive. · Determine when a function is injective, surjective or bijective. · Perform operations with permutations. · Use the axiom system for groups in determining group structures and their Use the chain rule to compute partial derivatives of functions of several variables; · Compute equations of tangent planes to two-dimensional surfaces; · Use partial derivatives to solve problems related to economic concepts such as partial elasticity, production and utility; · Solve unconstrained optimisation problems for functions of two variables and apply this knowledge to optimisation problems in economics; · Use the methods of Elimination of Variables and Lagrange Multipliers to solve constrained optimisation problems, including problems from economics State the basic concepts of the planar geometry; · Solve geometric problems using methods from linear algebra; · Perform computations involving 2x2-systems; · Formulate the basic properties of conic sections; · Apply the theory of various transformations of the plane overview of major developments in Mathematics. Module Content: The development of Geometry, Algebra and Calculus from ancient times to present day. Learning Outcomes: On successful completion of this module, students should be able to: · Describe the development of mathematics from ancient times to the present day. · Place mathematical events in chronological order. · Identify at least twenty of the outstanding personalities in the history of mathematics and be able to list their contributions to the subject. · Describe the social and political environments in which mathematics developed. · Describe the scientific,economic and military contexts which stimulated mathematicians to promote their subject. · Explain many mathematical concepts not encountered in undergraduate courses Compute the iterates of a real or complex valued function of a single variable and the orbits of points, · Determine the fixed and periodic points of such functions and the nature of these points, · Investigate the dynamics of families of functions of a real variable, · Determine the bifurcation points of such families of functions, in particular the families of logistic maps and tent maps, · Sketch the Julia sets of elementary quadratic maps of a complex variable whether passed or failed. Learning Outcomes: On successful completion of this module, students should be able to: · Calculate and interpret descriptive statistics such as the mean, median, standard deviation, quartiles, percentiles, etc. · Draw and interpret graphical summaries of data e.g. histograms, box plots, stem and leaf plots. · Calculate probabilities for discrete probability distributions e.g. Binomial distribution and Poisson distribution using probability mass function or statistical tables · Calculate probabilities for the Normal distribution using the approximation to the Standard Normal distribution. · Carry out hypothesis tests for one mean and one proportion and make conclusions based on the p-value for the test. · Compute descriptive statistics and construct graphs using SPSS. · Model the relationships between variables using Regression Analysis.). Module Objective: To provide an overview of combinatorics, graphs, trees and applications. Module Content: Induction, recurrence relations. Combinatorics: permutations, combinations, the pigeonhole principle. Discrete probability on finite sample spaces; conditioning and independence. Graph theory: Euler and Hamiltonian paths and cycles, weighted graphs, applications distance between two vertices in a weighted graph
Book DescriptionThis is a great book to learn linear algebra from. It introduce you for the basic of linear algebra, like matrices and how to work with them and what you can use them for. There are many matematics sentences in this book with proofs. Also you will learn how to manage determinants and eigenvectors, not forgetting the more difficult Normal Jordan Form (if you need that). The exercises are great to make you remember all the sentences. So if you want to learn more about vectors in dimensions, you can not imagine, such like 5-d, then here is your tool. Also the text is written so you can understand it, wihout beeing someone needing beeing perfect to English. Doing Maths for my Computer Science Degree wasn't really what I envisaged. So I bought this book to help me with my maths. Incredibly I didn't struggle with it at all. The theory is well explained and the examples make it easy to quickly use the newly learned in practical applications. The authors go into enough detail so you can understand the stuff, not just learn it. On the other hand, they don't warp your brain with unnecessary junk. The proofs were especially helpfull. All in all an excellent book about linear algebra today. I would recommend this book to anyone who wants to gain a good understanding of the basics of Linear Algebra. It has many examples to accompany the theory which help the student prepare for exams and also the added bonus of MATLAB-a computer program that comes with the book that allows the user to perform many of the calculations gieven in the book. Also answers given at the back.
AcademicsNumber Theory Course Outline: Number theory is primarily concerned with the properties of and relationships between whole numbers. Topics we will study: 1. Prime numbers 2. Modular arithmetic 3. Sums of squares 4. Pythagorean triples 5. Fermat's Last Theorem 6. Magic squares 7. Continued fractions 8. Approximation of reals by rationals We will also spend a couple of weeks studying cryptography. In particular, we will look at how the RSA system works. This relies heavily on some of the number theory we will have learnt and is behind almost all modern cryptographic systems. You will need two books for the course: "A Pathway Into Number Theory" by R.P. Burn and "An Introduction To Number Theory" by H. Stark. Burn's book will lead us to discover and prove for ourselves some of the main results of number theory. Stark's book is more traditional.
Miles Reid's Undergraduate Algebraic Geometry is an excellent topical (meaning it does not intend to cover any substantial part of the whole subject) introduction. In particular, it's the only undergraduate textbook that isn't commutative algebra with a few pictures thrown in.
098 Developmental Arithmetic (3-0-3). Credit not applicable toward degrees. Required of students whose ACT Mathematics Main score is less than 15 or COMPASS Math score of 30 or less. Fundamental topics in arithmetic, geometry, and pre-algebra. 099 Developmental Algebra (3-0-3). Credit not applicable toward degrees. Required of students whose ACT Mathematics Main score is at least 15 but less than 19 or COMPASS Math score of 31 to 58. Fundamental topics in algebra for students with insufficient knowledge of high school level mathematics. PR: ACT Mathematics Main score of 15 or grade of "S" in MATH 098. 109 Algebra (3-0-3). Real numbers, exponents, roots and radicals; polynomials, first and second degree equations and inequalities; functions and graphs. PR: ACT Mathematics main score of 19 or grade of "S" in MATH 099. 211 Informal Geometry (3-0-3). Theorems are motivated by using experiences with physical objects or pictures and most of them are stated without proof. Point approach is used with space as the set of all points; review elementary geometry, measurement, observation, intuition and inductive reasoning, distance, coordinate systems, convexitivity, separation, angles, and polygons. No field credit for math majors/minors. PR: MATH 101 or higher. 220 Calculus I (4-0-4). A study of elements of plane analytical geometry, including polar coordinates, the derivative of a function with applications, integrals and applications, differentiation of transcendental functions, and methods of integration. PR: MATH 109 and MATH 110, or GNET 116, or ACT Mathematics main score of 26 or COMPASS Trigonometry score of 46 or above. 250 Discrete Mathematics (3-0-3). Treats a variety of themes in discrete mathematics: logic and proof, to develop students' ability to think abstractly; induction and recursion, the use of smaller cases to solve larger cases of problems; combinatorics, mathematics of counting and arranging objects; algorithms and their analysis, the sequence of instructions; discrete structures, e.g., graphs, trees, sets; and mathematical models, applying one theory to many different problems. PR: MATH 109 and MATH 110 or GNET 116. 290 Topics in Mathematics (1-4 hours credit). Formal course in diverse areas of mathematics. Course may be repeated for different topics. Specific topics will be announced and indicated by subtitle on the student transcript. PR: Consent of instructor. 400 Introduction to Topology (3-0-3). A study of set theory; topological spaces, cartesian products, connectedness; separation axioms; convergences; compactness. Special attention will be given to the interpretation of the above ideas in terms of the real line and other metric spaces. PR: MATH 240. 490 Topics in Mathematics (1-4 hours credit per semester). Advanced formal courses in diverse areas of mathematics. Courses may be repeated for different topics. Specific topics will be announced and indicated by subtitle on transcript. PR: Consent of instructor.
Math homework help. Hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. Included in this website is access to live math tutors, unlimited math videos and lessons, graphing calculators and much much more.
MAC 1105 Review Sheet, Exam 3 April 18, 2011 When: Wednesday, April 27, 9:00-10:50, in our usual classroom (note the time is wrong on the syllabus) What Material: Cumulative, with an emphasis on new material (2.4-2.8, 3.1, 3.5, 4.1-4.5). Procedure: The test will be closed book. You will be allowed one page (8 1/2 - 11 in) of notes, front and back. You may put anything you want on this page of notes.Calculators will be necessary for some of the problems. No TI-89's or TI-92's. Suggested Review: First study the material from the first two tests. To make this material easier, I guarantee that any question based on previous material will be similar (though not exactly the same) as a question from the previous exams.Note that the questions that got cut from the first exam are fair game. For the new material, the advice is always the same: practice.Do all the problems listed below.Make sure you can do them in a test-type situation (mixed, no help from the book, no checking answers immediately after doing the problems, no help from MyMathLab, with a page of notes).If you read this, draw a smiley on the front of your exam. This will make me happy.You should also review material that is a prerequisite for the new stuff (exponent rules, domain, range, long division, etc).Keep in mind, having a cheat sheet does not guarantee you a better grade.Do not slack off on the studying just because you are allowed a bit of help. Thoughts on the cheat sheet:I have a couple reasons for allowing the sheet of notes.First, there is a lot of information on the final exam, including several long procedures from the last few sections, and I feel it is ok if you don't commit all of it to memory.Second, it keeps people from cheating.Not that any of you are cheating, but some people are known to put formulas in their graphing calculators or write answers on their hands, stuff like that.Now everyone is on a level playing field. DO NOT THINK THE CHEAT SHEET WILL HELP YOU!!!!Many students actually do worse on this exam because of the cheat sheet.People tend to study less, because they think all of the answers will be with them, so practice isn't necessary.This is just plain wrong.Treat this exam as if you wouldn't have any help.Study and practice as if the cheat sheet wasn't going to be there.Then use the cheat sheet when you get stuck.Writing up a cheat sheet is NOT the same thing as studying.You have been warned. MyMathLab Help: I will set up a MyMathLab assignment that contains practice problems for old material that is likely to show up on the final exam. This assignment will be optional, but it will probably be really really useful. Specific Topics from New Material: Linear Functions: You should be able to plot a line, and you should be able to figure out the equation of a line or the slope of a line based on different information. Practice Problems: 2.4, 7-24, 35-42, 45-50, 53-58; 2.5, 5-26, 31-39, 47-56. Piecewise Functions: You should be able to graph a piecewise function. Practice Problems: 2.6, 17-34. Graphing Techniques: Given a function, you need to know the transformations that perform reflections, translations, and stretches. You should also be able to determine if a function is even or odd, and if it is symmetric about the y-axis or symmetric about the origin. Practice Problems: 2.7, 3-14, 23-57. Function Operations: Given two functions, be able to add, subtract, multiply, divide, and compose them. Make sure you understand how the domain is affected by these operations. Practice Problems: 2.8, 1-14, 23-30, 33-54, 57-72, 77-80. Quadratic Functions: Given a quadratic function, be able to find its vertex, whether it is open up or down, its y-intercept, its x-intercepts (if they exist), its domain and range. Then be able to graph it. Practice Problems: 3.1, 13-26. Rational Functions: Given a rational function, be able to find its asymptotes, intercepts, and graph it. Make sure you can figure out what types of asymptotes exist from the degree of the polynomials in the numerator and denominator. Practice Problems: 3.1, 61-96. Inverse Functions: Given the graph of a function, determine if it is one-to-one (horizontal line test). Given a one-to-one function, find its inverse. Verify two functions are inverses of each other. Draw the graph of the inverse of a function. Practice Problems: 4.1, 3-17, 41-50, 55-76. Exponential Functions: Evaluate them (either exactly or approximately with a calculator), and solve exponential equations. Practice Problems: 4.2, 1-12, 49-69. Logarithmic Functions: Evaluate them (either exactly or approximately with a calculator), and then manimpulate them using log rules. Practice Problems: 4.3, 3-30, 59-88; 4.4, 11-26, 61-72.
Book Description: Includes a variety of interesting and engaging problems based on NCTM content standards. Students will learn to model and solve the same problem in different ways while developing and using their own problem-solving strategies and techniques
Lincoln AcresThe study of the laws of exponents, FOIL, equation of a straight line, and the solving one equation one unknown and the quadratic equation. Algebra 2 normally examines the students mastery of integrating mastery of basic number theory and the study of the unknown. This is exemplified the study ...I received A's on tests and quizzes in Pre Algebra.
Lynelle M. Weldon Education: Ph.D. University of California, Davis M.A. University of California, Davis B.S. Pacific Union College Biography: Lynelle Weldon loves explaining math concepts, looking for creative illustrations to help students organize their math ideas, and exploring the connections between math and Christianity. She believes it is of vital importance for you to connect with God. Lynelle and her husband Jerry are from Northern California and they both enjoy quiet, flying, piano, and hiking.
Nine PlanetsA Multimedia Tour of the Solar System: one star, eight planets, and more Search for South El Monte CalculusDiscrete topics include the Principles of Mathematical Induction, the Binomial Theorem, and sequences and series. In Trigonometry, students will analyze and graph trigonometric functions and inverse trigonometric functions. Students will learn and use the fundamental trigonometric identities and solve conditional trigonometric equations.
Expanded Skills and Practice: The 5th edition includes a number new of skill-building and practice exercises, as well as additional problems. Updated Data and Models: References to dates, prices, and other time-bound quantities have been updated for contemporary applied examples, problems, and projects. For example, Section 11.7 now introduces the current debate on Peak Oil production, underscoring the importance of mathematics in understanding the world's economic and social problems. New Projects: There are new projects in Chapter 1:Which way is the Wind Blowing?; Chapter 5: The Car and the Truck; Chapter 9: Prednisone; and Chapter 10: The Shape of Planets. More Problems: 10% more "problem"-type questions now included in the test banks and instructor's manuals. Chapter 4 Reorganization: This chapter has been reorganized to smooth the approach to optimization. New ConcepTests: Promote active learning in the classroom. These can be used with or without clickers, and have been shown to dramatically improve student learning. Available in a book or on theweb at Expanded Appendices: A new Appendix D introducing vectors in the plane has been added. This can be covered at any time, but may be particularly useful in the conjunction with Section 4.8 on parametric equations.
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Abstract The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar an electronic version of an article published in [International Journal of Mathematical Education in Science and Technology]. [International Journal of Mathematical Education in Science and Technology] is available online at informaworldTM with the open URL
A Catalog of Mathematics Resources on the WWW and the Internet (M. Maheswaran, University of Wisconsin, Marathon County). Comprehensive links to mathematics sites, organized into categories including pure and applied mathematics. Over 100 links under Activities for College and Pre-college Mathematics, although many are K-12. Other categories offer useful information and materials for college mathematics teachers. Teaching Tips (Thomas Garrity and Frank Morgan, American Mathematical Society). Tips for becoming a more successful teacher of mathematics, with an example from a lecture in calculus. Ted Panitz's Teaching/Learning Website. Scroll down the page for links to several articles by Panitz and others on using cooperative learning and writing to reduce math anxiety, increase learning, and create a student-centered learning environment in college math courses. MERLOT Mathematics Portal (Multimedia Educational Resource for Learning and Online Teaching). The portal for online teaching and learning materials from faculty and educators in higher education around the world. World Lecture Hall: Mathematics. Syllabi and course materials for a wide range of mathematics courses in higher education. Type mathematics or specific areas or courses into the search engine for these links. What Is QL/QR? (Bill Briggs, University of Colorado at Denver). This page offers definitions of quantitative literacy/reasoning set forth in various publications and describes its importance in contemporary life. Quantitative Reasoning for College Graduates: A Complement to the Standards, 1998. (Mathematical Association of America). An online handbook for implementing a QL program on college and university campuses. Includes goals and expectations (with thorough discussions of rationales); moves from "why" to "actions and strategies" to "assessment." Center for Mathematics and Quantitative Education at Dartmouth College. Offers links to resources for college and university QL education in a wide variety of disciplines including art, literature, the sciences, and mathematics. Most are downloadable at no cost. Colleges with QL/QR Programs: QuIRK, Carleton College's Quantitative Inquiry, Reasoning, and Knowledge Initiative. The material on this site, designed with grants from FIPSE, NSF, and the Keck Foundation, is intended to help institutions "better prepare students to evaluate and use quantitative evidence in their future roles." The site provides curricular materials for infusing quantitative reasoning throughout the curriculum, assessment, program design, and more. See also the QuIRK page of links to other quantitative reasoning programs and additional QR teaching resources. Quantitative Reasoning Across the Curriculum at Hollins University. Describes their QR Program instituted in 2001 with Basic and Applied requirements. Lists courses that fulfill these requirements. Links to brief descriptions of QL courses in a variety of disciplines. Mathematics Across the Curriculum at Dartmouth College. The MATC Project ended in 2000, but this site has their goals, principals, links to MATC courses, and the Evaluation Summary for this five-year project. The resources they compiled are described above with a link to those in higher education. Professor Freedman's Math Help (Camden County College, Blackwood New Jersey). This site addresses the learning needs in mathematics of the community college adult learner. Useful for both students and teachers, the site offers math tutorials, homework assignments, video snippets on math topics, information about learning styles, and much more.
Powderly, TX Algebra am knowledgeable at the TI-83 and TI-84 calculators since I coached UIL calculator applications. Algebra 2 revisits Algebra 1 and some Geometry. However, Algebra 2 goes into more depth and adding more concepts.
History of Mathematics 9780130190741 ISBN: 0130190748 Pub Date: 2001 Publisher: Prentice Hall Summary: For junior and senior level undergraduate courses, this text attempts to blend relevant mathematics and relevant history of mathematics, giving not only a description of the mathematics, but also explaining how it has been practiced through time.
Courses Mathematics MA 100. TOPICS IN CONTEMPORARY MATHEMATICS (3) (GEN. ED. #5) Selected topics to illustrate the nature of mathematics, its role in society, and its practical and abstract aspects. Applications of mathematics to business and social sciences are explored. Three hours lecture. Prerequisite: placement exam. Fall semester, repeated spring semester. MA 110. PROBLEM SOLVING AND MATHEMATICS: NUMBER SYSTEMS (4) (GEN. ED. #5) For students majoring in elementary education. Explores various approaches to problem solving by examining topics such as estimating numerical quantities, probability and statistics, the nature of numeric patterns, functions, and relations. The course focuses on the use of various tools, such as calculators and physical models, as aids in problem solving. Four hours lecture. Prerequisite: placement exam. Fall semester. Department. MA 113. PROBLEM SOLVING AND MATHEMATICS: GEOMETRY (4) (GEN. ED. #5) For students majoring in elementary education. Explores various approaches to problem solving by examining topics such as spatial sense and measurement with respect to various geometries, properties of curves and surfaces, coordinate geometry, and transformations. The course focuses on the use of various tools, such as calculators and physical models, as aids in problem solving. Four hours lecture. Prerequisite: placement exam. MA 110 is recommended but not required. Spring semester. Department. MA 140. INTRODUCTION TO STATISTICS (FORMERLY MA 105) (4) (GEN. ED. #5) Basic concepts of descriptive statistics, simple probability distributions, prediction of population parameters from samples. Problems chosen from the natural and social sciences. Use of the computer in the analysis and interpretation of statistical data. Four hours lecture. Prerequisite: placement exam. Credit will not be given for those who have received credit for MA 141. Fall semester, repeated spring semester. McKibben,Webster. MA 141. STATISTICAL DATA ANALYSIS WITH ENVIRONMENTAL ISSUES IN VIEW (4) (GEN. ED. #5 AND #11) Basic concepts of descriptive statistics, simple probability distributions, and prediction of population parameters from samples are developed as a means to analyze environmental issues and the debates centered on them. Use of computer in analysis and interpretation of statistical data. Four hours lecture. Prerequisite: placement exam. Credit will not be given for those who have received credit for MA 140. Fall semester, repeated spring semester. McKibben,Webster. MA 160. PRECALCULUS (FORMERLY MA 114) (4) (GEN. ED. #5) An applications-oriented, investigative approach to the study of the mathematical topics needed for further coursework in mathematics. The unifying theme is the study of functions, including polynomials; rational functions; and exponential, logarithmic, and trigonometric functions. Graphing calculators and/or the computer will be used as an integral part of the course. Four hours lecture. Prerequisite: placement exam. Fall semester, repeated spring semester. MA 170. CALCULUS I (FORMERLY MA 117) (4) (GEN. ED. #5) The concepts of limit and derivative are developed, along with their applications to the natural and social sciences 171. Fall semester, repeated spring semester. Department. MA 171. CALCULUS I-ENVIRONMENTAL (4) (GEN. ED. #5 AND #11) The concepts of limit and derivative are developed, along with their applications to planet and environmental sustainability issues 170. Fall semester.Webster. MA 180. CALCULUS II (FORMERLY MA 118) (4) (GEN. ED. #5) The concepts of Riemann sums and definite and indefinite integrals are developed, along with their applications to the natural and social sciences. A symbolic algebra system is used as both an investigative and computational tool. Three hours lecture, two hours laboratory. Prerequisite: placement exam or MA 170 or 171 with a minimum grade of C-. Prerequisite to MA 222. Fall semester, repeated spring semester. Department. MA 260. HISTORY OF MATHEMATICS (3) (GEN. ED. #4 AND #7) Selected topics in the history of mathematics chosen to show how mathematical concepts evolve. Topics include number, function, geometry, and calculus. Consideration of the cultural, social, and economic forces that have influenced the development of mathematics. Three hours lecture. Prerequisites: MA 221 and 222. Spring semester. Offered 2010-11 and alternate years. Lewand. MA 290. INTERNSHIP IN MATHEMATICS (3-4) Students interested in the application of mathematics to government, business, and industry are placed in various companies and agencies to work full time under the guidance of a supervisor. The director confers with individual students as needed. Students are selected for internships appropriate to their training and interest in mathematics and related fields. Prerequisites: junior standing and a major in mathematics. This course is graded pass/no pass only. Fall semester, repeated spring semester. Department. MA 299. INDEPENDENT WORK IN MATHEMATICS (1-4) Department. MA 311. INTRODUCTION TO HIGHER MATHEMATICS (3) An introduction to proof techniques within the context of the following topics: elementary set theory, functions and relations, and algebraic structures. Three hours lecture. Prerequisites: MA 221 and 222. Fall semester. Lewand, McKibben, Webster. MA 313. FUNDAMENTALS OF REAL ANALYSIS (3) A rigorous development of differential and integral calculus, beginning with the completeness of the real number system. The topological structure of the real number system is developed, followed by a rigorous notion of convergence of sequences. Limit, continuity, derivative, and integral are formally defined, culminating in the Fundamental Theorem of Calculus. Three hours lecture. Prerequisites: MA 311. Spring semester. Offered 2010-11 and alternate years. McKibben, Webster. Computer Science CS 105. EXPLORATIONS OF COMPUTER PROGRAMMING (3) (GEN. ED. #5) Introduction to the concepts of computer programming using 3-D virtual worlds. Programming constructs such as looping, selection, and data structures, along with the control of objects will be explored. No prior programming experience is required. Spring semester. Zimmerman. CS 116. INTRODUCTION TO COMPUTER SCIENCE (4) (GEN. ED. #5) Introduction to the discipline of computer science and its unifying concepts through a study of the principles of program specification and design, algorithm development, object-oriented program coding and testing, and visual interface development. Prerequisite: placement exam or CS 105 with a minimum grade of C-. Fall semester. Zimmerman. CS 220. COMPUTER ARCHITECTURE (3) Organization of contemporary computing systems: instruction set design, arithmetic circuits, control and pipelining, the memory hierarchy, and I/O. Includes topics from the ever-changing state of the art. Prerequisite: CS 119. Fall semester. Offered 2011-12 and alternate years. Kelliher. CS 230. ANALYSIS OF COMPUTER ALGORITHMS (3) The design of computer algorithms and techniques for analyzing the efficiency and complexity of algorithms. Emphasis on sorting, searching, and graph algorithms. Several general methods of constructing algorithms, such as backtracking and dynamic programming, will be discussed and applications given. Prerequisites: CS 119. Fall semester. Offered 2010-11 and alternate years. Zimmerman. CS 245. SOFTWARE ENGINEERING (3) This course emphasizes the application of tools of software engineering to programming. The focal point of the course is the design, implementation, and testing of a large programming project. Students gain familiarity with the standard programmer's tools, such as debugger, make facility, and revision control. Prerequisite: CS 119. Fall semester. Offered 2010-11 and alternate years. Kelliher. CS 290. INTERNSHIP IN COMPUTER SCIENCE (3-4) Students interested in the application of computer science to government, business, and industry are placed in various companies and agencies to work full time under the guidance of a supervisor. The director confers with individual students as needed. Students are selected for internships appropriate to their training and interest in computer science and related fields. Prerequisites: junior standing and a major in computer science. This course is graded pass/no pass only. Fall semester, repeated spring semester. Department.
Summary: Provides completely worked-out solutions to all odd-numbered exercises within the text, giving you a way to check your answers and ensure that you took the correct steps to arrive at an answer. 2007
The Saxon Difference Saxon is an integrated curriculum. Concepts in each math strand are broken into small increments that are interwoven together to create rich mathematical connections. Once taught, the increments are systematically distributed and practiced throughout a full year of instruction. No skills are ever dropped. It is this consistent review and practice that makes the difference in helping all students achieve long-term success Stephen Hake, Author of Saxon Math Intermediate 3-5 and Courses 1-3 When I read the published field-test results of John Saxon's Algebra 1 manuscript and learned that John Saxon taught students with the same methods I did, I placed an order and began using John's book with my eighth grade students with great success. Recognizing the country's need for an effective math program from grade school through high school, we joined our efforts and soon Saxon Math was helping millions of students across America succeed in math. Students can learn math and are advanced through the subject matter in a way that gently guides them step by step and provides the time and practice necessary to learn and remember the foundational concepts of mathematics. Saxon Math is written the way I taught–one bit of instruction each day with plenty of practice on previous instruction. The excellent performance of my students on problem-solving contests and on standardized tests convinced me that this method of instruction works. It produces excellent problem-solving skills and long-term learning of key math concepts. Pat Wrigley, Author of Adaptations for Saxon Math I began to adapt Saxon's intermediate grades series in 1991 for use with my own Special Ed students. When other resource specialists began to request copies of my work, I realized it might be of value to Saxon users. The Saxon approach is reliable. I've seen repeated proof that this program works for all students. Even as I reach my 40th year as a full-time resource specialist, seeing the happiness of my students' faces as they achieve beyond their dreams is still the greatest joy of my life. Dr. Frank Wang, Author of Saxon Calculus "My passion is for teaching and for helping students learn more mathematics than they ever thought possible. I am a fervent advocate for the Saxon pedagogy and highly recommend its mathematics textbooks as the best textbooks for providing students with a solid and firm foundation for further study in mathematics." Dr. Wang holds an undergraduate degree from Princeton University and a Ph. D. in pure mathematics from MIT. He began working for Saxon Publishers at age 16 as a high school student. He was president of the company from 1994 to 2001. Frank has taught at the University of Oklahoma and currently teaches at the Oklahoma School of Science and Mathematics.
Authors Document Type Contribution to Book Publication Date 2011 Source Publication Early Algebraization, Volume 2 Abstract This chapter highlights findings from the LieCal Project, a longitudinal project in which we investigated the effects of a Standards-based middle school mathematics curriculum (CMP) on students' algebraic development and compared them to the effects of other middle school mathematics curricula (non-CMP). We found that the CMP curriculum takes a functional approach to the teaching of algebra while non-CMP curricula take a structural approach. The teachers who used the CMP curriculum emphasized conceptual understanding more than did those who used the non-CMP curricula. On the other hand, the teachers who used non-CMP curricula emphasized procedural knowledge more than did those who used the CMP curriculum. When we examined the development of students' algebraic thinking related to representing situations, equation solving, and making generalizations, we found that CMP students had a significantly higher growth rate on representing-situations tasks than did non-CMP students, but both CMP and non-CMP students had an almost identical growth in their ability to solve equations. We also found that CMP students demonstrated greater generalization abilities than did non-CMP students over the three middle school years.
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Made of two mutually explanatory parts, this book provides information on the general, historical and cultural background, and the development of each subdiscipline that together comprise Chinese mathematics. It is organised topically rather than chronologically, and tells how to interpret the contextual setting, both mathematical and sinological. Dirac operators are used in physics, differential geometry, and group-theoretic settings. Using Dirac operators as a unifying theme, this work demonstrates how some of the important results in representation theory fit together when viewed from this perspective. It presents the important ideas on Dirac operators and Dirac cohomology. more... This monograph contains a selection of over 250 propositions which are equivalent to AC. The first part on set forms has sections on the well-ordering theorem, variants of AC, the law of the trichotomy, maximal principles, statements related to the axiom of foundation, forms from algebra, cardinal number theory, and a final section of forms from topology,... more... Universal Algebra has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices... more... This book examines the development of mathematics from the late 16th Century to the end of the 19th Century. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. - ;Aimed at students and researchers in Mathematics, History... more... This book is devoted to the spectral theory of commutative C*-algebras of Toeplitz operators on the Bergman space and its applications. For each such commutative algebra there is a unitary operator which reduces Toeplitz operators from this algebra to certain multiplication operators, thus providing their spectral type representations. This yields... more...... more...
Computational Mathematics . org Welcome to computationalmathematics.org, an educational website providing resources for students and researchers in the computational sciences. Follow the links above for information on specific topics. NEW!!! Register in the Forums and begin/join discussions on all things related to computational mathematics. Computational mathematics is the branch of mathematics which is concerned primarily with ways in which to compute results to various problems by applying the theory of numerical analysis. Through the development and execution of sophisticated computer programs, approximate numerical solutions are achieved as opposed to the analytic solutions of traditonal applied mathematics. The subject is the backbone of all computational science and engineering, which in recent decades has become vital to cutting edge research in almost every scientific field. As high speed computers provide ever increasing simulation capabilities, the importance of this field has never been greater.
Login Math 5329: Structure of Modeling with Rates of Change As suggested by the catalog description (below), this course bridges the major strands of secondary mathematics through a study of applications of rates of change. Topics include difference equations, curve-fitting, parameter-based simulation, and discrete and continuous dynamics. For detailed information and policies, please read the MATH 5329 Syllabus. A study of rates of change through modeling. Direct applications of rates of change to number concepts, algebra, geometry, probability, and statistics. Class Posts by Shermane King Interested math teachers, see the "Downloads" section in the sidebar → Description The activity is a general worksheet that can be utilize in any secondary mathematics classroom. At the beginning of the worksheet students are given a scenario that will be followed by 10 questions. The questions range from developing equations to calculating real-world… by Gwendolyn Walbey & Christopher Whiteneck Interested math teachers, see the "Downloads" section in the sidebar → »All information and corrections made in this document were updated as of 11-6-2011. The following guidelines are resource to instructors with objectives pertaining to: spreadsheet usages to create, analyze, and use data tables and graphs for linear, exponential or polynomial… Interested math teachers, see the "Downloads" section in the sidebar → by Paul Rodriguez and Vanessa Garza Sequence for Modeling Real Automobile Data Create a table using the data provided. Use the table to create a Scatter Plot. Use graph paper to plot the coordinates. Create appropriate intervals. Label the x- and y- axis. Title your… by Brittney Martinez & Amanda Raiborn Interested math teachers, see the "Downloads" section in the sidebar → Teacher's Guide for Speedy Delivery Route The Activity! This particular activity uses an algorithm called Dijkstra's Algorithm – or the shortest path algorithm. It takes a problem of delivery routes and solves for the quickest route from one point… Interested math teachers, see the "Downloads" section in the sidebar → Teacher's Guide for Choosing an Apartment Description: Individually, students will decide on four to six important criteria for selecting an apartment to live in once they graduate high school. Students will then select an appropriate amount (three to five) of local apartments to use for… For all the math teachers, see the "Downloads" section in the sidebar → Teacher's Guide to Does This Line Ever Move? Description Individually or group work, student(s) will work on Arm-and-a Leg Tickets Activity. Students will struggle thru the assignment, thus you will have to guide each group or individual's when they reach that point. Goal/Objectives… This collaborative quiz asks teachers to make sense of a recursively defined sequence that leads to the famous Collatz Problem. The main task of the quiz is to connect the sequences to the secondary mathematics classroom by using the Collatz problem as part of a lesson plan. Download Quiz 5: Cycles and Chaos Related Links… During an in-class activity on Modeling World Population, groups of mathematics teachers found data related to factors influencing population change. The spreadsheet generated during the activity is available below. Quiz 4 on Modeling World Population World Population Spreadsheet by Teachers The world population recently hit 7 billion people... or did it? The goal of this extended activity for math teachers is basic: Work as a group to develop a single mathematical model for the world population and use it to independently estimate the date when the world population equals 7 billion. Constraints The model must… This is just a follow-up on our in-class activity. I was hoping you can organize your thoughts about the Table you found interesting and any variables/equations you might use to model the data. In particular, it'd be great if you could make a testable hypothesis like "the increase in school enrollment is approximately exponential". (Nothing to turn-in here.)
Ease into algebra with clear instructions that simplify such concepts as real numbers, integers, properties, operations, exponents, square roots, and patterns. Geared toward struggling algebra students, these examples, practice problems, definitions, problem-solving strategies, and an assessment section make this a resource that teachers and parents can use to help students succeed! Answer keys and references are also provided.
Created for the independent, homeschooling student, Teaching Textbooks has helped thousands of high schoolers gain a firm foundation in upper-level math without constant parental or teacher involvement. Extraordinarily clear illustrations, examples, and graphs have a non-threatening, hand-drawn look, and engaging real life questions make learning pre-algebra practical and applicable. Textbook examples are clear while the audiovisual support includes lecture, practice and solution CDs for every chapter, homework, and test problem. The review-method structure helps students build problem solving skills as they practice core concepts and rote techniques. Teaching Textbooks' new Pre Algebra 2.0 includes the following new features: Automated grading A digital gradebook that can manage multiple student accounts and be easily edited by a parent. Over a dozen more lessons and hundreds of new problems and solutions Interactive lectures Hints and second chance options for many problems Animated buddies to cheer the student on Reference numbers for each problem so students and parents can see where a problem was first introduced What a blessing! Date:August 27, 2012 homeschoolingmomof2 Location:Douglas, GA Age:35-44 Gender:female This is our first year of Teaching Textbooks (pre-alg)... my 8th grade daughter has gone from crying, hating math last school year...to 'Math is my favorite subject!'...all because of Teaching Textbooks! Thank you so much for this curriculum, it makes it so much more understandable to a child that has always struggled in math, always cried about math because she didn't understand even the basics. Share this review: +1point 1of1voted this as helpful. Review 2 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 4out of5 Good foundation with slow pacing Date:March 21, 2012 ACT Tutor Location:Minnesota Age:35-44 Gender:female Quality: 3out of5 Value: 3out of5 Meets Expectations: 4out of5 I am a home school mom, a college professor and an ACT tutor. We just finished our first year with this curriculum. There are several good things to consider. The book sets a good foundation for high school math. The information is presented clearly if your student can handle the tone of voice of the speakers. My daughter decided the textbook was better as she felt the voice droned too much. The text and the CD's present the exact same information in the same way. I appreciated the automatic grading and being able to go in and change grades or erase certain problems for her to redo. I appreciate that multiple students can use the curriculum at the same time. I appreciate that it presents many of the skills needed later on high school standardized tests. For someone who is not confident in teaching upper level math to their students this would be a good choice. But listen to your kids and take their concerns seriously, they may need some slight modifications to make this work for them. The negatives. The voice and tone will be tough for some students. The graphics are childish and distracting for some. The sound can be turned off though and the text can be used instead of listening to the lectures for those students this is difficult for. My sixth grade daughter liked this better when we turned off sound completely. Data can not be saved to the cloud. We had a computer crash and lost her work, despite backups. I would advise printing out grade sheets periodically if you need complete records of grades. The most disappointing part of the curriculum is the pacing. It is glacial. Lessons are presented broken up over several days and we found the best way to make the curriculum work for us was to combine 2-3 lessons a day as only small parts of the information were presented and it was more clear when several lessons were combined for my daughter. In addition the daily review in the lessons is extensive and becomes tedious. Every lesson has extensive review and it does become too repetitive and unnecessary for some students. I suppose some students need the continual daily review and they will appreciate this feature. It was driving my daughter to hate the curriculum until we modified it for her. To make the book work for us we combined several lessons per day and I only required review work on the unit tests rather than daily. If she got the review questions wrong then the next unit she had to do the daily review on the topic. The software will grade based on what you choose to do rather than all the problems listed so that worked well for us. We finished the curriculum in 6 months following these strategies and are beginning the Algebra 1 curriculum now. Overall I still recommend the curriculum especially as an ACT tutor. Teaching textbooks presents the math skills needed for standardized tests in a clear foundational way. For this reason I have chosen to continue with the curriculum for Algebra despite the downsides. Listen to your kids though and consider making modifications if needed. Share this review: +21points 22of23voted this as helpful. Review 3 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 5out of5 Outstanding! Date:March 13, 2012 TolKat Location:Enterprise, AL Age:45-54 Gender:male Quality: 5out of5 Value: 5out of5 Meets Expectations: 5out of5 My wife and I just started homeschooling for the first time and needed some way to make math interesting for our 8th grade daughter - Teaching Textbooks fits the bill! Fun, easy to use, informative, very impressed with the way this is done, wish there were more offerings - Chemistry, History...Definately recommend! Share this review: +1point 1of1voted this as helpful. Review 4 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 4out of5 We anticipate great things! Date:February 15, 2012 Harborlights Location:Georgetown, SC Age:45-54 Gender:female Quality: 4out of5 Value: 4out of5 Meets Expectations: 4out of5 We are excited to begin using this math curriculum with our daughter. It has been highly recommended. Share this review: -4points 0of4voted this as helpful. Review 5 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 5out of5 Makes my children love math Date:February 8, 2012 Mamabearcw Location:Huntington, WV Age:35-44 Gender:female Quality: 5out of5 Value: 4out of5 Meets Expectations: 5out of5 My kids have really enjoyed this program. I have enjoyed not having to show them how to do everything, and having to relearn it at the same time. Math is not my favorite subject. Share this review: 0points 0of0voted this as helpful. Review 6 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 5out of5 Loving it so far! Date:February 3, 2012 mom2jjka Quality: 4out of5 Value: 4out of5 Meets Expectations: 5out of5 We are very pleased with TT Pre-Algebra. My daughter is doing well, and she actually likes math again! The one thing I was slightly disappointed in was the paper Textbook. The pages are rather thin and flimsy, and they look as if they were just photocopied rather than printed. My daughter mostly only uses the CD's however, so it's not too big of an issue for us. Share this review: +1point 1of1voted this as helpful. Review 7 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 5out of5 Excellent Product! Date:November 30, 2011 HSMom2six Quality: 5out of5 Value: 5out of5 Meets Expectations: 5out of5 I decided to upgrade from version 1 of this product because I wanted the auto-grading, but that wasn't the only improvement. This version has more content and, while it does review the previous year's work, it doesn't take a tedious amount to accomplish it. This makes the pace of the book better. I've tried several popular math curricula with my students, but TT text is the one we keep returning to. Overall, TT is an absolutely fantastic product for both the mathphobic and those who like math. Share this review: +1point 1of1voted this as helpful. Review 8 for Teaching Textbooks Pre-Algebra Kit, Version 2.0 Overall Rating: 5out of5 Perfect blend of review and new material. Date:November 1, 2011 Mom2Ike Location:Livonia, MI Age:45-54 Gender:female Quality: 5out of5 Value: 5out of5 Meets Expectations: 5out of5 My son loves Teaching Textbooks Pre-Algebra. The lessons are clearly explained and easy to understand. Each lesson has interactive practice with new concepts before the problem section begins. Also the "study buddy" feature is fun and encouraging. Overall this was an excellant fit for our 7th grade homeschool math program.
Algebra I, Part A (1 credit) Covers the material of the first semester of Algebra I over a full year (or a 35-lesson) time frame. This course is intended to assist those students who require additional time or practice to grasp algebraic concepts. (16 lessons and submissions, 4 exams) What People are Saying "With EdOptions you just have a lot of possibilities" - Jacqui Clay, Educator (AZ)
68660 / ISBN-13: 9780321268662 Mathematics for Elementary School Teachers Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers" leave the course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual skills upon which those ideas are built
1) "Geometry" by Serge Lang. This is essentially a high school geometry book done right. It doesn't bother with two column proofs and other silly things. It treats geometry like it should be treated. 2) "Introduction to Geometry" by Coxeter. Maybe use this as a second book. It's very rich and offers an introduction to many different fields of geometry. It's not an actual textbook though, but more an introduction. Not suitable as first book. 3) "Elements" by Euclid. The very first geometry book and a standard for over 1000's of years. It is a recommended read to everybody: it develops geometry from scratch. However, you must be aware that there are some errors and omissions (for example coordinate geometry).
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MITRES_18_001_strang_13 Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 C H A P T E R 13 Partial Derivatives This chapter is at the center of multidimensional calculus. Other chapters and other topics may be optional; this chapter and these topics are required. We are back to the basic idea of calculus-the derivative. There is a functionf, the variables move a little bit, and f moves. The question is how much f moves and how fast. Chapters . 1-4 answered this question for f(x), a function of one variable. Now we have f(x, y) orf(x, y, z)-with two or three or more variables that move independently. As x and y change,f changes. The fundamental problem of differential calculus is to connect Ax and Ay to Af. Calculus solves that problem in the limit. It connects dx and dy to df. In using this language I am building on the work already done. You know that dfldx is the limit of AflAx. Calculus computes the rate of change-which is the slope of the tangent line. The goal is to extend those ideas to fix, y) = x2 - y2 o r f(x, y) = Jm or f(x, y, z) = 2x + 3y + 42. These functions have graphs, they have derivatives, and they must have tangents. The heart of this chapter is summarized in six lines. The subject is diflerential calculus-small changes in a short time. Still to come is integral calculus-adding up those small changes. We give the words and symbols for f(x, y), matched with the words and symbols for f(x). Please use this summary as a guide, to know where calculus is going. Curve y =f(x) vs. Surface z =f(x, y) df af af becomes two partial derivatives - and - d~ ax ay - becomes four second derivatives - - - - d2{ a2f a2f a2f a2f dx ax2' axayY ay2 ayai Af % AX dx becomes the linear approximation Af % 9AX + a ax ay f ~ ~ tangent line becomes the tangent plane z - z, = a f ( x - x,) ax + a f ( y - yo) ay dy - dy dz az'ax ---- dx becomes the chain rule - = -- a~ +-- dy dt d~ dt dt a~ dt a~ dt df = 0 af - becomes two maximum-minimum equations - = 0 and af = 0. dx dx a~ 472 13 Partial Derivatives 13.1 Surfaces and Level Curves The graph of y =f(x) is a curve in the xy plane. There are two variables-x is independent and free, y is dependent on x. Above x on the base line is the point (x, y) on the curve. The curve can be displayed on a two-dimensional printed page. The graph of z =f(x, y) is a surface in xyz space. There are three variables-x and y are independent, z is dependent. Above (x, y) in the base plane is the point (x, y, z) on the surface (Figure 13.1). Since the printed page remains two-dimensional, we shade or color or project the surface. The eyes are extremely good at converting two- dimensional images into three-dimensional understanding--they get a lot of practice. The mathematical part of our brain also has something new to work on-two partial derivatives. This section uses examples and figures to illustrate surfaces and their level curves. The next section is also short. Then the work begins. EXAMPLE I Describe the surface and the level curves for z =f(x, y) = x2 + y2 . The surface is a cone. Reason: x 2 + y2 is the distance in the base plane from (0, 0) to (x, y). When we go out a distance 5 in the base plane, we go up the same distance 5 to the surface. The cone climbs with slope 1. The distance out to (x, y) equals the distance up to z (this is a 450 cone). The level curves are circles. At height 5, the cone contains a circle of points-all at the same "level" on the surface. The plane z = 5 meets the surface z = x2 + y 2 at those points (Figure 13.1b). The circle below them (in the base plane) is the level curve. DEFINITION A level curve or contour line of z =f(x, y) contains all points (x, y) that share the same valuef(x, y) = c. Above those points, the surface is at the height z = c. There are different level curves for different c. To see the curve for c = 2, cut through the surface with the horizontal plane z = 2. The plane meets the surface above the points where f(x, y) = 2. The level curve in the base plane has the equation f(x, y) = 2. Above it are all the points at "level 2" or "level c" on the surface. Every curve f(x, y) = c is labeled by its constant c. This produces a contour map (the base plane is full of curves). For the cone, the level curves are given by .x 2 + y2 = c, and the contour map consists of circles of radius c. Question What are the level curves of z =f(x, y) = x2 + y2 ? Answer Still circles. But the surface is not a cone (it bends up like a parabola). The circle of radius 3 is the level curve x2 + y2 = 9. On the surface above, the height is 9. N 2 z= 'x 2 +y =5 p A J 2 " Y Y 5- base plane .- Fig. 13.1 The surface for z =f(x, y) = x2 + y 2 is a cone. The level curves are circles. 13.1 Surfaces and Level Curves 473 EXAMPLE 2 For the linearfunction f(x, y) = 2x + y, the surface is a plane. Its level curves are straight lines. The surface z = 2x + y meets the plane z = c in the line 2x + y = c. That line is above the base plane when c is positive, and below when c is negative. The contour lines are in the base plane. Figure 13.2b labels these parallel lines according to their height in the surface. Question If the level curves are all straight lines, must they be parallel? Answer No. The surface z = y/x has level curves y/x = c. Those lines y = cx swing around the origin, as the surface climbs like a spiral playground slide. y 2 x 2x+y=O \2x+y=1\2x+y=2 y = 1 2 3 Wol x Fig. 13.2 A plane has parallel level lines. The spiral slide z = y/x has lines y/x = c. EXAMPLE 3 The weather map shows contour lines of the temperaturefunction. Each level curve connects points at a constant temperature. One line runs from Seattle to Omaha to Cincinnati to Washington. In winter it is painful even to think about the line through L.A. and Texas and Florida. USA Today separates the contours by color, which is better. We had never seen a map of universities. -- - j Fig. 13.3 The temperature at many U.S. and Canadian universities. Mt. Monadnock in New Hampshire is said to be the most climbed mountain (except Fuji?) at 125,000/year. Contour lines every 6 meters. 13 Pattial Derhrcttiwes Question From a contour map, how do you find the highest point? Answer The level curves form loops around the maximum point. As c increases the loops become tighter. Similarly the curves squeeze to the lowest point as c decreases. EXAMPLE 4 A contour map of a mountain may be the best example of all. Normally the level curves are separated by 100 feet in height. On a steep trail those curves are bunched together-the trail climbs quickly. In a flat region the contour lines are far apart. Water runs perpendicular to the level curves. On my map of New Hampshire that is true of creeks but looks doubtful for rivers. Question Which direction in the base plane is uphill on the surface? Answer The steepest direction is perpendicular to the level curves. This is important. Proof to come. EXAMPLE 5 In economics x2y is a utility function and x2y = c is an indiference c u m . The utility function x2y gives the value of x hours awake and y hours asleep. Two hours awake and fifteen minutes asleep have the value f = (22)(4). This is the same as one hour of each: f = (12)(1).Those lie on the same level curve in Figure 13.4a. We are indifferent, and willing to exchange any two points on a level curve. The indifference curve is "convex." We prefer the average of any two points. The line between two points is up on higher level curves. Figure 13.4b shows an extreme case. The level curves are straight lines 4 x + y = c. Four quarters are freely substituted for one dollar. The value is f = 4x + y dollars. Figure 13.4~ shows the other extreme. Extra left shoes or extra right shoes are useless. The value (or utility) is the smaller of x and y. That counts pairs of shoes. asleep y quarters right shoes hours awake I ; ; ; * left shoes 1 2 Fig. 13.4 Utility functions x2y, 4x + y, min(x, y). Convex, straight substitution, complements. 13.1 EXERCISES Read-through questions The graph of z =Ax, y) is a a in b -dimensional For z =f(x, y) = x2 - y2, the equation for a level curve is space. The c curvef(x, y) = 7 lies down in the base plane. I . This curve is a i . For z = x - y the curves are Above this level curve are all points at height d in the k . Level curves never cross because I . They crowd surface. The z = 7 cuts through the surface at those together when the surface is m . The curves tighten to a points. The level curves f(x, y) = f are drawn in the xy point when n . The steepest direction on a mountain is plane and labeled by g . The family of labeled curves is 0 to the P . a h map. 13.2 ParHal Derivatives 475 1 Draw the surface z =f(x, y) for these four functions: 22 Sketch a map of the US with lines of constant temperature (isotherms) based on today's paper. fl=Jp f2=2-JZ7 f3=2-&x2+y2) f4= 1 +e-X2-y2 23 (a) The contour lines of z = x2 + y2 - 2x - 2y are circles around the point , where z is a minimum. 2 The level curves of all four functions are . They (b)The contour lines of f = are the circles enclose the maximum at . Draw the four curves x2 + Y2 = c + 1 on which f = c. flx, y) = 1 and rank them by increasing radius. 24 Draw a contour map of any state or country (lines of 3 Set y = 0 and compute the x derivative of each function constant height above sea level). Florida may be too flat. at x = 2. Which mountain is flattest and which is steepest at 25 The graph of w = F(x, y, z) is a -dimensional sur- that point? face in xyzw space. Its level sets F(x, y, z) = c are 4 Set y = 1 and compute the x derivative of each function dimensional surfaces in xyz space. For w = x - 2y + z those at x = 1. level sets are . For w = x2 + Y2 + z2 those level sets are For f5 to f10 draw the level curvesf = 0, 1,2. Alsof = - 4. 26 The surface x2 + y2 - z2 = - 1 is in Figure 13.8. There is empty space when z2 is smaller than 1 because + + 27 The level sets of F = x2 y2 qz2 look like footballs when q is , like basketballs when q is , and like frisbees when q is 11 Suppose the level curves are parallel straight lines. Does 28 Let T(x, y) be the driving time from your home at (0,O) the surface have to be a plane? to nearby towns at (x, y). Draw the level curves. 12 Construct a function whnse level curve f = 0 is in two 29 (a) The level curves offlx, y) = sin(x - y) are separate pieces. (b)The level curves of g(x, y) = sin(x2- y2) are 13 Construct a function for which f = 0 is a circle and f = 1 (c) The level curves of h(x, y) = sin(x - y2) are is not. 30 Prove that if xly, = 1 and x2y2= 1 then their average 14 Find a function for which f = 0 has infinitely many pieces. + + x = g x l x2), y = g y , y2) has xy 2 1. The function f = xy 15 Draw the contour map for f = xy with level curves f = has convex level curves (hyperbolas). -2, -1,0, 1, 2. Describe the surface. 31 The hours in a day are limited by x + y = 24. Write x2y 16 Find a function f(x, y) whose level curve f = 0 consists of as x2(24-x) and maximize to find the optimal number of a circle and all points inside it. hours to stay awake. 32 Near x = 16 draw the level curve x2y = 2048 and the line Draw two level curves in 17-20. Are they ellipses, parabolas, or hyperbolas? Write r- before squaring both sides. 2x = c as = c + 2x x + y = 24. Show that the curve is convex and the line is tangent. 33 The surface z = 4x + y is a . The surface z = min(x, y) is formed from two . We are willing to exchange 6 left and 2 right shoes for 2 left and 4 right shoes but better is the average 21 The level curves of f = (y - 2)/(x- 1) are through the point (1, 2) except that this point is not 34 Draw a contour map of the top of your shoe. Partial Derivatives The central idea of differential calculus is the derivative. A change in x produces a change in$ The ratio Af/Ax approaches the derivative, or slope, or rate of change. What to do iff depends on both x and y? The new idea is to vary x and y one at a timk. First, only x moves. If the function is x + xy, then Af is Ax + yAx. The ratio Af/Ax is 1 + y. The "x derivative" of x + xy 13 Partial Derhratives is 1 + y. For all functions the method is the same: Keep y constant, change x, take the firnit of AflAx: DEFINITION df(x, y) = lim - = lim f (x + Ax, Y)-f (x, Y) Af ax AX-OAX AX-o Ax On the left is a new symbol af/dx. It signals that only x is allowed to vary-afpx is a partial derivative. The different form a of the same letter (still say "d") is a reminder that x is not the only variable. Another variable y is present but not moving. Do not treat y as zero! Treat it as a constant, like 6. Its x derivative is zero. If f(x) = sin 6x then dfldx = 6 cos 6x. If f(x, y) = sin xy then af/ax = y cos xy. Spoken aloud, af/ax is still "d f d x." It is a function of x and y. When more is needed, call it "the partial off with respect to x." The symbolf ' is no longer available, since it gives no special indication about x. Its replacement fx is pronounced "fx" or "fsub x," which is shorter than af/ax and means the same thing. We may also want to indicate the point (x,, yo) where the derivative is computed: EXAMPLE 2 f(x, y) = sin 2x cos y fx = 2 cos 2x cos y (cos y is constant for a/dx) The particular point (x,, yo) is (0,O). The height of the surface is f(0,O) = 0. The slope in the x direction is fx = 2. At a different point x, = n, yo = n we find fx(n, n) = - 2. Now keep x constant and vary y. The ratio Af/Ay approaches aflay: f,(x, y) = lim AY+O f = Alim Of(x, Y + BY)-f(x, Ay ~+ AY Y) This is the slope in the y direction. Please realize that a surface can go up in the x direction and down in the y direction. The plane f(x, y) = 3x - 4y has fx = 3 (up) and f , = - 4 (down). We will soon ask what happens in the 45" direction. /Zy The x derivative of , x + 'is really one-variable calculus, because y is constant. 4 The exponent drops from to - i,and there is 2x from the chain rule. This distance function has the curious derivative af/ax = xlf. The graph is a cone. Above the point (0,2) the height is ,- /= 2. The partial derivatives are fx = 012 and f, = 212. At that point, Figure13.5 climbs in the y direction. It is level in the x direction. An actual step Ax will increase O2 + 22 to AX)^ + 22. But this change is of order (Ax)2 and the x derivative is zero. Figure 13.5 is rather important. It shows how af@x and af/dy are the ordinary derivatives of f(x, yo) and f(x,, y). It is natural to call these partial functions. The first has y fixed at yo while x varies. The second has x fixed at xo while y varies. Their graphs are cross sections down the surface-cut out by the vertical planes y = yo and x = x,. Remember that the level curve is cut out by the horizontal plane z = c. 13.2 Partial Derivatives 477 2 2 f(Oy) =-0 +y 2 f(x, 2)= 4x 2 +2 2 • X Fig. 13.5 Partial functions x• + 22 and /02 y2 of the distance functionf= / + y2. The limits of Af/Ax and Af/Ay are computed as always. With partial functions we are back to a single variable. The partial derivative is the ordinary derivative of a partial function (constant y or constant x). For the cone, af/ay exists at all points except (0, 0). The figure shows how the cross section down the middle of the cone produces the absolute value function:f(0, y) = lyl. It has one-sided derivatives but not a two-sided derivative. Similarly Of/ax will not exist at the sharp point of the cone. We develop the idea of a continuous function f(x, y) as needed (the definition is in the exercises). Each partial derivative involves one direction, but limits and continuity involve all direc- tions. The distance function is continuous at (0, 0), where it is not differentiable. EXAMPLE 4 f(x, y) = y 2 af/Ox = - 2x Of/ay = 2y Move in the x direction from (1, 3). Then y 2 - x 2 has the partial function 9 - x 2 . With y fixed at 3, a parabola opens downward. In the y direction (along x = 1) the partial function y 2 - 1 opens upward. The surface in Figure 13.6 is called a hyperbolic paraboloid, because the level curves y 2 -_ 2 = c are hyperbolas. Most people call it a saddle, and the special point at the origin is a saddle point. The origin is special for y 2 - x 2 because both derivatives are zero. The bottom of the y parabolaat (0, 0) is the top of the x parabola.The surface is momentarily flat in all directions. It is the top of a hill and the bottom of a mountain range at the same 2 0 1 =2 _ 1 0 f= y2 _ x2 -1 0 y 1 1 -l 0 01 1 0 Fig. 13.6 A saddle function, its partial functions, and its level curves. 13 Partial Derivatives time. A saddle point is neither a maximum nor a minimum, although both derivatives are zero. Note Do not think that f(x, y) must contain y2 and x2 to have a saddle point. The function 2xy does just as well. The level curves 2xy = c are still hyperbolas. The partial functions 2xyo and 2xoy now give straight lines-which is remarkable. Along the 45" line x = y, the function is 2x2 and climbing. Along the - 45" line x = - y, the function is -2x2 and falling. The graph of 2xy is Figure 13.6 rotated by 45". EXAMPLES 5-6 f(x, y, z) = x2 + y2 + z2 P(T, V) = nRT/V Example 5 shows more variables. Example 6 shows that the variables may not be named x and y. Also, the function may not be named f! Pressure and temperature and volume are P and T and V. The letters change but nothing else: aP/aT = nR/V dP/aV = - ~ R T / V ~ (note the derivative of 1/V). There is no dP/aR because R is a constant from chemistry-not a variable. Physics produces six variables for a moving body-the coordinates x, y, z and the momenta p,, p,, p,. Economics and the social sciences do better than that. If there are 26 products there are 26 variables-sometimes 52, to show prices as well as amounts. The profit can be a complicated function of these variables. The partial derivatives are the marginalprofits, as one of the 52 variables is changed. A spreadsheet shows the 52 values and the effect of a change. An infinitesimal spreadsheet shows the derivative. SECOND DERIVATIVE Genius is not essential, to move to second derivatives. The only difficulty is that two first derivatives f, and f , lead to four second derivativesfxx and fxy and f , and f,. (Two subscripts: f,, is the x derivative of the x derivative. Other notations are d2 flax2 and a2f/axdy and aflayax and d2flay2.) Fortunately fxy equals f,, as we see first by example. EXAMPLE 7 f = x/y has f, = l/y, which has fxx =0 and f, = - l/y2. The function x/y is linear in x (which explainsfxx = 0). Its y derivative isf, = - xly2. This has the x derivative f,,, = -l/y2. The mixed derivativesfxy and fyx are equal. In the pure y direction, the second derivative isf, = 2x/y3. One-variable calculus is sufficient for all these derivatives, because only one variable is moving. EXAMPLE 8 f = 4x2 + 3xy + y2 has f, = 8x + 3y and f , = 3x + 2y. Both "cross derivatives" f,, andf,, equal 3. The second derivative in the x direction is a2f/ax2 = 8 or fxx = 8. Thus "fx x" is "d second f d x squared." Similarly a2flay2 = 2. The only change is from d to a. . Iff(x, y) has continuous second derivatives thenf,, =&, Problem 43 sketches a proof based on the Mean Value Theorem. For third derivatives almost any example shows that f,, =fxyx =f,, is different from fyyx =fyxy =fxyy . Question How do you plot a space curve x(t), y(t), z(t) in a plane? One way is to look parallel to the direction (1, 1, 1). On your XY screen, plot X = (y - x ) / d and Y = (22 - x - y)/$. The line x = y = z goes to the point (0, O)! How do you graph a surface z =f (x, y)? Use the same X and Y. Fix x and let y vary, for curves one way in the surface. Then fix y and vary x, for the other partial function. For a parametric surface like x = (2 + v sin i u ) cos u, y = (2 + v sin f u) sin u, z = v cos iu, vary u and then u. Dick Williamson showed how this draws a one-sided "Mobius strip." 13.2 EXERCISES Read-through questions 25 xl"' Why does this equal tl""? 26 cos x The h derivative a f / a ~ e comes from fixing b and 27 Verify f,, =fyx for f = xmyn.If fxy = 0 then fx does not moving c . It is the limit of d . Iff = e2, sin y then depend on and& is independent of . The af/ax = and a f / a ~ = Iff = (x2+ y2)'12 thenfx = function must have the form f (x, y) = G(x) + cr and f , = h . At (x,, yo) the partial derivativef, is the ordinary derivative of the I function Ax, yo). Simi- : 28 In tmns of 0, computef, and.&forf (x, Y)= J aft) tit. First larly f, comes from f( 1 ). Those functions are cut out by vary x. Then vary Y. vertical planes x = xo and k , while the level curves are cut out by I planes. 29 Compute af/ax for f = IT v(t)dt. Keep y constant. The four second derivatives are f,,, m , n , o . 30 What is f (x, y) = : 1 dtlt and what are fx and fy? For f = xy they are P . For f = cos 2x cos 3y they are q . In those examples the derivatives r and s fxxy,... off 31 Calculate all eight third derivatives fxXx, = are the same. That is always true when the second derivatives x3y3. HOW many are different? are f . At the origin, cos 2x cos 3y is curving u in the x and y directions, while xy goes v in the 45" direc- 32-35, ,.hoosc g(y) so that f(x,Y)= ecxdy) the tion and w in the -45" direction. equation. Find aflax and af/ay for the functions in 1-12. 32 fx+fy=O 33 fx= 7 & 35 f x x = 4fyy 36 Show that t - '12e-x214t satisfies the heat equation f; =f,, . 3 x3y2- x2 - e y 4 ~ e " + ~ Thisflx, t) is the temperature at position x and time t due to 5 (x + Y)/(x- Y) 6 1 / J M a point source of heat at x = 0, t = 0. 37 The equation for heat flow in the xy plane isf, =f,, +hY. Show thatflx, y, t) = e-2t sin x sin y is a solution. What expo- nent in f = e - sin 2x sin 3y gives a solution? 11 tan-'(ylx) 12 ln(xy) 38 Find solutions Ax, y) = e - sin mx cos ny of the heat equation /, = +f,. Show that t - 'e-x214re-"214r also a / , is Compute fxx,fx, =A,, and&, for the functions in 13-20. solution. 39 The basic wave equation is f,, =f,,. Verify that flx, t) = + sin(x t) and f (x, t) = sin(x - t) are solutions. Draw both graphs at t = 4 4 . Which wave moved to the left and which moved to the right? 40 Continuing 39, the peaks of the waves moved a distance 19 cos ax cos by 20 l/(x + iy) Ax = in the time step At = 1114. The wave velocity is AxlAt = Find the domain and range (all inputs and outputs) for the 41 Which of these satisfy the wave equation f;, = c2fxx? functions 21-26. Then compute fx, fy ,fz,f;. + sin(x - ct), COS(X ct), ex- ect, ex cos ct. 23 (Y- x)l(z - t) 24 In(x + t) 42 Suppose aflat = afjax. show that a2flat2 = a2flax2. 480 13 Partial Derhrathres 43 The proof of fxy =fy, studies f(x, y) in a small rectangle. distance from (x,, yn)to (a, b) is and it approaches The top-bottom difference is g(x) =f(x, B) -f(x, A). The 4 For any E > 0 there is an N such that the distance difference at the corners 1, 2, 3, 4 is: < E for all n > . Q = C -f31 f 4 -Cf2 -f1l 46 Find (x,, y2) and (x,, y,) and the limit (a, b) if it exists. Start from (x,, yo)= (1, 0). = g(b) - g(a) (definition of g) (a) (xn, yn) = (lib + I), nl(n + 1)) = (b - a)g,(c) (Mean Value Theorem) (b)(xn, yn) =(xn-l, yn-1) (c) ( x n , ~ n ) = ( ~ n - l , ~ n - l ) = (6 - a)(B - A)fxy(c,C) (MVT again) (a) The right-left difference is h(y) =f (b, y) -f (a, y). The 47 (Limit o f (x, y)) 1 Informal definition: the numbers f same Q is h(B) - h(A). Change the steps to reach Q = f(x,, yn)approach L when the points (x,, y,) approach (a, b). (B - A)@- alfyxk, C). 2 Epsilon-delta dejinition: For each E > 0 there is a 6 > 0 such (b)The two forms of Q make fxy at (c, C) equal to f,, at that I f(x, y) - LI is less than when the distance from (c, C). Shrink the rectangle toward (a, A). What assump- (x, Y) to (a, b) is . The value off at (a, b) is not tion yields fxy =fy, at that typical point? involved. 48 Write down the limit L as (x, y) + (a, b). At which points (a, b) does f(x, y) have no limit? (a)f(x, Y)= JW (b)f(x, Y)= XIY ( 4 f b , Y)= ll(x + Y) (d)f(x, Y)= xyl(xZ+ y2) In (d) find the limit at (0,O) along the line y = mx. The limit changes with m, so L does not exist at (0,O). Same for xly. f 49 Dejinition o continuity: f(x, y) is continuous at (a, b) if f(a, b) is defined and f(x, y) approaches the limit as (x, y) approaches (a, b). Construct a function that is not con- tinuous at (1, 2). 44 Find df/dx and dfldy where they exist, based on equations + 50 Show that xZy/(x4 yZ)-+ 0 along every straight line (1) and (2). y = mx to the origin. But traveling down the parabola y = xZ, (a)f=lxyl ( b ) f = x Z + y 2 ifx#O, f = O i f x = O the ratio equals 51 Can you definef (0,O) so that f (x, y) is continuous at (0, O)? Questions 45-52 are about limits in two dimensions. + (a)f = 1 1 Iy- 1 (b)f = ( l + x ) ~ (c)f = ~ ' + ~ . - x 1 f 45 Complete these four correct dejinitions o limit: 1 The points (xn,yn) approach the point (a, b) if xn converges to a 52 Which functions zero as (x, Y)- (0, O and ) and 2 For any circle around (a, b), the points (x,, y,) xy2 x~~~ xmyn eventually go the circle and stay . 3 The (a) (b) (c) 13.3 Tangent Planes and Linear Approximations Over a short range, a smooth curve y =f(x) is almost straight. The curve changes direction, but the tangent line y - yo =f '(xo)(x - xo) keeps the same slope forever. The tangent line immediately gives the linear approximation to y=f(x): Y = Yo +f'(xo)(x - xo). What happens with two variables? The function is z =f(x, y), and its graph is a surface. We are at a point on that surface, and we are near-sighted. We don't see far away. The surface may curve out of sight at the horizon, or it may be a bowl or a saddle. To our myopic vision, the surface looks flat. We believe we are on a plane (not necessarily horizontal), and we want the equation of this tangent plane. 13.3 Tangent Planes and Linear Approximations 481 Notation The basepoint has coordinates x0 and Yo. The height on the surface is zo =f(xo, Yo). Other letters are possible: the point can be (a, b) with height w. The subscript o indicates the value of x or y or z or 8f/Ox or aflay at the point. With one variable the tangent line has slope df/dx. With two variables there are two derivatives df/8x and Of/Oy. At the particular point, they are (af/ax)o and (af/ay)o. Those are the slopes of the tangent plane. Its equation is the key to this chapter: 43A The tangent plane at (xo, Yo, zo) has the same slopes as the surface z = f(x, y). The equation of the tangent plane (a linear equation) is z - zo = (x- Xo) + A (yo). y- (1) The normal vector N to that plane has components (af/ax) , (0f/ly)o, -1. 0 EXAMPLE 1 Find the tangent plane to z = 14 - x 2 - y2 at (xo, Yo, zo) = (1, 2, 9). Solution The derivatives are af/ax = - 2x and Ofl/y = - 2y. When x = 1 and y = 2 those are (af/ax)o = - 2 and (df/ay)o = - 4. The equation of the tangent plane is z - 9 = - 2(x - 1)- 4(y - 2) or z+2x+4y= 19. This z(x, y) has derivatives - 2 and - 4, just like the surface. So the plane is tangent. The normal vector N has components -2, -4, -1. The equation of the normal line is (x, y, z) = (1, 2, 9) + t(- 2, - 4, - 1). Starting from (1, 2, 9) the line goes out along N-perpendicular to the plane and the surface. N = Fig. 13.7 The tangent plane contains the x and y tangent lines, perpendicular to N. Figure 13.7 shows more detail about the tangent plane. The dotted lines are the x and y tangent lines. They lie in the plane. All tangent lines lie in the tangent plane! These particular lines are tangent to the "partial functions"--where y is fixed at Yo = 2 or x is fixed at x0 = 1. The plane is balancing on the surface and touching at the tangent point. More is true. In the surface, every curve through the point is tangent to the plane. Geometrically, the curve goes up to the point and "kisses" the plane.t The tangent T to the curve and the normal N to the surface are perpendicular: T . N = 0. tA safer word is "osculate." At saddle points the plane is kissed from both sides. 482 13 Partial Derivatives EXAMPLE 2 Find the tangent plane to the sphere z 2 = 14 - x 2 - y 2 at (1, 2, 3). Solution Instead of z = 14 - x 2 - y 2 we have z = 14- x 2 - y 2 . At xo = 1, yo = 2 the height is now zo = 3. The surface is a sphere with radius 1/4. The only trouble from the square root is its derivatives: 2 2 - 1z = 2(- 2x) a z _ (- 2y) 2 ax ax 114 - x 2 - y 2 -y /14- 2- y At (1, 2) those slopes are - 4 and - S. The equation of the tangent plane is linear: z - 3 = - ½(x - 1) - 1(y - 2). I cannot resist improving the equation, by multiplying through by 3 and moving all terms to the left side: tangent plane to sphere: l(x - 1) + 2(y - 2) + 3(z - 3) = 0. (4) If mathematics is the "science of patterns," equation (4) is a prime candidate for study. The numbers 1, 2, 3 appear twice. The coordinates are (xo, Yo, zo) = (1, 2, 3). The normal vector is ii + 2j + 3k. The tangent equation is lx + 2y + 3z = 14. None of this can be an accident, but the square root of 14 - x 2 - y2 made a simple pattern look complicated. This square root is not necessary. Calculus offers a direct way to find dz/dx- implicit differentiation. Just differentiate every term as it stands: 2 y2 Z2 = 14 leads to 2x + 2z az/ax = 0 and 2y + 2z az/ay = 0. (5) Canceling the 2's, the derivatives on a sphere are - x/z and - y/z. Those are the same as in (3). The equation for the tangent plane has an extremely symmetric form: Z - Zo = (x - xo)- (y - yo) or xo(x - xo) + yo(y - yo) + zo(z - zo)=O. (6) Z0 Z0 Reading off N = xoi + yoj + zok from the last equation, calculus proves something we already knew: The normal vector to a sphere points outward along the radius. Z \ oj - zok 0 N = x0i +y( x Y 2 r x 2 + y2 _ 2 =1 X +y z 2 = -1 Fig. 13.8 Tangent plane and normal N for a sphere. Hyperboloids of 1 and 2 sheets. THE TANGENT PLANE TO F(x, y, z)= c The sphere suggests a question that is important for other surfaces. Suppose the equation is F(x, y, z) = c instead of z =f(x, y). Can the partial derivatives and tangent plane be found directly from F? The answer is yes. It is not necessary to solve first for z. The derivatives of F, 13.3 Tangent Planes and Linear Approximations 483 computed at (xo, Yo, zo), give a second formula for the tangent plane and normal vector. 13B The tangent plane to the surface F(x, y, z)= c has the linear equation (OF (x - X0) + ( (7 - F ) + OF (z - )= 0 (7) The normal vector is a- = N +( ij + ( k. (Tx o ayo (Tzo Notice how this includes the original case z =f(x, y). The function F becomes f(x, y) - z. Its partial derivatives are Of/Ox and Of/Oy and -1. (The -1 is from the derivative of - z.) Then equation (7)is the same as our original tangent equation (1). EXAMPLE 3 The surface F = x 2 + y 2 - z 2 = c is a hyperboloid.Find its tangent plane. Solution The partial derivatives are Fx = 2x, F, = 2y, Fz = - 2z. Equation (7) is tangent plane: 2xo(x - xo) + 2 yo(y - Yo) - 2zo(z - zo)= 0. (8) We can cancel the 2's. The normal vector is N = x 0 i + yoj - z0 k. For c > 0 this hyperboloid has one sheet (Figure 13.8). For c = 0 it is a cone and for c < 0 it breaks into two sheets (Problem 13.1.26). DIFFERENTIALS Come back to the linear equation z - zo = (Oz/Ox) 0(x - x0 ) + (Oz/Oy)o(y - Yo) for the tangent plane. That may be the most important formula in this chapter. Move along the tangent plane instead of the curved surface. Movements in the plane are dx and dy and dz-while Ax and Ay and Az are movements in the surface. The d's are governed by the tangent equation- the A's are governed by z =f(x, y). In Chapter 2 the d's were differentials along the tangent line: dy = (dy/dx)dx (straight line) and Ay, (dy/dx)Ax (on the curve). (9) Now y is independent like x. The dependent variable is z. The idea is the same. The distances x - x0 and y - yo and z - zo (on the tangent plane) are dx and dy and dz. The equation of the plane is dz = (Oz/Ox) 0dx + (Oz/Oy)ody or df=fxdx +fdy. (10) This is the total differential. All letters dz and df and dw can be used, but Oz and Of are not used. Differentials suggest small movements in x and y; then dz is the resulting movement in z. On the tangent plane, equation (10) holds exactly. A "centering transform" has put x0 , Yo, zo at the center of coordinates. Then the "zoom transform" stretches the surface into its tangent plane. EXAMPLE 4 The area of a triangle is A = lab sin 0. Find the total differential dA. Solution The base has length b and the sloping side has length a. The angle between them is 0. You may prefer A = ½bh, where h is the perpendicular height a sin 0. Either way we need the partial derivatives. If A = ½absin 0, then OA 1 OA 1 dA 1 -b sin0 - a sin 6 - ab cos 0. (11) Oa 2 Ob 2 06 2 484 13 Partial Derivatives These lead immediately to the total differential dA (like a product rule): (dAd (DAN (DAN 1 1 1 dA = Ida + I db + ± -ab dO= b sin 0 da + a sin 8 db + cos 8 dO. \Da/ \b 00 2 2 2 EXAMPLE 5 The volume of a cylinder is V = nr2 h. Decide whether V is more sensitive to a change from r = 1.0 to r = 1.1 or from h = 1.0 to h = 1.1. Solution The partial derivatives are V/Or = 2n7rh and DV/ah = irr2 . They measure the sensitivity to change. Physically, they are the side area and base area of the cylinder. The volume differential dV comes from a shell around the side plus a layer on top: dV = shell + layer = 2nrh dr + rr 2dh. (12) Starting from r = h = 1, that differential is dV= 2rndr + 7rdh. With dr = dh = .1, the shell volume is .21t and the layer volume is only .17r. So V is sensitive to dr. For a short cylinder like a penny, the layer has greater volume. Vis more sensitive to dh. In our case V= rTr 2h increases from n(1) 3 to ~n(1.1)3 . Compare AV to dV: AV= n(1.1) 3 - 7(1) 3 = .3317r and dV= 27r(.1)+ 7n(.1)= .3007r. The difference is AV- dV= .0317. The shell and layer missed a small volume in Figure 13.9, just above the shell and around the layer. The mistake is of order (dr)2 + (dh)2 . For V= 7rr 2 h, the differential dV= 27rrh dr + 7rr 2 dh is a linearapproxima- tion to the true change A V. We now explain that properly. LINEAR APPROXIMATION Tangents lead immediately to linear approximations. That is true of tangent planes as it was of tangent lines. The plane stays close to the surface, as the line stayed close to the curve. Linear functions are simpler than f(x) or f(x, y) or F(x, y, z). All we need are first derivatives at the point. Then the approximation is good near the point. This key idea of calculus is already present in differentials. On the plane, df equals fxdx +fydy. On the curved surface that is a linear approximation to Af: 43C The linear approximation to f(x, y) near the point (xo, Yo) is f(x, y) ýf(xo, Yo) + ( (x - Xo) + ( y(y - Yo). (13) In other words Af fxAx +fAy, as proved in Problem 24. The right side of (13) is a linear function fL(x, y). At (xo, yo), the functions f and fL have the same slopes. Then f(x, y) curves away fromfL with an error of "second order:" If(x, y) -fL(x, Y)I < M[(x - Xo) 2 + (y - yo) 2 ]. (14) This assumes thatfx,,,fx, and fy are continuous and bounded by M along the line from (xo, Yo) to (x, y). Example 3 of Section 13.5 shows that If,,I < 2M along that line. A factor ½ comes from equation 3.8.12, for the error f-fL with one variable. For the volume of a cylinder, r and h went from 1.0 to 1.1. The second derivatives of V = lrr 2 h are V, , = 27rh and Vh = 27rr and Vhh = 0. They are below M = 2.27r. Then (14) gives the error bound 2.27r(.1 2 + .12) = .0447r, not far above the actual error .03 17r. The main point is that the error in linear approximation comes from the quadratic terms-those are the first terms to be ignored by fL. 13.3 Tangent Planes and Linear Approximations 485 layer dh area n r 2 shell dr area 2nrh Fig. 13.9 Shiell plus layer gives d V = .300n. Fig. 13.10 Quantity Q and price P move with the lines. Including top ring gives A V = .33In. EXAMPLE 6 / Find a-linear approximation to the distance function r = , =. Solution The partial derivatives are x/r and ylr. Then Ar z(x/r)Ax + (y/r)Ay. For (x, y, r) near (1, 2, &): , /z ,/m - I)/& + 2(y - 2)/fi. , = + (x If y is fixed at 2, this is a one-variable approximation to d m . If x is fixed at 1, it is a linear approximation in y. Moving both variables, you might think dr would involve dx and dy in a square root. It doesn't. Distance involves x and y in a square root, but: change of distance is linear in Ax and Ay-to a first approximation. There is a rough point at x = 0, y = 0. Any movement from (0,O) gives Ar = J k(Ay)2.The square root has returned. The reason is that the partial deriva- m tives x/r and y/r are not continuous at (0,O). The cone has a sharp point with no tangent plane. Linear approximation breaks down. The next example shows how to approximate Az from Ax and Ay, when the equation is F(x, y, z) = c. We use the implicit derivatives in (7) instead of the explicit derivatives in (1). The idea is the same: Look at the tangent equation as a way to find Az, instead of an equation for z. Here is Example 6 with new letters. EXAMPLE 7 From F = - x2 - y2 + z2 = 0 find a linear approximation to Az Solution (implicit derivatives) Use the derivatives of F: - 2xAx - 2yAy + 2zAz z 0. Then solve for Az, which gives Az z (x/z)Ax + (y/z)Ay-the same as Example 6. EXAMPLE 8 How does the equilibrium price change when the supply curve changes? The equilibrium price is at the intersection of the supply and demand curves (supply =: demand). As the price p rises, the demand q drops (the slope is - .2): demand line DD: p = - .2q + 40. (15) The supply (also q) goes up with the price. The slope s is positive (here s = .4): supply line SS: p = sq + t = .4q + 10. Those lines are in Figure 13.10. They meet at the equilibrium price P = $30. The quantity Q = 50 is available at P (on SS) and demanded at P (on DD). So it is sold. Where do partial derivatives come in? The reality is that those lines DD and SS are not fixed for all time. Technology changes, and competition changes, and the value of money changes. Therefore the lines move. Therefore the crossing point (Q, P) also moves. Please recognize that derivatives are hiding in those sentences. 13 Partial Derivatives Main point: The equilibrium price P is a function of s and t. Reducing s by better technology lowers the supply line to p = .3q + 10. The demand line has not changed. The customer is as eager or stingy as ever. But the price P and quantity Q are different. The new equilibrium is at Q = 60 and P = $28, where the new line XX crosses DD. If the technology is expensive, the supplier will raise t when reducing s. Line YY is p = .3q + 20. That gives a higher equilibrium P = $32 at a lower quantity Q = 40- the demand was too weak for the technology. Calculus question Find dP/ds and aP/at. The difficulty is that P is not given as a function of s and t. So take implicit derivatives of the supply = demand equations: supply = demand: P = - .2Q + 40 = sQ + t (16) s derivative: P, = - .2Q, = sQ, + Q (note t, = 0) t derivative: P, = - .2Q, = sQ, + 1 (note t, = 1) Now substitute s = .4, t = 10, P = 30, Q = 50. That is the starting point, around which we are finding a linear approximation. The last two equations give P, = 5013 and P, = 113 (Problem 25). The linear approximation is Comment This example turned out to be subtle (so is economics). I hesitated before including it. The equations are linear and their derivatives are easy, but something in the problem is hard-there is no explicit formula for P. The function P(s, t) is not known. Instead of a point on a surface, we are following the intersection of two lines. The solution changes as the equation changes. The derivative of the solution comes from the derivative of the equation. Summary The foundation of this section is equation (1) for the tangent plane. Every- thing builds on that-total differential, linear approximation, sensitivity to small change. Later sections go on to the chain rule and "directional derivatives" and "gradients." The central idea of differential calculus is A z f,Ax +f,,Ay. f I N W O N ' S METHOD F O R M0 EQUATIONS Linear approximation is used to solve equations. To find out where a function is zero, look first to see where its approximation is zero. To find out where a graph crosses the xy plane, look to see where its tangent plane crosses. Remember Newton's method for f(x) = 0. The current guess is x,. Around that point, f(x) is close to f(x,) + (x - x,)f'(x,). This is zero at the next guess x,,, = x, -f(x,)/f'(x,). That is where the tangent line crosses the x axis. With two variables the idea is the same- but two unknowns x and y require two equations. We solve g(x, y) = 0 and h(x, y) = 0. Both functions have linear approxi- mations that start from the current point (x,, y,)-where derivatives are computed: The natural idea is to set these approximations to zero. That gives linear equations for x - x, and y - y,. Those are the steps Ax and Ay that take us to the next guess 13.3 Tangent Planes and Linear Approxlmations 487 in Newton's method: 13D Newton's method to solve g(x, y)= 0 and h(x, y)= 0 has linear equations for the steps Ax and Ay that go from (xe, yJ)to (x, + 1, y,, +1) Ax + Ay= -g(x, y.) and Ax + Ay= - h(x., yJ). (19) ax sy ~ ~ •/ •/ • ••• '!• • ! •/ 3D• • ,i•,//• • t •ii • s 1 • •• // •• • • ,!, (ax ~ • Q i•ii••!ii ,i~~i•li,,•i!• i •••,,ii,••i•i,•••i ~ ~ y • / _y _ •ii~~~ii,,!~ii iiii•,i••i•• ,,••• : h s •• •/ • ll~ /••/a ••q • a~on • x i · / _ ·iiiiii i iii Iii i i! iiii ¸ i ? in: ::(/ ii ii EXAMPLE 9 g = x 3 - y = 0 and h = y3 - x = 0 have 3 solutions (1, 1), (0, 0), (-1, -1). I will start at different points (xo, yo). The next guess is x, = xo + Ax, yl = Yo + Ay. It is of extreme interest to know which solution Newton's method will choose-if it converges at all. I made three small experiments. 1. Suppose (xo, yo) = (2, 1). At that point g = 2 - 1 = 7 and h = 13 - 2 = -1. The derivatives are gx = 3x2 = 12, gy = - 1, hx = - 1, hy = 3y 2 = 3. The steps Ax and Ay come from solving (19): 12Ax - Ay= -7 Ax = - 4/7 x = xo + Ax= 10/7 -Ax+3Ay= +1 Ay= + 1/7 = yo + Ay= 8/7. This new point (10/7, 8/7) is closer to the solution at (1, 1). The next point is (1.1, 1.05) and convergence is clear. Soon convergence is fast. 2. Start at (xo, Yo) = (½, 0). There we find g = 1/8 and h = - 1/2: (3/4)Ax - Ay= -1/8 Ax = - 1/2 x = xo + =Ax =0 - Ax + OAy= + 1/2 Ay = + 1/4 y, = yo + Ay = - 1/4. Newton has jumped from (½, on the x axis to (0, - f) on the y axis. The next step 0) goes to (1/32, 0), back on the x axis. We are in the "basin of attraction" of (0, 0). 3. Now start further out the axis at (1, 0), where g = 1 and h = - 1: 3Ax- Ay= -1 Ax= -1 x= xo+Ax=0O -Ax+OAy= +1 Ay=-2 yl=yo+Ay=-2. Newton moves from (1, 0) to (0, -2) to (16, 0). Convergence breaks down-the method blows up. This danger is ever-present, when we start far from a solution. Please recognize that even a small computer will uncover amazing patterns. It can start from hundreds of points (xo, Yo), and follow Newton's method. Each solution has a basin of attraction,containing all (xo, Yo) leading to that solution. There is also a basin leading to infinity. The basins in Figure 13.11 are completely mixed together- a color figure shows them asfractals.The most extreme behavior is on the borderline between basins, when Newton can't decide which way to go. Frequently we see chaos. Chaos is irregular movement that follows a definite rule. Newton's method deter- mines an iteration from each point (x,, y,) to the next. In scientific problems it normally converges to the solution we want. (We start close enough.) But the com- puter makes it posible to study iterations from faraway points. This has created a new part of mathematics-so new that any experiments you do are likely to be original. 488 13 Partial Derivatives Section 3.7 found chaos when trying to solve x 2 + 1 = 0. But don't think Newton's method is a failure. On the contrary, it is the best method to solve nonlinear equations. The error is squared as the algorithm converges, because linear approximations have errors of order (Ax) 2 + (Ay) 2 . Each step doubles the number of correct digits, near the solution. The example shows why it is important to be near. Fig. 13.11 The basins of attraction to (1, 1), (0, 0), (-1, -1), and infinity. 13.3 EXERCISES Read-through questions next point E . Each solution has a basin of F. Those basins are likely to be G The tangent line to y =f(x) is y - Yo = a . The tangent plane to w =f(x, y) is w - wo = b . The normal vector is In 1-8 find the tangent plane and the normal vector at P. N= c . For w = x 3 + y 3 the tangent equation at (1, 1, 2) 2 is d . The normal vector is N = .For a sphere, the 1 z= +y 2, P = (0, 1, 1) direction of N is f 2 x+y+z=17,P=(3, 4, 10) The surface given implicitly by F(x, y, z) = c has tangent 3 z = x/y, P = (6, 3, 2) equation (OF/Ox)o(x - xo) + g . For xyz = 6 at (1, 2, 3) 4 z = ex +2, P = (0, 0, 1) the tangent plane is h . On that plane the differentials satisfy I dx + i dy + k dz = 0. The differential 5 X2 + y 2 + Z2 = 6, P = (1, 2, 1) of z =f(x, y) is dz = I . This holds exactly on the tangent = 6 x 2 + y2 + 2Z2 7, P = (1, 2, 1) plane, while Az m m holds approximately on the n y The height z = 3x + 7y is more sensitive to a change in 0 7 z = x , P = (1, 1, 1) than in x, because the partial derivative P is larger than 8 V = r 2 h, P= (2, 2, 87x). 9 Show that the tangent plane to z 2 -_x2 -y 2 =0 goes The linear approximation to f(x, y) is f(xo, Yo) + r through the origin and makes a 450 angle with the z axis. This is the same as Af s Ax + t Ay. The error is of order u . For f= sin xy the linear approximation 10 The planes z = x + 4y and z = 2x + 3y meet at (1, 1, 5). around (0, 0) is fL = v . We are moving along the w The whole line of intersection is (x, y, z) = (1, 1, 5) + vt. instead of the x . When the equation is given as Find v= N1 x N 2. F(x, y, z) = c, the linear approximation is Y Ax + 11 If z = 3x - 2y find dz from dx and dy. If z = x31y 2 find dz z Ay + A Az = 0. from dx and dy at xo = 1, yo = 1. If x moves to 1.02 and y Newton's method solves g(x, y)= 0 and h(x, y)= 0 by a moves to 1.03, find the approximate dz and exact Az for both B approximation. Starting from x,, y, the equations are functions. The first surface is the to the second replaced by c and D . The steps Ax and Ay go to the surface. 13.3 Tangent Planes and Linear Approximations 489 12 The surfaces z = x2 + 4y and z = 2x + 3y2 meet at (1, 1, 5). 1 (3) a = f x k yo -+fx(xo,YO) provided fx is - Find the normals N, and N, and also v = N, x N,. The line in this direction v is tangent to what curve? + (4) b =fy(xo Ax, C) -+fy(xo,yo) provided f, is . 25 If the supplier reduces s, Figure 13.10 shows that P 13 The normal N to the surface F(x, y, z) = 0 has components decreases and Q . F,, F,, F,. The normal line has x = xo + Fxt, y = yo + F,t, (a) Find P, = 5013 and P, = 113 in the economics equation z= . For the surface xyz - 24 = 0, find the tangent (17) by solving the equations above it for Q, and Q,. plane and normal line at (4, 2, 3). (b) What is the linear approximation to Q around s = .4, 14 For the surface x2y2-- z = 0, the normal line at (1, 2,4) t = 10, P = 30, Q = 50? hasx= ,y= ,z= . 26 Solve the equations P = - .2Q + 40 and P = sQ + t for P 15 For the sphere x2 + y'' + z2 = 9, find the equation of the and Q. Then find aP/as and aP/dt explicitly. At the same tangent plane through (2, 1,2). Also find the equation of the s, t, P, Q check 5013 and 113. normal line and show that it goes through (O,0,0). 27 If the supply = demand equation (16) changes to P = 16 If the normal line at every point on F(x, y, z) = 0 goes s Q + t = - Q + 5 0 , find P, and P, at s = 1, t = 10. through (0, 0, 0), show that Fx= cx, F, = cy, F, = cz. The sur- face must be a sphere. 28 To find out how the roots of x2 + bx + c = 0 vary with b, 17 For w = xy near (x,, y,,), the linear approximation is dw = take partial derivatives of the equation with respect to . This looks like the rule for derivatives. . Compare axlab with ax/ac to show that a root at The difference between Aw = xy - xoyo and this approxima- x = 2 is more sensitive to b. tion is . 29 Find the tangent planes to z = xy and z = x2 - y2 at x = 18 Iff = xyz (3 independent variables) what is df? 2, y = 1. Find the Newton point where those planes meet the xy plane (set z = 0 in the tangent equations). 19 You invest P = $4000 at R = 8% to make I = $320 per year. If the numbers chan,ge by dP and dR what is dl? If the 30 (a) To solve g(x, y) = 0 and h(x, y) = 0 is to find the meeting rate drops by dR = .002 (to 7.8%) what change dP keeps d l = point of three surfaces: z = g(x, y) and z = h(x, y) and ? O Find the exact interest I after those changes in R and P. (b) Newton finds the meeting point of three planes: the 20 Resistances R, and R:! have parallel resistance R, where tangent plane to the graph of g, , and . 1/R = 1/R, + 1/R2. Is R more sensitive to AR, or AR, if R, = 1 and R, = 2? (a) If your batting average is A = (25 hits)/(100 at bats) = Problems 31-36 go further with Newton's method for g = x3 - y and h = Y3 - X. This is Example 9 with solutions (1, I), .250, compute the increase (to 261101) with a hit and the decrease (to 251101) w:ith an out. (0, 01, (-1, -1). (b) If A = xly then dA == dx + dy. A hit 31 Start from xo = 1, yo = 1 and find Ax and Ay. Where are (dx = dy = 1) gives dA = (1 - A)/y. An out (dy = 1) gives x, and y,, and what line is Newton's method moving on? dA = - Aly. So at A ==.250 a hit has times the effect of an out. 32 Start from (3,i) and find the next point. This is in the basin of attraction of which solution? (a) 2 hits and 3 outs (dx = 2, dy = 5) will raise your average (dA > 0) provided A is less than . 33 Starting from (a, -a) find Ay which is also -Ax. Newton (b)A player batting A = .500 with y = 400 at bats needs goes toward (0, 0). But can you find the sharp point in dx = hits to raise his average to .505. Figure 13.11 where the lemon meets the spade? If x and y change by Ax and Ay, find the approximate 34 Starting from (a, 0) show that Newton's method goes to change A0 in the angle 8 == tan - '(y/x). (0, -2a3) and find the next point (x,, y,). Which numbers a lead to convergence? Which special number a leads to a cycle, 24 The Fundamental Lernma behind equation (13) writes in which (x2, y2) is the same as the starting point (a, O)? A = aAx + bAy. The Lernma says that a +fx(xo, yo) and f b +fy(xo,yo) when Ax + 0 and Ay + 0. The proof takes A.x 35 Show that x3 = y, y3 = x has exactly three solutions. first and then Ay: 36 Locate a point from which Newton's method diverges. (l)f(xo + Ax, yo) -f(x,, yo) = Axfx(c, yo) where c is between and (by which theorem?) 37 Apply Newton's method to a linear problem: g = (2)f(xo + Ax, Yo + AY)--f(x0 + Ax, yo) = Ayf,(xo + Ax, C ) + x 2y - 5 = 0, h = 3x - 3 = 0. From any starting point show where C is between and . that (x,, y,) is the exact solution (convergence in one step). 490 13 Partial Derivatives 38 The complex equation (x + i ~= )1 contains two real equ- ~ 41 The matrix in Newton's method is the Jacobian: ations, x3 - 3xy2 = 1 from the real part and 3x2y - y 3 = 0 from the imaginary part. Search by computer for the basins of attraction of the three solutions (1, O), (- 112, fi/2), and (- 112, - &2)-which give the cube roots of 1. Find J and Ax and Ay for g = ex - 1, h = eY+ x. 42 Find the Jacobian matrix at (1, 1) when g = x2 + y2 and 39 In Newton's method the new guess comes from (x,, y,) by h = xy. This matrix is and Newton's method fails. , an iteration: x, + = G(x,, y,) and y, + = H(x,, y,). What are The graphs of g and h have tangent planes. G and H f o r g = x 2 - y = O , h = x - y = O ? First find Ax and + + Ay; then x, Ax gives G and y, Ay gives H. + 43 Solve g = x2 - y2 1 = 0 and h = 2xy = 0 by Newton's method from three starting points: (0, 2) and (- 1, 1) and (2,O). Take ten steps by computer or one by hand. The solution 40 In Problem 39 find the basins of attraction of the solution (0, 1) attracts when yo > 0. If yo = 0 you should find the chaos (0, 0) and (1, 1). iteration x, + = 4(xn- xn- I). 13.4 Directional Derivatives and Gradients As x changes, we know how f(x, y) changes. The partial derivative dfldx treats y as constant. Similarly df/dy keeps x constant, and gives the slope in the y direction. But east-west and north-south are not the only directions to move. We could go along a 45" line, where Ax = Ay. In principle, before we draw axes, no direction is preferred. The graph is a surface with slopes in all directions. On that surface, calculus looks for the rate of change (or the slope). There is a directional derivative, whatever the direction. In the 45" case we are inclined to divide f A by Ax, but we would be wrong. Let me state the problem. We are given f(x, y) around a point P = (x,, yo). We are also given a direction u (a unit vector). There must be a natural definition of D,f- the derivative off in the direction u. To compute this slope at P, we need a formula. Preferably the formula is based on df/dx and dfldy, which we already know. Note that the 45" direction has u = i/$ + j/$. The square root of 2 is going to f enter the derivative. This shows that dividing A by Ax is wrong. We should divide by the step length As. EXAMPLE 1 Stay on the surface z = xy. When (x, y) moves a distance As in the 45" direction from (1, I), what is Az/As? Solution The step is As times the unit vector u. Starting from x = y = 1 the step ends at x = y = 1 + AS/$. (The components of "As are AS/$.) Then z = xy is r = (1 + ~ s / f i )= 1 + $AS ~ + %As)', which means Az = $AS + $(As)2. The ratio AzlAs approaches fi as As + 0. That is the slope in the 45" direction. DEFINITION The derivative off'in the direction u at the point P is D,f ( P ) : The step from P = (x,, yo) has length As. It takes us to (x, + ulAs, yo + u2As). We f compute the change A and divide by As. But formula (2) below saves time. 13.4 Directional Derivatives and Gradients 491 The x direction is u = (1, 0). Then uAs is (As, 0) and we recover af/ax: Af f(xo + As, Yo) -f(xo, Yo) approaches D(1, 0 )f As As ax Similarly Df= aflay, when u = (0, 1) is in the y direction. What is D,f when u= (0, -1)? That is the negative y direction, so Df= - aflay. CALCULATING THE DIRECTIONAL DERIVATIVE D,f is the slope of the surface z =f(x, y) in the direction u. How do you compute it? From af/ax and af/ay, in two special directions, there is a quick way to find Df in all directions. Remember that u is a unit vector. 13E The directionalderivative D,f in the direction u = (u1 , u 2) equals af ,f Df= - ua+ - u 2 . (2) The reasoning goes back to the linear approximation of Af: Af4Ax+f f Af" Ax + Ay= ulAs+ u2 As. ax ay ax ay Divide by As and let As approach zero. Formula (2) is the limit of Af/As, as the approximation becomes exact. A more careful argument guarantees this limit pro- vided f and fy are continuous at the basepoint (xo,Yo). Main point: Slopes in all directions are known from slopes in two directions. EXAMPLE 1 (repeated) f= xy and P = (1,1)and u = (1/,i, 1//-2). Find Df(P). The derivatives f = y and fy = x equal 1 at P. The 450 derivative is D.f(P) =fuI +fyu 2= 1(1/./) + 1(1//2) = /2 as before. EXAMPLE 2 The linear function f= 3x + y + 1 has slope Df= 3u, + u2 . The x direction is u = (1, 0), and D.f= 3. That is af/ax. In the y direction Df= 1. Two other directions are special--along the level lines off(x, y) and perpendicular: Level direction: D.f is zero because f is constant Steepest direction: D.f is as large as possible (with u2 + u2 = 1). To find those directions, look at D,f= 3u, + u2 . The level direction has 3u, + u2 = 0. Then u is proportional to (1, - 3). Changing x by 1 and y by - 3 produces no change in f= 3x + y + 1. In the steepest direction u is proportional to (3, 1). Note the partial derivatives f = 3 and fy = 1. The dot product of (3, 1) and (1, -3) is zero-steepest direction is perpendicular to level direction.To make (3, 1) a unit vector, divide by 1/0. Steepest climb: D,f= 3(3/_0) + l(1//10) = 10//10 = /10 Steepest descent: Reverse to u= (-3//10, -1//10) and Df= -/10. The contour lines around a mountain follow Df= 0. The creeks are perpendicular. On a plane like f= 3x + y + 1, those directions stay the same at all points (Figure 13.12). On a mountain the steepest direction changes as the slopes change. 492 13 Partial Derivatives \| \| , = 'A _lr ,n Y i level direction O,I/'1-0) y steep Du ion n 3U 1 t U2 -U Fig. 13.12 Steepest direction is along the gradient. Level direction is perpendicular. THE GRADIENT VECTOR Look again at ful +fu 2 , which is the directional derivative Duf. This is the dot product of two vectors. One vector is u = (u1 , u2 ), which sets the direction. The other vector is (f,,f,), which comes from the function. This second vector is the gradient. af aT DEFINITION The gradient off(x, y) is the vector whose components are and . ax Oy grad f f=Vf 8af i + 83ff j add kf k in three dimensions . The space-saving symbol V is read as "grad." In Chapter 15 it becomes "del." For the linear function 3x + y + 1, the gradient is the constant vector (3, 1). It is the way to climb the plane. For the nonlinear function x 2 + xy, the gradient is the non-constant vector (2x + y, x). Notice that gradf shares the two derivatives in N = (f£,fy, -1). But the gradient is not the normal vector. N is in three dimensions, pointing away from the surface z =f(x, y). The gradient vector is in the xy plane! The gradient tells which way on the surface is up, but it does that from down in the base. The level curve is also in the xy plane, perpendicular to the gradient. The contour map is a projection on the base plane of what the hiker sees on the mountain. The vector grad f tells the direction of climb, and its length Igradfl gives the steepness. 13F The directional derivative is Df= (grad f) u. The level direction is per- pendicular to gradf, since D,f= 0. The slope Df is largest when u is parallel to gradf. That maximum slope is the length Igradfl = Xf +fy: grad f Igradf 12 for u grad f the slope is (gradf)u- gradf Igradfl. Igrad fl jgradfl The example f= 3x + y + 1 had grad f= (3, 1). Its steepest slope was in the direc- tion u = (3, 1)/!10. The maximum slope was F10. That is Igradf I= S + 1. Important point: The maximum of (gradf) u is the length Igradf1.In nonlinear examples, the gradient and steepest direction and slope will vary. But look at one particular point in Figure 13.13. Near that point, and near any point, the linear picture takes over. On the graph off, the special vectors are the level direction L = (fy, -fx, 0) and the uphill direction U = (,,f x +f 2) and the normal N = (f,fy, - 1). Problem 18 checks that those are perpendicular. 13.4 Directional Derivatives and Gradients EXAMPLE 3 The gradient of f(x, y) = (14 - x2 - y2)/3 is Vf = (- 2x13, - 2~13). On the surface, the normal vector is N = (- 2x13, - 2~13, 1). At the point (1,2, 3), - this perpendicular is N = (- 213, - 413, - 1). At the point (1, 2) down in the base, the gradient is (- 213, - 413). The length of grad f is the slo e ,/%/3. Probably a hiker does not go straight up. A "grade" of &/3 is fairly steep (almost 150%). To estimate the slope in other directions, measure the distance along the path between two contour lines. If A = 1 in a distance As = 3 the slope is about 113. This f calculation is not exact until the limit of AflAs, which is DJ vel Fig. 13.13 N perpendicular to surface and grad f perpendicular to level line (in the base). EXAMPLE 4 The gradient of f(x, y, z) = xy + yz + xz has three components. The pattern extends fromf(x, y) tof(x, y, z). The gradient is now the three-dimensional vector ( j ; , fy ,f,). For this function grad f is (y + z, x + z, x + y). To draw the graph of w =f(x, y, z) would require a four-dimensional picture, with axes in the xyzw directions. Notice: the dimensions. The graph is a 3-dimensional "surface" in 4-dimensional space. The gradient is down below in the 3-dimensional base. The level sets off come from xy -tyz + zx = c-they are 2-dimensional. The gradient is perpendicular to that level set (still down in 3 dimensions). The gradient is not N! The normal vector is (fx ,fy ,fz :, - I), perpendicular to the surface up in 4-dimensional space. EXAMPLE!5 + Find grad z when z(x, y) is given implicitly: F(x, y, z) = x2 y2 - z2 = 0. In this case we find z = f Jm. The derivatives are & and f y/,/? + y2,which go into grad z. But the point is this: To find that gradient faster, differentiate F(x, y, z) as it stands. Then divide by F,: The gradient is (- Fx/Fz, - Fy/F,). Those derivatives are evaluated at (xo, yo). The computation does not need the explicit function z =f(x, y): F = x2 + y2 - z2 =. Fx = 2x, Fy = 2y, Fz = - 2z grad z = (xlz, ylz). To go uphill on the cone, move in. the direction (xlz, ylz). That gradient direction goes radially outward. The steepness of the cone is the length of the gradient vector: lgrad zl = J(x/z)~ + ( y l ~= 1 because z2 = x2 + y2 on the cone. )~ 13 Partial Derivatives DERIVATIVES ALONG CURVED PATHS On a straight path the derivative off is D, = (gradf ) u. What is the derivative on f a curved path? The path direction u is the tangent vector T. So replace u by T, which gives the "direction" of the curve. The path is given by the position vector R(t) = x(t)i + y(t)j. The velocity is v = (dx/dt)i + (dy/dt)j. The tangent vector is T = vllvl. Notice the choice-to move at any speed (with v) or to go at unit speed (with T). There is the same choice for the derivative of.f(x, y) along this curve: df afdx rateofchange --(gradf)v=--+-- af dy dt ax dt ay dt df slope -=(gradf)T=--+--dx af af dy ds ax ds ay ds The first involves time. If we move faster, dfldt increases. The second involves distance. If we move a distance ds, at any speed, the function changes by df. So the slope in that direction is dflds. Chapter 1 introduced velocity as dfldt and slope as dyldx and mixed them up. Finally we see the difference. Uniform motion on a straight line has R = R, + vt. The velocity v is constant. The direction T = u = vllvl is also constant. The directional derivative is (grad f ) u, but the rate of change is (grad f ) v. Equations (4) and (5) look like chain rules. They are chain rules. The next section extends dfldt = (df/dx)(dx/dt) to more variables, proving (4) and (5). Here we focus on the meaning: dflds is the derivative off in the direction u = T along the curve. EXAMPLE 7 Find dfldt and dflds for f = r. The curve is x = t2, y = t in Figure 13.14a. Solution The velocity along the curve is v = 2ti + j. At the typical point t = 1 it is v = 2i + j. The unit tangent is T = v/&. The gradient is a unit vector i l f i j / f i + pointing outward, when f (x, y) is the distance r from the center. The dot product with v is dfldt = 3 / d . The dot product with T is dflds = 3 / a . When we slow down to speed 1 (with T), the changes in f(x, y) slow down too. EXAMPLE 8 Find dflds for f = xy along the circular path x = cos t, y = sin t. First take a direct approach. On the circle, xy equals (cos t)(sin t).Its derivative comes from the product rule: dfldt = cos2t - sin2t. Normally this is different from dflds, because the time t need not equal the arc length s. There is a speed factor dsldt to divide by-but here the speed is 1. (A circle of length s = 2 1 is completed at t = 2n.) 7 Thus the slope dflds along the roller-coaster in Figure 13.14 is cos2t - sin2t. A D= distance to (xo,yo) Fig. 13.14 The distance f = r changes along the curve. The slope of the roller-coaster is (grad f ) T. The distance D from (x,, y o ) has grad D = unit vector. 13.4 Directional DerhrcrHves and Gradients The second approach uses the vectors grad f and T. The gradient off = xy is (y, x) = (sin t, cos t). The unit tangent vector to the path is T = (- sin t, cos t). Their dot product is the same dflds: slope along path = (grad f ) T = - sin2t + cos2t. R DE T I H U G A I N S WT O T COORDINAJES This section ends with a little "philosophy." What is the coordinate-free dejnition of the gradient? Up to now, grad f = (fx,f,,) depended totally on the choice of x and y axes. But the steepness of a surface is independent of the axes. Those are added later, to help us compute. The steepness dflds involves only f and the direction, nothing else. The gradient should be a "tensorw-its meaning does not depend on the coordinate system. The gradient has different formulas in different systems (xy or re or ...), but the direction and length of gradf are determined by dflds-without any axes: The drrection of grad f is the one in which dflds is largest. The length Igrad f 1 is that largest slope. The key equation is (change inf ) x (gradient off) (changein position). That is another way to write Af x fxAx +@y. It is the multivariable form-we used two variables- of the basic linear approximation Ay x (dy/dx)Ax. EXAMPLE 9 D(x, y) = distance from (x, y) to (x,, yo). Without derivatives prove lgrad Dl = 1. The graph of D(x, y) is a cone with slope 1 and sharp point (x,, yo). First question In which direction does the distance D(x, y) increase fastest? Answer Going directly away from (x,, yo). Therefore this is the direction of grad D. Second question How quickly does D increase in that steepest direction? ~nswer A step of length As increases D by As. Therefore ]grad Dl = AslAs = 1. Conclusion grad D is a unit vector. The derivatives of D in Problem 48 are (x - xo)/D and (y - yo)/D. The sum of their squares is 1, because (x - x,)~+ (y - yo)equals D ~ . 13.4 EXERCISES Read-through questions The gradient of f(x, y, z) is s . This is different from the gradient on the surface F(x, y, z) = 0, which is -(F,/F,)i + D,f gives the rate of change of a in the direction b . . - Traveling with velocity v on a curved path, the rate t It can be computed from the two derivatives c in the of change off is dfldt = u . When the tangent direction special directions d . In terms of u,, u2 the formula is D,f = e . This is a f product of u with the vector is T, the slope off is dflds = v . In a straight direction u, ' dflds is the same as w . g , which is called the h . For the linear functionf = ax + by, the gradient is gradf = 1 the directional and Compute . then Du = (gradf ) u, then Du at PP. f f derivative is D,f = i k . 1 f(x, y) = x2 - y2 u = (&2, 112) P = (1, 0) The gradient V = (fx,f,) is not a vector in I dimen- f sions, it is a vector in the m . It is perpendicular to the 2 f(x, y) = 3x + 4y + 7 u = (315, 415) P = (0, 7112) n lines. It points in the direction of o climb. Its 3 f(x, y) = ex cos y magnitude Igrad f ( is P . For f = x2 + y2 the gradient points q and the slope in that steepest direction is r . 4 f(x, Y)=Y'O u=(O, -1) P = ( l , -1) 5 f(x, y) = distance to (0, 3) u = (1, 0) P = (1, 1) 20 Compute N, U, L for x2 + y2 - z2 = 0 and draw them at a typical point on the cone. Find grad f = (f,, fy,f,) for the functions 6 8 from physics. 6 1/Jx2 + y2 + z2 (point source at the origin) With gravity in the negative z direction, in what direction - U will water flow down the roofs 21-24? 7 ln(x2+ y2) (line source along z axis) 21 z = 2x (flat roof) 22 z = 4x - 3y (flat roof) 8 l/J(x - + y2 + z2 - l/J(x + + y2 + z2 (dipole) 9 For f = 3x2 + 2y2 find the steepest direction and the level 23 z = / ,- (sphere) 24 z = - ,/= (cone) direction at (1,2). Compute D, f in those directions. 25 Choose two functions f(x, y) that depend only on x + 2y. Their gradients at (1, 1) are in the direction . Their 10 Example 2 claimed that f = 3x + y + 1 has steepest slope level curves are Maximize Duf = 3u1 + u2 = 3ul +,/-. 26 The level curve off = y/x through (1, 1) is . The 11 True or false, when f(x, y) is any smooth function: direction of the gradient must be . Check grad f. (a) There is a direction u at P in which D, f = 0. (b) There is a direction u in which D, f = gradf: 27 Grad f is perpendicular to 2i + j with length 1, and grad g is parallel to 2i +j with length 5. Find gradf, grad g,f, and g. (c) There is a direction u in which D, f = 1. + (d) The gradient of f(x)g(x) equals g grad f f grad g. 28 True or false: (a) If we know gradf, we know f: 12 What is the gradient of f(x)? (One component only.) What (b) The line x = y = - z is perpendicular to the plane z = are the two possible directions u and the derivatives Duf ? What is the normal vector N to the curve y=f(x)? (Two + x y. components.) (c) The gradient of z = x + y lies along that line. 29 Write down the level direction u for 8 = tan-'(ylx) at the In 13-16 find the direction u in whichf increases fastest at P = point (3,4). Then compute grad 8 and check DUB 0. = (1, 2). How fast? 30 On a circle around the origin, distance is As = rAO. Then 13 f(x, y) = ax + by 14 f(x, y) = smaller of 2x and y dO/ds= llr. Verify by computing grad 8 and T and (grad 8) T. 15 f(x, y) = ex-Y 16 fix, y) = J5 - x2 - y2 (careful) 31 At the point (2, 1,6) on the mountain z = 9 - x - y2, 17 (Looking ahead) At a point where f(x, y) is a maximum, which way is up? On the roof z = x + 2y + 2, which way is what is grad f ? Describe the level curve containing the maxi- down? The roof is to the mountain. mum point (x, y). 32 Around the point (1, -2) the temperature T = e-"-y2 has 18 (a) Check by dot products that the normal and uphill and level directions on the graph are perpendicular: N = AT z AX + Ay. In what direction u does it get hot fastest? (fxyfy, - 1 ) J =(fx,fy,fx2 +f:W =(fy, -fx, 0). (b) N is to the tangent plane, U and L are 33 Figure A shows level curves of z =f(x, y). to the tangent plane. (a) Estimate the direction and length of grad f at P, Q, R. (c) The gradient is the xy projection of and also (b) Locate two points where grad f is parallel to i + j. of . The projection of L points along the (c) Where is Igrad f ( largest? Where is it smallest? (d) What is your estimate of, ,z on this figure? , 19 Compute the N, U, L vectors for f = 1 - x + y and draw (e) On the straight line from P to R, describe z and esti- them at a point on the flat surface. mate its maximum. 13.5 The Chain Rule + + + 34 A quadratic function ax2 by2 cx dy has the gradi- 42 f = x x = cos 2t y = sin 2t ents shown in Figure B. Estimate a, b, c, d and sketch two level curves. 43 f = x 2 - y 2 x=xo+2t y=yo+3t 35 The level curves of f(x, y) are circles around (1, 1). The 44 f = x y x=t2+1 y=3 curve f = c has radius 2c. What is f ? What is grad f at (0, O)? 45 f = l n xyz x = e' y = e2' = e-' 36 Suppose grad f is tangent to the hyperbolas xy = constant 46 f=2x2+3y2+z2 x = t y=t2 Z=t3 in Figure C. Draw three level curves off(x, y). Is lgrad f 1 larger 47 (a) Find df/ds and df/dt for the roller-coasterf = xy along at P or Q? Is lgrad f 1 constant along the hyperbolas? Choose + a function that could bef: x2 y2, x2 - y2, xy, x2y2. the path x = cos 2t, y = sin 2t. (b) Change to f = x2 + y2 and explain why the slope is zero. 37 Repeat Problem 36, if grad f is perpendicular to the hyper- 48 The distance D from (x, y) to (1, 2) has D2 = bolas in Figure C. (x - + (y - 2)2. Show that aD/ax = (X- l)/D and dD/ay = 38 Iff = 0, 1, 2 at the points (0, I), (1, O), (2, I), estimate grad f (y - 2)/D and [gradDl = 1. The graph of D(x, y) is a + by assumingf = Ax By + C. with its vertex at . 39 What functions have the following gradients? 49 Iff = 1 and grad f = (2, 3) at the point (4, 5), find the tan- + (a) (2x y, x) (b) (ex- Y,- ex- Y, (c) (y, -x) (careful) gent plane at (4, 5). Iff is a linear function, find f(x, y). 40 Draw level curves of f(x, y) if grad f = (y, x). 50 Define the derivative of f(x, y) in the direction u = (ul, u2) f at the point P = (x,, yo). What is A (approximately)? What In 41-46 find the velocity v and the tangent vector T. Then is D, f (exactly)? compute the rate of change df/dt = grad f v and the slope 51 The slope off along a level curve is dflds = = 0. df/ds = grad f T. This says that grad f is perpendicular to the vector in the level direction. 13.5 The Chain Rule Calculus goes back and forth between solving problems and getting ready for harder problems. The first is "application," the second looks like "theory." If we minimizef to save time or money or energy, that is an application. If we don't take derivatives to find the minimum-maybe because f is a function of other functions, and we don't have a chain rule-then it is time for more theory. The chain rule is a fundamental working tool, because f(g(x)) appears all the time in applications. So do f(g(x, y)) and f(x(t), y(t)) and worse. We have to know their derivatives. Otherwise calculus can't continue with the applications. You may instinctively say: Don't bother with the theory, just teach me the formulas. That is not possible. You now regard the derivative of sin 2x as a trivial problem, unworthy of an answer. That was not always so. Before the chain rule, the slopes of sin 2x and sin x2 and sin2x2were hard to compute from Af/Ax. We are now at the same point for f(x, y). We know the meaning of dfldx, but iff = r tan B and x = r cos 8 and y = r sin 8, we need a way to compute afldx. A little theory is unavoidable, if the problem-solving part of calculus is to keep going. To repeat: The chain rule applies to a function of a function. In one variable that was f(g(x)). With two variables there are more possibilities: 1. f ( ~ ) withz=g(x,y) Find df/dx and afldy 2. f(x, y) with x = x(t), y = y(t) Find dfldt 3. f(x, y) with x = x(t, u), y = y(t, u) Find dfldt and afldu 13 Partial Derhrattves All derivatives are assumed continuous. More exactly, the input derivatives like ag/ax and dxldt and dx/au are continuous. Then the output derivatives like af/ax and dfldt and df/au will be continuous from the chain rule. We avoid points like r = 0 in polar coordinates-where ar/dx = x/r has a division by zero. A Typical Problem Start with a function of x and y, for example x times y. Thus f(x, y) = xy. Change x to r cos 8 and y to r sin 8. The function becomes (r cos 8) times (r sin 8). We want its derivatives with respect to r and 8. First we have to decide on its name. To be correct, we should not reuse the letter5 The new function can be F : f(x, y) = x y f(r cos 8, r sin 8) = (r cos 8)(r sin 8) = F(r, 8). W h y not call it f(r, 8)? Because strictly speaking that is r times 8! If we follow the rules, then f(x, y) is x y and f(r, 8) should be re. The new function F does the right thing-it multiplies (r cos 8)(r sin 8). But in many cases, the rules get bent and the letter F is changed back to 5 This crime has already occurred. The end of the last page ought to say dFlat. Instead the printer put dfldt. The purpose of the chain rule is to find derivatives in the new variables t and u (or r and 8). In our example we want the derivative of F with respect to r. Here is the chain rule: d~ - d ----f a x + g? = (y)(cos8) + (x)(sin8) = 2r sin 8 cos 8. dr dx ar dyer I substituted r sin 8 and r cos 8 for y and x. You immediately check the answer: F(r, 8) = r2 cos 8 sin 8 does lead to ZF/dr = 2r cos 8 sin 8. The derivative is correct. The only incorrect thing-but we do it anyway-is to write f instead of F. af ae + --. ae Question What is -? af ax Answer It is -- af ay ae ax ay THE DERIVATIVES O f(g(x, y)) F Here g depends on x and y, and f depends on g. Suppose x moves by dx, while y stays constant. Then g moves by dg = (ag/ax)dx. When g changes, f also changes: T df = (df/dg)dg. Now substitute for dg to make the chain: df = (df/dg)(ag/dx)dx. his is the first rule: J l o c r ?f 8f df dg and -=--dfag 13G C a i rulefovf(g(x, y)): - = -- (11 dx dgdic a~ dg ad* EXAMPLE 1 Every f ( x + cy) satisfies the l-way wave equation df/ay = c af/ax. The inside function is g = x + cy. The outside function can be anything, g2 or sin g or eg. The composite function is ( x + cy)2 or sin(x + cy) or ex+cy. In each separate case we could check that df/dy = c dfldx. The chain rule produces this equation in all cases at once, from aglax = 1 and i?g/ay = c: This is important: af/ay = c afldx is our first example of a partial dierential equation. The unknown f(x, y) has two variables. Two partial derivatives enter the equation. 13.5 The Chain Rule Up to now we have worked with dyldt and ordinary di$ercntial equations. The independent variable was time or space (and only one dimension in space). For partial differential equations the variables are time and space (possibly several dimensions in space). The great equations of mathematical physics-heat equation, wave equa- tion, Laplace's equation-are partial differential equations. Notice how the chain rule applies to f = sin xy. Its x derivative is y cos xy. A patient reader would check that f is sing and g is xy and f, is &g,. Probably you are not so patient-you know the derivative of sin xy. Therefore we pass quickly to the next chain rule. Its outside function depends on two inside functions, and each of those depends on t. We want dfldt. F T E DERIVATIVE O f(x(t), y(t)) H Before the formula, here is the idea. Suppose t changes by At. That affects x and y; $ they change by Ax and Ay. There is a domino effect onfi it changes by A Tracing backwards, A f z d f ~ x + - af y A dx and Ax=-At d~ and Ayz-At. ax dy dt dt f Substitute the last two into the first, connecting A to At. Then let At - 0: , This is close to the one-variable rule dzldx = (dz/dy)(dy/dx).There we could "cancel" dy. (We actually canceled Ay in (Az/Ay)(Ay/Ax), and then approached the limit.) f Now At affects A in two ways, through x and through y. The chain rule has two terms. If we cancel in (af/ax)(dx/dt) we only get one of the terms! We mention again that the true name for f(x(t), y(t)) is F(t) not f(t). For f(x, y, z) the rule has three terms: fxx, +fyyt +fiz, isf, (or better dF/dt). EXAMPLE 2 How quickly does the temperature change when you drive to Florida? Suppose the Midwest is at 30°F and Florida is at 80°F. Going 1000 miles south increases the temperaturef(x, y) by 50°, or .05 degrees per mile. Driving straight south at 70 miles per hour, the rate of increase is (.05)(70)= 3.5 degrees per hour. Note how (degreeslmile) times (miles/hour)equals (degrees/hour). That is the ordinary chain rule (df/dx)(dx/dt) = (df/dt)- there is no y variable going south. If the road goes southeast, the temperature is f = 30 + .05x + .Oly. Now x(t) is distance south and y(t) is distance east. What is dfldt if the speed is still 70? df af dx af dy + 70 Solution - = -- -- - .05 -+ .01- 70 z 3 degrees/hour. dt ax dt ay dt Ji Ji In reality there is another term. The temperature also depends directly on t, because of night and day. The factor cos(2?ct/24)has a period of 24 hours, and it brings an extra term into the chain rule: df af dx af dy af For f(x, y, t) the chain rule is - = -- +--+-. dt ax dt ay dt at This is the total derivative dfldt, from all causes. Changes in x, y, t all affect J The partial derivative af/dt is only one part of dfldt. (Note that dtldt = 1.) If night and 13 Partlal Derivatives day add 12 cos(2nt/24)tof, the extra term is df/at = - n sin(2nt124). At nightfall that is - n degrees per hour. You have to drive faster than 70 mph to get warm. SECOND DERIVATIVES What is d2f/dt2? We need the derivative of (4), which is painful. It is like acceleration in Chapter 12, with many terms. So start with movement in a straight line. + Suppose x = xo t cos 9 and y = yo + t sin 9. We are moving at the fixed angle 9, with speed 1. The derivatives are x, = cos 9 and y, = sin 9 and cos29 + sin29 = 1. Then dfldt is immediate from the chain rule: f, =fxx, +fyyt=fx cos 9 +f, sin 9. For the second derivativef,,, apply this rule to f,. Then f,, is cos 9 + (f,), sin 9 = (fxx cos 9 +Ax sin 9) cos 9 + (f, cos 9 +fyy sin 9) sin 9. Collect terms: f,, =fxx cos26 + 2fxy cos 6 sin 6 +fYy sin26. (6) In polar coordinates change t to r. When we move in the r direction, 9 is fixed. Equation (6) givesf, from fxx,fxy,fyy. Second derivatives on curved paths (with new terms from the curving) are saved for the exercises. EXAMPLE 3 If fxx,fxy,fyy are all continuous and bounded by M, find a bound onf;,. This is the second derivative along any line. Solution Equation (6) gives If,l < M cos26 + M sin 29 + M sin29< 2M. This upper bound 2M was needed in equation 13.3.14, for the error in linear approximation. T E DERIVATIVES O f(x(t, u), y(t, u)) H F Suppose there are two inside functions x and y, each depending on t and u. When t moves, x and y both move: dx = x,dt and dy = y,dt. Then dx and dy force a change inf d =fxdx +fydy. The chain rule for af/& is no surprise: f 131 Chain rule for f(x(t, u), y(t, u)): af af ax af - = -- +-- ay at ax at ay a t ' (7) This rule has a/at instead of dldt, because of the extra variable u. The symbols remind us that u is constant. Similarly t is constant while u moves, and there is a second chain rule for aflau: fu =fxxu+f,yu. EXAMPLE 4 In polar coordinates findf, andf,,. Start from f(x, y) =f(r cos 9, r sin 9). The chain rule uses the 6 derivatives of x and y: ' a -a af ---- ax +---ay - 89 ax 89 ay 89 (z) (- r sin 9) + ($)~(r cos 0). The second 9 derivative is harder, because (8) has four terms that depend on 6. Apply the chain rule to the first term af/ax. It is a function of x and y, and x and y are "(32 "(3 functions of 9: ae 9 ax = (212+ ay ax ax a6 ax ae =fxX(- r sin 9) +fxy(rcos 9). 13.5 The Chain Rule The 8 derivative of af/dy is similar. So apply the product rule to (8): = [fxx(- + r sin 8) fx,(r cos 8)] (- r sin 8) +fx(- r cos 8) + [fYx(- r sin 8) +fyy(r cos 8)](r cos 8) +f,(- r sin 8). (9) This formula is not attractive. In mathematics, a messy formula is almost always a + signal of asking the wrong question. In fact the combination f,, f,, is much more special thian the separate derivatives. We might hope the same forf,, +f,,, but dimen- sionally that is impossible-since r is a length and 8 is an angle. The dimensions of f,, andf,, are matched byf,, andf,/r and f,,/r2. We could even hope that 1 1 fxx + +f,, =f,r + ;f, This equation is true. Add (5) + (6) + (9) with t changed to r. Laplace's equation +&, = 0 is now expressed in polar coordinates:f,, +f,/r +f,,/r2 = 0. fxx A PARADOX Before leiaving polar coordinates there is one more question. It goes back to drldx, which wals practically the first example of partial derivatives: My problem is this. We know that x is r cos 8. So x/r on the right side is cos 8. On the other hand r is xlcos 8. So &-/ax is also l/cos 8. How can drldx lead to cos 8 one way and l/cos 8 the other way? I will admit that this cost me a sleepless night. There must be an explanation- we cannot end with cos 8 = l/cos 8. This paradox brings a new respect for partial derivatives. May I tell you what I finally noticed? You could cover up the next paragraph and think about the puzzle first. The key to partial derivatives is to ask: Which variable is held constant? In equation (11), y is constant. But when r = xlcos 8 gave &/ax = l/cos 8 , 8 was constant. In both cases we change x and look at the effect on r. The movement is on a horizontal line (constant y) or on a radial line (constant 8). Figure 13.15 shows the difference. Remark This example shows that drldx is different from l/(dx/ar). The neat formula (dr/dx)(dx/dr)= 1 is not generally true. May I tell you what takes its place? We have to includle (dr/dy)(ay/dr). With two variables xy and two variables re, we need 2 by 2 matrices! Section 14.4 gives the details: ,. / - : ar = ax cos u r I / d.r Fig. 13.15 dr = dx cos 0 when y is constant, dr = dxlcos 8 when 0 is constant. 13 Partial Deriwthres NON-INDEPENDENT VARIABLES This paradox points to a serious problem. In computing partial derivatives off(x, y, z), we assumed that x, y, z were independent. Up to now, x could move while y and z were fixed. In physics and chemistry and economics that may not be possible. If there is a relation between x, y, z, then x can't move by itself. EXAMPLE 5 The gas law PV = nRT relates pressure to volume and temperature. P, V T are not independent. What is the meaning of dV/aP? Does it equal l/(dP/aV)? , Those questions have no answers, until we say what is held constant. In the paradox, &/ax had one meaning for fixed y and another meaning for fixed 8. To inrlicate what is held constant, use an extra subscript (not denoting a derivative): (af/aP), has constant volume and (af/aP), has constant temperature. The usual af/dP has both V and T constant. But then the gas law won't let us change P. EXAMPLE 6 Let f = 3x + 2y + Z. Compute af/ax on the plane z = 4x + y. Solution 1 Think of x and y as independent. Replace z by 4x + y: f = 3x + 2~ + ( 4 + y) so (af/ax), = 7. ~ Solution 2 Keep x and y independent. Deal with z by the chain rule: (aflax), = aflax + (aflaz)(az/ax)= 3 + (I)(+ = 7. Solution 3 (di$evnt) Make x and z independent. Then y = z - 4x: Without a subscript, af/ax means: Take the x derivative the usual way. The answer is af/ax = 3, when y and z don't move. But on the plane z = 4x + y, one of them must move! 3 is only part of the total answer, which is (aflax), = 7 or (af/ax), = - 5. Here is the geometrical meaning. We are on the plane z = 4x + y. The derivative (afldx),, moves x but not y. To stay on the plane, dz is 4dx. The change in f = 3~+2y+zisdf=3dx+O+dz=7dx. EXAMPLE 7 On the world line x2 + y2 + z2 = t2 find (af/dy),,, for f = xyzt. The subscripts x, z mean that x and z are fixed. The chain rule skips af/dx and aflaz : (af1a~)X.z aflay + (aflat)(at/ay)= xzt + (xyz)(y/t). Why ylt? = EXAMPLE 8 From the law PV = T, compute the product (aP/aV),(aV,/aT),(aT/aP),. Any intelligent person cancels aV's, aT's, and aP's to get 1. The right answer is - 1: (a la v), = - TI v2 (a v,aT), = 1/P (a TIaP), = v. The product is - TIPV. This is -1 not + l! The chain rule is tricky (Problem 42). EXAMPLE 9 Implicit differentiation was used in Chapter 4. The chain rule explains it: If F(x, y) = 0 then F, + Fyyx= 0 so dyldx = - Fx/Fy. (13) 13.5 The Chain Rule 503 13.5 EXERCISES Read-through questions 12 Equation (10) gave the polar formf, +J/r +fee/r2 = 0 of Laplace's equation. (a) Check that f = r2e2" and its real part The chain rule applies to a function of a a . The x deriva- r2 cos 28 and its imaginary part r2 sin 28 all satisfy Laplace's tive of f(g(x, y)) is dflax = . b . The y derivative is dfldy = equation. (b) Show from the chain rule that any functionf(reie) c . The example f = (x + y)" has g = d . Because satisfies this equation if i2 = - 1. dgldx = dgldy we know ithat e = f . This g differential equation is satiisfied by any function of x + y. Along a path, the derivaiive of f(x(t), y(t)) is dfldt = h . In Problems 13-18 find dfldt from the chain rule (3). The derivative of f(x(t), y(r:), z(t)) is i . Iff = xy then the chain rule gives dfldt = i . That is the same as the k rule! When x = ult and y I u2t the path is I . The chain = rule for f(x, y) gives dfldt == m . That is the n deriva- tive DJ The chain rule for f(x(t, u), y(t, u)) is df/at = 0 . We don't write dfldt because P . If x = r cos 0 and y = r sin 0, 17 f = ln(x + y), x = et, y = et the variables t, u change to q . In this case afldr = r and df/d8= s . That connects the derivatives in + and u coordinates. The difference between 19 If a cone grows in height by dhldt = 1 and in radius by &/ax = x/r and drldx = l/cos 0 is because v is constant drldt = 2, starting from zero, how fast is its volume growing in the first and w is c'onstant in the second. at t = 3? With a relation like xyz = 1, the three variables are x 20 If a rocket has speed dxldt = 6 down range and dyldt = independent. The derivatives (afldx), and (dflax), and (af/ax) 2t upward, how fast is it moving away from the launch point mean Y and z and A . For f = x2 + y2 + z2 with at (0, O)? How fast is the angle 8 changing, if tan 8 = ylx? xyz = 1, we compute (afldx), from the chain rule B . In 21 If a train approaches a crossing at 60 mph and a car that rule dz/dx = c from the relation xyz = 1. approaches (at right angles) at 45 mph, how fast are they Find f, and& in Problems '1-4. What equation connects them? coming together? (a) Assume they are both 90 miles from the 1 f(x, y) = sin(x + cy) 2 f(x, y) = (ax + by)'' crossing. (b) Assume they are going to hit. 22 In Example 2 does the temperature increase faster if you 3 f(x, y) = ex+7y 4 f(x, Y)= In(x + 7 ~ ) drive due south at 70 mph or southeast at 80 mph? 5 Find both terms in the: t derivative of (g(x(t),~ ( t ) ) ~ . 23 On the line x = u,t, y = u2t, z = u,t, what combination of 6 Iff(x, y) = xy and x = ul(t)and y = v(t), what is dfldt? Prob- f,,f,, f, gives dfldt? This is the directional derivative in 3D. ably all other rules for deriivatives follow from the chain rule. 24 On the same line x = u, t, y = u2t, z = u3t, find a formula 7 The step function f(x) is zero for x < 0 and one for x > 0. for d f/dt 2. Apply it to f = xyz. Graph f(x) and g(x) =f(x -t2) and h(x) =f(x + 4). If f(x + 2t) represents a wall of water (a tidal wave), which way is it 25 For f(x, y, t) = x + y + t find afldt and dfldt when x = 2t moving and how fast? and y = 3t. Explain the difference. 8 The wave equation is J;, = c2f,,. (a) Show that (x + ct)" is 26 ~f z = (X+ y)2 then x = Jr - y. Does ( a ~ j a x ) ( a x j a ~ )I? = a solution. (b) Find C different from c so that (x + Ct)" is also 27 Suppose x, = t and y, = 2t, not constant as in (5-6). For a solution. f(x, y) find f, and f,,. The answer involves fx ,fy ,fxx ,fxy ,fyy. 9 Iff = sin(x - t), draw two lines in the xt plane along which 28 Suppose x, = t and y, = t 2. For f = (x + y)3 findf, and then f = 0. Between those lines sketch a sine wave. Skiing on top f,, from the chain rule. of the sine wave, what is your speed dxldt? 29 Derive d f p = (afldx) cos 0 + (afldy) sin 8 from the chain 10 If you float at x = 0 in Problem 9, do you go up first or rule. Why do we take ax/& as cos 8 and not l/cos O ? down first? At time t = 4 what is your height and upward velocity? 30 Compute f,, for f(x, y) = (ax + by + c)". If x = t and y = t computef,,. True or false: (af/dx)(ax/at) = afpt. 11 Laplace's equation is fx, +fyy = 0. Show from the chain rule that any function f(x + iy) satisfies this equation if i2 = 31 Show that a2r/dx2= y2/r3 in two ways: - 1. Check that f = (x + i ! ~ )and its real part ~ and (1) Find the x derivative of drldx = x/ Jm its imaginary part all satisfy Laplace's equation. (2) Find the x derivative of drldx = xlr by the chain rule. 504 13 Partla1 ~erivatives 32 Reversing x and y in Problem 31 gives ryy= x2/r3. But 41 For f = ax + by on the plane z = 3x + 5y, find (aflax), and show that r, = - xy/r3. (aflax), and (aflaz),. 33 If sin z = x + y find (az/ax), in two ways: 42 The gas law in physics is PV = nRT or a more general relation F(P, T) = 0. Show that the three derivatives in (1)Write z = sin- '(x + y) and compute its derivative. Example 8 still multiply to give - 1. First find (aP/aV), from (2)Take x derivatives of sin z = x + y. Verify that these aF/av + (aFIaP)(aP/av), = 0. answers, explicit and implicit, are equal. 43 If Problem 42 changes to four variables related by 34 By direct computation find f, and f,, and f,, for F(x, y, z, t) = 0, what is the corresponding product of four f = JW. derivatives? 35 Find a formula for a2f/arae in terms of the x and y deriva- 44 Suppose x = t + u and y = tu. Find the t and u derivatives tives of f(x, y). offlx, y). Check when f(x, y) = x2 - 2y. 36 Suppose z =f(x, y) is solved for x to give x = g(y, z). Is it 45 (a) For f = r2 sin28 find f, and f,. true that az/ax = l/(ax/az)? Test on examples. (b) For f = x2 + y2 findf, andf,. " 37 Suppose z = e, and therefore x = (In z)/y. Is it true or not that (az/ax) = i/(ax/az)? 46 On the curve sin x + sin y = 0, find dy/dx and d 2 y / d ~by 2 implicit differentiation. 38 If x = x(t, u, v) and y = y(t, u, v) and z = z(t, u, v), find the t derivative offlx, y, z). + 47 (horrible) Suppose f,, +f,, = 0. If x = u v and y = u - v + and f(x, y) = g(u, v), find g, and g,. Show that g,, g,, = 0. 39 The t derivative of f(x(t, u), y(t, u)) is in equation (7). What 48 A function has constant returns to scale if f(cx, cy) = is frt? cf(x, y ) When x and y are doubled so are f = 40 (a) For f = x2 + y2 + z2 compute af/ax (no subscript, and f = fi. In economics, input/output is constant. In x, y, z all independent). mathematics f is homogeneous of degree one. (b) When there is a further relation z = x2 + y2, use it to Prove that x af/ax + y if/ay =f(x, y), by computing the c remove z and compute (aflax),. derivative at c = 1. Test this equation on the two examples (c) Compute (aflax), using the chain rule (af/dx)+ and find a third example. (aflaz)(azlax). 49 True or false: The directional derivative of f(r, 8) in the (d) Why doesn't that chain rule contain (af/ay)(ay/ax)? direction of u is af/a8. , The outstanding equation of differential calculus is also the simplest: dfldx = 0. The slope is zero and the tangent line is horizontal. Most likely we are at the top or bottom of the graph-a maximum or a minimum. This is the point that the engineer or manager or scientist or investor is looking for-maximum stress or production or velocity or profit. With more variables in f(x, y) and f(x, y, z), the problem becomes more realistic. The question still is: How to locate the maximum and minimum? The answer is in the partial derivatives. When the graph is level, they are zero. Deriving the equations f, = 0 and f, = 0 is pure mathematics and pure pleasure. Applying them is the serious part. We watch out for saddle points, and also for a minimum at a boundary point-this section takes extra time. Remember the steps for f(x) in one-variable calculus: 1. The leading candidates are stationary points (where dfldx = 0). 2. The other candidates are rough points (no derivative) and endpoints (a or b). 3. Maximum vs. minimum is decided by the sign of the second derivative. In two dimensions, a stationary point requires af/dx = 0 and df/ay = 0. The tangent line becomes a tangent plane. The endpoints a and b are replaced by a boundary curve. In practice boundaries contain about 40% of the minima and 80% of the work. 13.6 Maxima, Minima, and Saddle Points 505 Finally there are three second derivativesfxx,fxy, and fy,. They tell how the graph bends away from the tangent plane-up at a minimum, down at a maximum, both ways at a saddle point. This will be determined by comparing (fxx)(fyy) with (fx) 2 . STATIONARY POINT -+ HORIZONTAL TANGENT -- ZERO DERIVATIVES Supposef has a minimum at the point (xo, Yo). This may be an absolute minimum or only a local minimum. In both casesf(xo, yo) <f(x, y) near the point. For an absolute minimum, this inequality holds wherever f is defined. For a local minimum, the inequality can fail far away from (xo, yo). The bottom of your foot is an absolute minimum, the end of your finger is a local minimum. We assume for now that (xo, Yo) is an interior point of the domain off. At a boundary point, we cannot expect a horizontal tangent and zero derivatives. Main conclusion: At a minimum or maximum (absolute or local) a nonzero deriva- tive is impossible. The tangent plane would tilt. In some direction f would decrease. Note that the minimum point is (xo, yo), and the minimum value is f(xo, yo). 13J If derivatives exist at an interior minimum or maximum, they are zero: Of/lx = 0 and Oflay = 0 (together this is grad f= 0). (1) For a function f(x, y, z) of three variables, add the third equation af/az = 0. The reasoning goes back to the one-variable case. That is because we look along the lines x = x0 and y = Yo. The minimum off(x, y) is at the point where the lines meet. So this is also the minimum along each line separately. Moving in the x direction along y = yo, we find Of/Ox = 0. Moving in the y direction, Of/Oy = 0 at the same point. The slope in every direction is zero. In other words grad f= 0. Graphically, (xo, Yo) is the low point of the surface z =f(x, y). Both cross sections in Figure 13.16 touch bottom. The phrase "if derivatives exist" rules out the vertex of a cone, which is a rough point. The absolute value f= IxI has a minimum without df/dx = 0, and so does the distance f= r. The rough point is (0, 0). 1 y fixed at - . _ - /-2 = x+ y+ -- + 1 • - - -. .... - - - , x fixed at 3 I / I / /1 x '(Xo, Yo) = (-,--) 1 ,3 . Fig. 13.16 af/Ox = 0 and afl/y = 0 at the minimum. Quadratic f has linear derivatives. EXAMPLE 1 Minimize the quadratic f(x, y) = x 2 + xy + y 2 - x - y + 1. To locate the minimum (or maximum), set f = 0 and fy = 0: - fx=2x+y 1 =0 and f= x+2y-1=0. 13 Partial Derivatives Notice what's important: There are two equations for two unknowns x and y. Since f is quadratic, the equations are linear. Their solution is xo = 3, yo = $ (the stationary point). This is actually a minimum, but to prove that you need to read further. The constant 1 affects the minimum value f = :-but not the minimum point. The graph shifts up by 1. The linear terms - x - y affectfx and fy . They move the minimum + away from (0,O). The quadratic part x2 xy + y2 makes the surface curve upwards. Without that curving part, a plane has its minimum and maximum at boundary points. EXAMPLE 2 (Steiner'sproblem) Find the point that is nearest to three given points. This example is worth your attention. We are locating an airport close to three cities. Or we are choosing a house close to three jobs. The problem is to get as near as possible to the corners of a triangle. The best point depends on the meaning of "near." The distance to the first corner (x, , y,) is dl = ,/(x - x,), + (y - y,),. The dis- tances to the other corners (x,, y,) and (x,, y,) are d; and d,. Depending on whether or our cost equals (distance) or (di~tance)~ (di~tance)~, problem will be: Minimize d,+d,+d, or d : + d i + d : oreven d ~ + d ~ + d ~ The second problem is the easiest, when d: and d t and d i are quadratics: a ~ j a x = 2 ~ ~ - x l + x - x 2 + x - x 3 ~ = ~a f / a y = 2 [ y - y l + y - y 2 + y - y 3 1 = o . Solving iflax = 0 gives x = i ( x l + x, + x,). Then af/dy = 0 gives y = i(y, + y, + y,). The best point is the centroid of the triangle (Figure 13.17a). It is the nearest point to the corners when the cost is (distance),. Note how squaring makes the derivatives linear. Least squares dominates an enormous part of applied mathematics. U3 Fig. 13.17 The centroid minimizes d : + d $ + d 3 . The Steiner point minimizes dl + d2 + d3 The real "Steiner problem" is to minimize f(x, y) = dl + d, + d, . We are laying down roads from the corners, with cost proportional to length. The equations f = 0 and, f , = 0 look complicated because of square roots. But the nearest point in Figure 13.17b has a remarkable property, which you will appreciate. Calculus takes derivatives of d : = (x - xl), + (y - y,),. The x derivative leaves 2dl(ddl/dx) = 2(x - x,). Divide both sides by 2d1: adl - x - x, ad1 - Y and - -- - Y l so grad dl = x-Xl .; - Y l Y )j (T7 (3) dx dl 8~ dl and This gradient is a unit vector. The sum of (x - ~ , ) ~ / d : (y - yJ2/d: is d:/d: = 1. This was already in Section 13.4: Distance increases with slope 1 away from the center. The gradient of dl (call it u,) is a unit vector from the center point (x,, y,). 13.6 Maxima, Minima, and Saddle Points Similarly the gradients of d, and d, are unit vectors u2 and u3. They point directly away from the other corners of the triangle. The total cost is f(x, y) = dl + d , + d3, so we add the gradients. The equations f, = 0 and f, = 0 combine into the vector equation grad f = u, + u2 + u3 = 0 at the minimum. The three unit vectors add to zero! Moving away from one corner brings us closer to another. The nearest point to the three corners is where those movements cancel. This is the meaning of "grad f = 0 at the minimum." It is unusual for three unit vectors to add to zero-this can only happen in one way. The three directions must form angles of 120". The best point has this property, which is .repeated in Figure 13.18a. The unit vectors cancel each other. At the "Steiner point," the roads to the corners make 120" angles. This optimal point solves the problem,, except for one more possibility. - - I - - - - - - u2 ,( x , y ) has rough point> angle > 120" '"3 n.=o d, + + Fig. 13.181 Gradients ul u2 u, = 0 for 120" angles. Corner wins at wide angle. Four corners. In this case two branchpoints are better-still 120". The other possibility is a minimum at a rough point. The graph of the distance function d,(x, y) is a cone. It has a sharp point at the center (x,, y,). All three corners of the triangle are rough points for dl + d, + d,, so all of them are possible minimizers. Suppo,se the angle at a corner exceeds 120". Then there is no Steiner point. Inside the triangle, the angle would become even wider. The best point must be a rough point-one of the corners. The winner is the corner with the wide angle. In the figure that mea.ns dl = 0. Then the sum d, + d, comes from the two shortest edges. sum mar.^ The solution is at a 120" point or a wide-angle corner. That is the theory. The real problem is to compute the Steiner point-which I hope you will do. Remark 1 Steiner's problem for four points is surprising. We don't minimize dl + d2 4- d3 + d4-there is a better problem. Connect the four points with roads, minimizing their total length, and allow the roads to branch. A typical solution is in Figure 1 . 3 . 1 8 ~The angles at the branch points are 120". There are at most two branch . points (two less than the number of corners). Remark 2 For other powers p, the cost is + (d2)P+ (d3)P.The x derivative is The key equations are still dfldx = 0 and df/ay = 0. Solving them requires a computer and an algorithm. To share the work fairly, I will supply the algorithm (Newton's method) if you supply the computer. Seriously, this is a terrific example. It is typical of real problems-we know dfldx and dflay but not the point where they are zero. You can calculate that nearest point, which changes as p changes. You can also discover new mathematics, about how that point moves. I will collect all replies I receive tlo Problems 38 and 39. 13 Partial Derivatives R H MINIMUM O MAXIMUM ON T E BOUNDARY Steiner's problem had no boundaries. The roads could go anywhere. But most appli- cations have restrictions on x and y, like x 3 0 or y d 0 or x2 + y 2 2 1. The minimum with these restrictions is probably higher than the absolute minimum. There are three possibilities: (1) stationary point fx = 0, fy = 0 (2) rough point (3) boundary point That third possibility requires us to maximize or minimize f(x, y) along the boundary. EXAMPLE 3 Minimize f(x, y) = x2 + xy + y2 - x - y + 1 in the halfplane x 2 0. The minimum in Example 1 was 3 . It occurred at x, = 3, yo = 3. This point is still allowed. It satisfies the restriction x 3 0. So the minimum is not moved. EXAMPLE 4 Minimize the same f (x, y) restricted to the lower halfplane y < 0. Now the absolute minimum at (3, i)is not allowed. There are no rough points. We look for a minimum on the boundary line y = 0 in Figure 13.19a. Set y = 0, so f depends only on x. Then choose the best x: f(x, 0) = x2 + 0 - x - 0 + 1 and fx = 2x - 1 = 0. The minimum is at x = and y = 0, where f = 2. This is up from 5. Fig. 13.19 The boundaries y = 0 and x2 + y 2 = 1 contain the minimum points. EXAMPLE 5 Minimize the same f(x, y) on or outside the circle x2 + y 2 = 1. One possibility is fx = 0 and f,, 0. But this is at ( , = i inside the circle. The other k), possibility is a minimum at a boundary point, on the circle. To follow this boundary we can set y = Jm. The function f gets complicated, and dfldx is worse. There is a way to avoid square roots: Set x = cos t and y = sin t. Then f = x2 + xy + y 2 - x - y + 1 is a function of the angle t: + cos t sin t - cos t - sin t + 1 f(t) = 1 dfldt = cos2t sin2t + sin t - cos t = (cos t - - sin t)(cos t + sin t- 1). Now dfldt = 0 locates a minimum or maximum along the boundary. The first factor (cos t - sin t ) is zero when x = y. The second factor is zero when cos t + sin t = 1, or x + y = 1. Those points on the circle are the candidates. Problem 24 sorts them out, and Section 13.7 finds the minimum in a new way-using "Lagrange multipliers." 13.6 Maxima, Minima, and Saddle Points 509 Minimization on a boundary is a serious problem-it gets difficult quickly-and multipliers are ultimately the best solution. MAXIMUM VS. MINIMUM VS. SADDLE POINT How to separate the maximum from the minimum? When possible, try all candidates and decide. Computef at every stationary point and other critical point (maybe also out at infinity), and compare. Calculus offers another approach, based on second derivatives. With one variable the second derivative test was simple: fxx > 0 at a minimum, fxx = 0 at an inflection point, fxx < 0 at a maximum. This is a local test, which may not give a global answer. But it decides whether the slope is increasing (bottom of the graph) or decreasing (top of the graph). We now find a similar test for f(x, y). The new test involves all three second derivatives. It applies where fx = 0 and f, = 0. The tangent plane is horizontal. We ask whether the graph off goes above or below that plane. The tests fxx > 0 and fy, > 0 guarantee a minimum in the x and y directions, but there are other directions. EXAMPLE 6 f(x, y) = x 2 + lOxy + y2 has fxx = 2, fx = 10, fyy, = 2 (minimum or not?) All second derivatives are positive-but wait and see. The stationary point is (0, 0), where af/ax and aflay are both zero. Our function is the sum of x2 + y2, which goes upward, and 10xy which has a saddle. The second derivatives must decide whether x 2 + y 2 or lOxy is stronger. Along the x axis, where y = 0 and f= x 2, our point is at the bottom. The minimum in the x direction is at (0, 0). Similarly for the y direction. But (0, 0) is not a minimum point for the whole function, because of lOxy. Try x = 1, y = - 1. Then f= 1 - 10 + 1, which is negative. The graph goes below the xy plane in that direction. The stationary point at x = y = 0 is a saddle point. 2 2 f= -x +y f= -x 2 -_y2 y a..y -0 Y 2 f- X + y a>O ac>b2 x x a<O ac>b2 x ac<b 2 Fig. 13.20 Minimum, maximum, saddle point based on the signs of a and ac - b2 . EXAMPLE 7 f(x, y) = x 2 + xy + y2 has fxx = 2, fx, = 1, fyy = 2 (minimum or not?) The second derivatives 2, 1, 2 are again positive. The graph curves up in the x and y directions. But there is a big difference from Example 6: fx, is reduced from 10 to 1. It is the size offx (not its sign!) that makes the difference. The extra terms - x - y + 4 in Example 1 moved the stationary point to (-, -). The second derivatives are still 2, 1, 2, and they pass the test for a minimum: 13K At (0, 0) the quadratic function f(x, y)= ax2 + 2bxy + cy2 has a a>0 a<0 ac > b2 ac > b2 510 13 Partial Derivatives For a direct proof, split f(x, y) into two parts by "completing the square:" ax2 + 2bx y + cy 2= a x+ y + ac - b2 a a That algebra can be checked (notice the 2b). It is the conclusion that's important: if a > 0 and ac > b2 , both parts are positive: minimum at (0, 0) 2 if a < 0 and ac > b , both parts are negative: maximum at (0, 0) if ac < b2 , the parts have opposite signs: saddle point at (0, 0). Since the test involves the square of b, its sign has no importance. Example 6 had b = 5 and a saddle point. Example 7 had b = 1 and a minimum. Reversing to 2 2 - x2 - xy - y2 yields a maximum. So does - x + xy - y Now comes the final step, from ax 2 + 2bxy + cy 2 to a general functionf(x, y). For all functions, quadratics or not, it is the second order terms that we test. EXAMPLE 8 f(x, y) = ex - x - cos y has a stationary point at x = 0, y = 0. The first derivatives are ex - 1 and sin y, both zero. The second derivatives arefxx ex = 1 and fry = cos y = 1 and fxy = 0. We only use the derivatives at the stationary point. The first derivatives are zero, so the second order terms come to the front in the series for ex - x - cos y: 2 2 (1+ x + ½x ... _ 2 ... 2 + higher order terms. (7) There is a minimum at the origin. The quadratic part ½x2 + ½y 2 goes upward. The x 3 and y 4 terms are too small to protest. Eventually those terms get large, but near a stationary point it is the quadratic that counts. We didn't need the whole series, because from fxx =f,, = 1 and fxy = 0 we knew it would start with ½x 2 + ½y 2. 13L The test in 43K applies to the second derivatives a =fxx, b =fx,, c =fy of any f(x, y) at any stationary point. At all points the test decides whether the graph is concave up, concave down, or "indefinite." EXAMPLE 9 f(x, y) = exy has fx = yexy and f, = xexy. The stationary point is (0, 0). The second derivatives at that point are a =fxx = 0, b =fxy = 1, and c =fy, = 0.The test b 2 > ac makes this a saddle point. Look at the infinite series: exY = 1 + xy + x 2y 2 + ... No linear term becausefx =f,= 0: The origin is a stationarypoint. No x 2 or y 2 term (only xy): The stationary point is a saddle point. At x = 2, y = - 2 we find fxxfry > (fxy) 2 . The graph is concave up at that point- but it's not a minimum since the first derivatives are not zero. The series begins with the constant term-not important. Then come the linear terms-extremely important. Those terms are decided by first derivatives, and they give the tangent plane. It is only at stationary points-when the linear part disappears and the tangent plane is horizontal-that second derivatives take over. Around any basepoint, these constant-linear-quadratic terms are the start of the Taylor series. 13.6 Maxima, Minima, and Saddle Points 511 THE TAYLOR SERIES We now put together the whole infinite series. It is a "Taylor series"-which means it is a power series that matches all derivatives off (at the basepoint). For one variable, the powers were x" when the basepoint was 0. For two variables, the powers are x" times y' when the basepoint is (0, 0). Chapter 10 multiplied the nth derivative d"f/dx n by xl/n! Now every mixed derivative (d/dx)"(d/8y)mf(x, y) is computed at the basepoint (subscript o). We multiply those numbers by x"y m/n!m! to match each derivative of f(x, y): 13M When the basepoint is (0, 0), the Taylor series is a double sum 1ya,,mxp. The term anmxnym has the same mixed derivative at (0, 0) asf(x, y). The series is f + fý + X ( a2f- + y2 (a2.. f(O, 0)+ x t2+ +y ax)yo o 2 n+M>2 n!m!\dx"~~o The derivatives of this series agree with the derivatives off(x, y) at the basepoint. The first three terms are the linear approximation to f(x, y). They give the tangent plane at the basepoint. The x2 term has n = 2 and m = 0, so n!m! = 2. The xy term has n = m = 1, and n!m! = 1. The quadraticpart-ax 2 + 2bxy + cy 2 ) is in control when the linearpart is zero. EXAMPLE 10 All derivatives of ex+Y equal one at the origin. The Taylor series is x2 y2 nm = ex + Y 1 +x + - + xy+ - + 2 2 n!m! This happens to have ac = b2 , the special case that was omitted in 13M and 43N. It is the two-dimensional version of an inflection point. The second derivatives fail to decide the concavity. When fxxfy, = (fxy) 2, the decision is passed up to the higher derivatives. But in ordinary practice, the Taylor series is stopped after the quadratics. If the basepoint moves to (xo, Yo), the powers become (x - xo)"(y - yo)m"-and all derivatives are computed at this new basepoint. Final question: How would you compute a minimum numerically? One good way is to solve fx = 0 and fy = 0. These are the functions g and h of Newton's method (Section 13.3). At the current point (x,, yn), the derivatives of g =fx and h =f, give linear equations for the steps Ax and Ay. Then the next point x,. 1 = x, + Ax, y,, + = y, + Ay comes from those steps. The input is (x,, y,), the output is the new point, and the linear equations are (gx)Ax + (gy)Ay = - g(xn, y,) (fxx)Ax + (fxy)Ay = -fx(xn, y,,) or (5) (hx)Ax + (hy)Ay = - h(x,, y,) (fxy)Ax + (fyy)Ay = -fy(Xn, y,). When the second derivatives of f are available, use Newton's method. When the problem is too complicated to go beyond first derivatives, here is an alternative-steepestdescent. The goal is to move down the graph of f(x, y), like a boulder rolling down a mountain. The steepest direction at any point is given by the gradient, with a minus sign to go down instead of up. So move in the direction Ax = - s af/ax and Ay = - s aflay. 13 Partial Derivatives The question is: How far to move? Like a boulder, a steep start may not aim directly toward the minimum. The stepsize s is monitored, to end the step when the function f starts upward again (Problem 54). At the end of each step, compute first derivatives and start again in the new steepest direction. 13.6 EXERCISES Read-through questions A minimum occurs at a a point (where fx =f, = 0) or a 17 A rectangle has sides on the x and y axes and a corner on b point (no derivative) or a c point. Since f = the line x + 3y = 12. Find its maximum area. x2 - xy + 2y has fx = d and f, = e , the stationary 18 A box has a corner at (0, 0, 0) and all edges parallel to the point is x = f , y = . This is not a minimum, axes. If the opposite corner (x, y, z ) is on the plane because f decreases when h . 3x + 2y + z = 1, what position gives maximum volume? Show + o The minimum of d = (x - x , ) ~ (y - Y , ) ~ ccurs at the first that the problem maximizes xy - 3x2y - 2xy2. rough point 1 . The graph of d is a i and grad d 19 (Straight line fit, Section 11.4) Find x and y to minimize is a k vector that points I . The graph off = lxyl the error touches bottom along the lines m . Those are "rough lines" because the derivative n . The maximum of d and E = (x + Y)2+ (X+ 2y - 5)2+ (x + 3y - 4)2. f must occur on the 0 of the allowed region because it Show that this gives a minimum not a saddle point. doesn't occur P . 20 (Least squares) What numbers x, y come closest to satisfy- When the boundary curve is x = x(t), y = y(t), the derivative ing the three equations x - y = 1, 2x + y = - 1, x + 2y = l? of f(x, y) along the boundary is q (chain rule). Iff = Square and add the errors, (x - y - + + x2 + 2y2 and the boundary is x = cos t, y = sin t, then df/dt = . Then minimize. r . It is zero at the points s . The maximum is at t and the minimum is at u . Inside the circle f has 21 Minimize f = x2 + xy + y2 - x - y restricted by an absolute minimum at v . (a)x 6 0 (b) Y 3 1 (c) x 6 0 and y 3 1. To separate maximum from minimum from w , com- 22 Minimize f = x2 + y2 + 2x + 4y in the regions pute the x derivatives at a Y point. The tests for a (a) all x, Y (b) y 3 0 (c) x 3 0, y 3 0 minimum are 2 . The tests for a maximum are A . In 23 Maximize and minimize f = x + f i y on the circle x = case ac < B or fxx f,, < C , we have a D . At all cos t, y = sin t. points these tests decide between concave up and E and "indefinite." Forf = 8x2 - 6xy + y2, the origin is a F . The 24 Example 5 followed f = x2 + xy + y2 - x - y + 1 around signs off at (1, 0) and (1, 3) are G . the circle x2 + Y2 = 1. The four stationary points have x = y or x + y = 1. Compute f at those points and locate the The Taylor series for f(x, y) begins with the six terms H . minimum. The coefficient of xnymis I . To find a stationary point numerically, use J or K . 25 (a) Maximize f = ax + by on the circle x2 + y 2 = 1. (b) Minimize x2 + y 2 on the line ax + by = 1. 26 For f(x, y) = ax4 - xy + $y4, what are the equations fx = Find all stationary points (fx =f, = 0) in 1-16. Separate mini- 0 and f, = O What are their solutions? What is fmi,? ? mum from maximum from saddle point. Test 13K applies to a =fxx, b =fx,, c =f,,. 27 Choose c > 0 so that f = x2 + xy + cy2 has a saddle point at (0,O). Note that f > 0 on the lines x = 0 and y = 0 and y = 1 x2 + 2xy+ 3y2 2 xy-x+y x and y = - x, so checking four directions does not confirm 3 x2 + 4xy + 3 ~ - 6x - 12y 4 x2 - y2 + 4y ' a minimum. 5 x~~~- x 6 xeY- ex Problems 28-42 minimize the Steiner distancef = dl + d2 + d3 7 - x2 + 2xy - 3y2 8 (x + y)2 + (X + 2y - 6)2 and related functions. A computer is needed for 33 and 36-39. 9 X ~ + ~ ~ + Z ~ - ~ Z 10 (x+y)(x+2y-6) 28 Draw the triangle with corners at (0, O), (1, I), and (1, -1). By symmetry the Steiner point will be on the x axis. Write 11 ( x - Y ) ~ 12 (1 + x2)/(1+ y2) down the distances d l , d2, d3 to (x, 0) and find the x that ~ 13 (x + Y ) -(x +2 ~ ) ~ 14 sin x - cos y minimizes dl + d2 + d,. Check the 120" angles. 13.6 Maxima, Minima, and Saddle Points 513 29 Suppose three unit vectors add to zero. Prove that the Find all derivatives at (0, Construct the Taylor series: 0). angles between them must be 120". 30 In three dimensions, Steiner minimizes the total distance 45 f(x, y) = In(1- xy) + + + Ax, y, z) = dl d2 d3 d, from four points. Show that grad dl is still a unit vector (in which direction?) At what Find f,, fy, f,,, fxy,fyy at the basepoint. Write the quadratic angles do four unit vectors add to zero? approximation to f(x, y) - the Taylor series through second- 31 With four points in a plane, the Steiner problem allows order terms: branches (Figure 13.18~). Find the shortest network connect- ing the corners of a rectangle, if the side lengths are (a) 1 and 2 (b) 1 and 1 (two solutions for a square) (c) 1 and 0.1. 32 Show that a Steiner point (120" angles) can never be out- 50 The Taylor series around (x, y) is also written with steps side the triangle. hand k:Jx + h, y + k)=f(x,y)+ h +k + 33 Write a program to minimize f(x, y) = dl + d2 + d3 by 3h2- +hk + --..Fill in those four blanks. Newton's method in equation (5). Fix two corners at (0, O), 51 Find lines along which f(x, y) is constant (these functions (3, O), vary the third from (1, 1) to (2, 1) to (3, 1) to (4, l), and have f,, fyy=fa or ac = b2): compute Steiner points. + (a)f = x2 - 4xy 4y2 (b)f = eXeY 34 Suppose one side of the triangle goes from (- 1,0) to (1,O). 52 For f(x, y, z) the first three terms after f(O, 0,0) in the Tay- Above that side are points from which the lines to (- 1, 0) and lor series are . The next six terms are (1, 0) meet at a 120" angle. Those points lie on a circular arc- draw it and find its center and its radius. 53 (a) For the error f -f, in linear approximation, the Taylor series at (0, 0) starts with the quadratic terms 35 Continuing Problem 34, there are circular arcs for all three (b)The graph off goes up from its tangent plane (and sides of the triangle. On the arcs, every point sees one side of the triangle at a 120" angle. Where is the Steiner point? f > f d if- . Then f is concave upward. (c) For (0,O) to be a minimum we also need (Sketch three sides with their arcs.) 36 Invent an algorithm to converge to the Steiner point based 54 The gradient of x2 + 2y2 at the point (1, 1) is (2,4). Steepest descent is along the line x = 1 - 2s, y = 1 - 4s (minus on Problem 35. Test it on the triangles of Problem 33. sign to go downward). Minimize x2 + 2y2 with respect to the 37 Write a code to minimize f =d: +d: +d: by solving f, =0 stepsize s. That locates the next point , where and fy = 0. Use Newton's method in equation (5). steepest descent begins again. 38 Extend the code to allow all powers p 2 1, not only p = 55 Newton's method minimizes x2 + 2y2 in one step. Starting 4. Follow the minimizing point from the centroid at p = 2 to at (xo,yo) = (1, I), find AX and Ay from equation (5). the Steiner point at p = 1 (try p = 1.8, 1.6, 1.4, 1.2). 56 Iff,, +f,, = 0, show that f(x, y) cannot have an interior 39 Follow the minimizing point with your code as p increases: maximum or minimum (only saddle points). p = 2, p = 4, p = 8, p = 16. Guess the limit at p = rn and test 57 The value of x theorems and y exercises isf = x2y (maybe). whether it is equally distant from the three corners. The most that a student or author can deal with is 4x y = + 40 At p = c we are making the largest of the distances o 12. Substitute y = 12 - 4x and maximize5 Show that the line dl, d2, d, as small as possible. The best point for a 1, 1, fi 4x + y = 12 is tangent to the level curve x2y=f,,,. right triangle is . 58 The desirability of x houses and y yachts is f(x, y). The 41 Suppose the road from corner 1 is wider than the others, + constraint px qy = k limits the money available. The cost of and the total cost is f(x, y) =fi dl + d2 + d,. Find the gradi- a house is , the cost of a yacht is . Substi- ent off and the angles at which the best roads meet. tute y = (k - px)/q into f(x, y) = F(x) and use the chain rule for dF/dx. Show that the slope -f,& at the best x is -p/q. + d2 42 Solve Steiner's problem for two points. Where is d , a minimum? Solve also for three points if only the three 59 At the farthest point in a baseball field, explain why the corners are allowed. fence is perpendicular to the line from home plate. Assume it is not a rough point (corner) or endpoint (foul line). 514 13 Partial ~erivut~ves 13.7 Constraints and Lagrange Multipliers This section faces up to a practical problem. We often minimize one function f(x, y) while another function g(x, y) is fixed. There is a constraint on x and y, given by g(x, y) = k. This restricts the material available or the funds available or the energy available. With this constraint, the problem is to do the best possible (f,, or fmin). At the absolute minimum off(x, y), the requirement g(x, y) = k is probably violated. In that case the minimum point is not allowed. We cannot use f, = 0 and f,, = O- those equations don't account for g. Step 1 Find equations for the constrained minimum or constrained maximum. They will involve f, andf,, and also g, and g,, which give local information about f and g. To see the equations, look at two examples. EXAMPLE 1 Minimize f = x2 + y2 subject to the constraint g = 2x + y = k. Trial runs The constraint allows x = 0, y = k, where f = k2. Also ($k, 0) satisfies the constraint, and f = $k2 is smaller. Also x = y = $k gives f = $k2 (best so far). Idea of solution Look at the level curves of f(x, y) in Figure 13.21. They are circles x2 + y2 = C. When c is small, the circles do not touch the line 2x + y = k. There are no points that satisfy the constraint, when c is too small. Now increase c. Eventually the growing circles x2 + y2 = c will just touch the line x + 2y = k. The point where they touch is the winner. It gives the smallest value of c that can be achieved on the line. The touching point is (xmin, ymi,), and the value of c is fmin. What equation describes that point? When the circle touches the line, they are tangent. They have the same slope. The perpendiculars to the circle and he line go in the same direction. That is the key fact, which you see in Figure 13.21a. The direction perpendicular to f = c is given by grad f = (f,, f,). The direction perpendicular to g = k is given by grad g = (g,, g,). The key equation says that those two vectors are parallel. One gradient vector is a multiple of the other gradient vector, with a multi- plier A (called lambda) that is unknown: I 13N At the minimum of f(x, y) subject to gjx, y) = k, the gradient off is parallel to the gradient of g-with an unknown number A as the multiplier: Step 2 There are now three unknowns x, y, A. There are also three equations: In the third equation, substitute 2 for 2x and fi. for y. Then 2x + y equals A 3). equals k. Knowing = $k, go back to the first two equations for x, y, and fmin: The winning point (xmin, ymin)is ($k, f k). It minimizes the "distance squared," f = x2 + y2 = 3k2, from the origin to the line. 13.7 Constmints and Lagrange Muliipllen Question What is the meaning of the Lagrange multiplier A? Mysterious answer The derivative of *k2 is $k, which equals A. The multipler A is the devivative of fmin with respect to k. Move the line by Ak, and fmin changes by about AAk. Thus the Lagrange multiplier measures the sensitivity to k. Pronounce his name "Lagronge" or better "Lagrongh" as if you are French. If =fmin Fig. 13.21 Circlesf = c tangent to line g = k and ellipse g = 4: parallel gradients. + EXAMPLE 2 Maximize and minimize f = x2 y2 on the ellipse g = (x -1)' + 44' = 4. Idea and equations The circles x2 + y2 = c grow until they touch the ellipse. The touching point is (x,,,, ymi,) and that smallest value of c is fmin. As the circles grow they cut through the ellipse. Finally there is a point (x,,,, y,,,) where the last circle touches. That largest value of c is f,,, . The minimum and maximum are described by the same rule: the circle is tangent to the ellipse (Figure 13.21b). The perpendiculars go in the same direction. Therefore (fx, 4)is a multiple of (g,, gy), and the unknown multiplier is A: a. Solution The second equation allows two possibilities: y = 0 or A = Following up y = 0, the last equation gives (x - 1)' = 4. Thus x = 3 or x = - 1. Then the first + equation gives A = 312 or A = 112. The values of f are x2 y2 = 3' + 0' = 9 and ~ ~ + ~ ~ = ( - 1 )1.~ + 0 ~ = Now follow A = 114. The first equation yields x = - 113. Then the last equation + requires y2 = 5/9. Since x2 = 119 we find x2 y2 = 619 = 213. This is f,,,. Conclusion The equations (3) have four solutions, at which the circle and ellipse are tangent. The four points are (3, O), (- 1, O), (- 113, &3), and (- 113, -&3). The four values off are 9, 1,3,3. Summary The three equations are fx = Agx and fy = Ag,, and g = k. The unknowns are x, y, and A. There is no absolute system for solving the equations (unless they are linear; then use elimination or Cramer's Rule). Often the first two equations yield x , and y in terms of A and substituting into g = k gives an equation for A . At the minimum, the level curve f(x, y) = c is tangent to the constraint curve g(x, y) = k. If that constraint curve is given parametrically by x(t) and y(t), then 13 Partial Derlvclthres minimizing f(x(t), y(t)) uses the chain rule: df - af ---- dx af dy + -- = 0 or (grad f ) (tangent to curve) = 0. dt ax dt dy dt This is the calculus proof that grad f is perpendicular to the curve. Thus grad f is parallel to grad g. This means (fx , f,) = A(g, ,gy)- We have lost f, = 0 and fy = 0. But a new function L has three zero derivatives: 130 The Lagrange function is y x , y, A =f(x, y) - I(g(x, y) - k). Its three I ) derivatives are L, = L, = LA= 0 at the solution: Note that dL/aA = 0 automatically produces g = k. The constraint is "built in" to L. Lagrange has included a term A(g - k), which is destined to be zero-but its derivatives are absolutely needed in the equations! At the solution, g = k and L = f and k . a ~ / a =A What is important is fx = Ag, andf, = Agy,coming from L, = Ly = 0. In words: The constraint g = k forces dg = g,dx + gydy= 0. This restricts the movements dx and dy. They must keep to the curve. The equations say that d =fxdx +fydy is equal to Adg. f Thus df is zero in the aElowed direction-which is the key point. IH W MAXIMUM AND MINIMUM WT T O CONSTRAINTS The whole subject of min(max)imization is called optimization. Its applications to business decisions make up operations research. The special case of linear functions is always important -in this part of mathematics it is called linear programming. A book about those subjects won't fit inside a calculus book, but we can take one more step-to allow a second constraint. The function to minimize or maximize is now f(x, y, z). The constraints are g(x, y, z) = k, and h(x, y, z) = k,. The multipliers are A, and A,. We need at least three variables x, y, z because two constraints would completely determine x and y. 13P To minimizef(x, y, z) subject to g(x, y, z) = k, and h(x, y, z) = k2,solve five equations for x, y, z, A,, 2,. Combine g = k, and h = k2 with I Figure 13.22a shows the geometry behind these equations. For convenience f is x2 + y2 + z2, SO we are minimizing distance (squared). The constraints g = x + y + z = 9 and h = x + 2y + 32 = 20 are linear-their graphs are planes. The constraints keep (x, y, z) on both planes-and therefore on the line where they meet. We are finding the squared distance from (0, 0, 0) to a line. What equation do we solve? The level surfaces x2 + y2 + z2 = c are spheres. They grow as c increases. The first sphere to touch the line is tangent to it. That touching point gives the solution (the smallest c). All three vectors gradf, grad g, grad h are perpendicular to the line: line tangent to sphere => grad f perpendicular to line line in both planes grad g and grad h perpendicular to line. 13.7 Constraints and Lagmnge Multipliers 517 Thus gradf, grad g, grad h are in the same plane-perpendicular to the line. With three vectors in a plane, grad f is a combination of grad g and grad h: This is the key equation (5). It applies to curved surfaces as well as planes. EXAMPLE 3 Minimize x2 + y2 + z2 when x + y + z = 9 and x + 2y + 32 = 20. In Figure 13.22b, the normals to those planes are grad g = (1, 1, 1) and grad h = (1, 2, 3). The gradient off = x2 + y2 + z2 is (2x, 2y, 22). The equations (5)-(6) are Substitute these x, y, z into the other two equations g = x + y + z = 9 and h = 20: A1+A2 Al+2A2 A1+3A2 Al+A2 2 + ------- + ------- - 9 2 2 and 2 + Al+2A2 -=A1+3A2 20. - 2 ------- + 3 2 2 + After multiplying by 2, these simplify to 3A1 6A2 = 18 and 61, 14A2= 40. The + solutions are A, = 2 and A, = 2. Now the previous equations give (x, y, z) = (2,3,4). The Lagrange function with two constraints is y x , y, z, A,, A,) = f - A,(g - kl) - A2(h - k,). Its five derivatives are zero-those are our five equations. Lagrange has increased the number of unknowns from 3 to 5, by adding A, and A,. ,, The best point (2, 3,4) gives f = 29. The 2 s give af/ak-the sensitivity to changes in 9 and 20. grad h plane Fig. 13.22 Perpendicular vector grad f is a combination R , grad g + & grad h. INEQUALITY CONSTRAINTS In practice, applications involve inequalities as well as equations. The constraints might be g < k and h 2 0. The first means: It is not required to use the whole resource k, but you cannot use more. The second means: h measures a quantity that cannot be negative. At the minimum point, the multipliers must satisfy the same inequalities: R1 ,< 0 and A2 3 0.There are inequalities on the A's when there are inequalities in the constraints. Brief reasoning: With g < k the minimum can be on or inside the constraint curve. Inside the curve, where g < k, we are free to move in all directions. The constraint is , , not really constraining. This brings back f = 0 and f = 0 and 3, = 0-an ordinary minimum. On the curve, where g = k constrains the minimum from going lower, we have 1 < 0. We don't know in advance which to expect. " 13 Partial Derivatives For 100 constraints gi < k,, there are 100 A's. Some A's are zero (when gi < k,) and some are nonzero (when gi = k,). It is those 2'' possibilities that make optimization interesting. In linear programming with two variables, the constraints are x 0, y 0: The constraint g = 4 is an equation, h and H yield inequalities. Each has its own Lagrange multiplier-and the inequalities require A, 2 0 and A,> 0. The derivatives off, g, h, H are no problem to compute: Those equations make A, larger than A,. Therefore A, > 0, which means that the constraint on H must be an equation. (Inequality for the multiplier means equality for the constraint.) In other words H = y = 0. Then x + y = 4 leads to x = 4. The ymin) (4, O), where fmin = 20. solution is at (xmin, = At this minimum, h = x = 4 is above zero. The multiplier for the constraint h 2 0 must be A, = 0. Then the first equation gives 2, = 5. As always, the multiplier mea- sures sensitivity. When g = 4 is increased by Ak, the cost fmin = 20 is increased by 5Ak. In economics 2, = 5 is called a shadow price-it is the cost of increasing the constraint. Behind this example is a nice problem in geometry. The constraint curve x + y = 4 is a line. The inequalities x 2 0 and y 2 0 leave a piece of that line-from P to Q in Figure 13.23. The level curves f = 5x + 6y = c move out as c increases, until they touch the line. Thefivst touching point is Q = (4, O), which is the solution. It is always an endpoint-or a corner of the triangle PQR. It gives the smallest cost fmin,which is c = 20. 5s + 6y = c c too small .=R Fig. 13.23 Linear programming: f and g are linear, inequalities cut off x and y. 13.7 EXERCISES Read-through questions A restriction g(x, y) = k is called a a . The minimizing fmi, is f to the constraint curve g = k. The number E. equations for f(x, y) subject to g = k are b . The number turns out to be the derivative of s with respect to h . A is the Lagrange c . Geometrically, grad f is d to The Lagrange function is L = i and the three equations grad g at the minimum. That is because the e curve f = for x, y, j are i and k and . 1 . 13.7 Constmints and Lagrange Multipliers 519 To minimize f = x2 - y subject to g = x - y = 0, the three 13 Draw the level curves off = x2 + y2 with a closed curve C equations for x, y, d are m . The solution is n . In this across them to represent g(x, y) = k. Mark a point where C example the curve f(x, y) =fmin = 0 is a P which is crosses a level curve. Why is that point not a minimum off q ymin). to the line g = 0 at (xmin, on C? Mark a point where C is tangent to a level curve. Is With two constraints g(x, y, z) = kl and h(x, y, z) = k2 there that the minimum off on C? are r multipliers. The five unknowns are s . The five 14 On the circle g = x2 + y2 = 1, Example 5 of 13.6 mini- equations are f . The level surfacef =fmin is u to the mized f = xy - x - y. (a) Set up the three Lagrange equations curve where g = k, and h = k2. Then gradf is v to this for x, y, A. (b) The first two equations give x = y = curve, and so are grad g and w . Thus x is a combina- (c) There is another solution for the special value A = - 4, tion of grad g and v . With nine variables and six con- when the equations become . This is easy to miss straints, there will be' 2 multipliers and eventually A but it gives fmin = - 1 at the point equations. If a constraint is an B g < k, then its multiplier must satisfy A ,< 0 at a minimum. Problems 15-18 develop the theory of Lagrange multipliers. 15 (Sensitivity) Certainly L =f - d(g - k) has aL/ak = A. + 1 Example 1 minimized f = x2 y2 subject to 2x + y = k. Since L =fmin and g = k at the minimum point, this seems to Solve the constraint equation for y = k - 2x, substitute into prove the key formula dfmin/dk A. But xmin, = A ymin, , and fmin f, and minimize this function of x. The minimum is at (x, y) = all change with k. We need the total derivative of L(x, y, 1,k): , where f = . Note: This direct approach reduces to one unknown x. Lagrange increases to x, y, A. But Lagrange is better when the Equation (1) at the minimum point should now yield the first step of solving for y is difficult or impossible. = . sensitivity formula dfmin/dk 1 Minimize and maximizef(x, y) in 2-6. Find x, y, and A. ) 16 (Theory behind A When g(x, y) = k is solved for y, it gives a curve y = R(x). Then minimizing f(x, y) along this 2 f=x2y with g = x 2 +y2 = 1 curve yields af - ; af dR -0,-+--=o. ag agdR ax ay dx ax ay Those come from the rule: dfldx = 0 at the mini- , mum and dgldx = 0 along the curve because g = Multiplying the second equation by A= (af/ay)/(ag/ay) and 6 f = x + y with g = x1i3y2I3 k. With x = capital and y = = subtracting from the first gives aflay = = 0. ~ l s o labor, g is a Cobb-Douglas function in economics. Draw two laglay. These are the equations (1) for x, y, 1. of its level curves. 17 (Example o failure) A =f,/gy breaks down if g,, = 0 at the f + 7 Find the point on the circle x2 y2 = 13 wheref = 2x - 3y minimum point. is a maximum. Explain the answer. (a) g = x2 - y3 = 0 does not allow negative y because + + + + 8 Maximize ax by cz subject to x2 y2 z2 = k2. Write + (b) When g = 0 the minimum off = x2 y is at the point your answer as the Schwarz inequality for dot products: (a, b, c) (x, Y,z) < - k. (c) At that point f , = AgY becomes which is + 9 Find the plane z =ax +by c that best fits the points impossible. (x, y, Z)= (0, 0, l), (1,0, O), (1, 1, 2), (0, 1, 2). The answer a, b, c (d) Draw the pointed curve g = 0 to see why it is not tan- minimizes the sum of (z - ax - by - c ) at the four points. ~ gent to a level curve of5 10 The base of a triangle is the top of a rectangle (5 sides, 18 (No maximum) Find a point on the line g = x y = 1 + combined area = 1). What dimensions minimize the distance where f(x, y) = 2x + y is greater than 100 (or 1000). Write out around? gradf = A grad g to see that there is no solution. 11 Draw the hyperbola xy = - 1 touching the circle g = 19 Find the minimum of f = x2 + 2y2+ z2 if (x, y, Z) is x2 + y2 = 2. The minimum off = xy on the circle is reached restricted to the planes g = x + y + z = 0 and h = x - z = 1. at the points . The equationsf, = Agx and f , = dgY 20 (a) Find by Lagrange multipliers the volume V = xyz of are satisfied at those points with A = . the largest box with sides adding up to x + y + z = k. (b) 12 Find the maximum off = xy on the circle g = x2 + y2 = 2 Check that A = dVmax/dk. United Airlines accepts baggage (c) by solvingf = ilg, and f, = A , and substituting x and y into , g with x + y + z = 108". If it changes to 11I", approximately J: Draw the level curve f =fmax that touches the circle. ) , how much (by A and exactly how much does V increase? 520 13 Partial ~ e r h r o ~ v e s 21 The planes x = 0 and y = 0 intersect in the line x = y = 0, 27 With an inequality constraint g < k, the multiplier at a which is the z axis. Write down a vector perpendicular to the maximum point satisfies A >, 0. Change the reasoning in 26. plane x = 0 and a vector perpendicular to the plane y = 0. 28 When the constraint h 2 k is a strict inequality h > k at , Find A, times the first vector plus 1 times the second. This the minimum, the multiplier is A = 0. Explain the reasoning: combination is perpendicular to the line . For a small increase in k, the same minimizer is still available 22 Minimizef = x2 + y2 + z2 on the plane ax + by + cz = d- (since h > k leaves room to move). Therefore fmin is one constraint and one multi lier. Compare fmin with the (changed)(unchanged), and A = dfmin/dkis . distance formula - J in Section 11.2. 29 Minimize f = x2 + y2 subject to the inequality constraint 23 At the absolute minimum of flx, y), the derivatives x + y < 4. The minimum is obviously at , where f , are zero. If this point happens to fall on the curve and f, are zero. The multiplier is A = . A small , g(x, y) = k then the equations f = AgXand fy = AgYhold with change from 4 will leave fmin = so the sensitivity A= . dfmi,/dk still equals A. Problems 24-33 allow inequality constraints, optional but good. 30 Minimize f = x2 + y2 subject to the inequality constraint x + y $4. Now the minimum is at and the multi- 24 Find the minimum off = 3x + 5y with the constraints g = plier is A = andfmin = - small change to .A x + 2y = 4 and h = x 2 0 and H = y 30, using equations like 4 + dk changes fmin by what multiple of dk? (7). Which multiplier is zero? 31 M i n i m i z e f = 5 ~ + 6 y w i t h g = x + y = 4 a n d h = x b O a n d 25 Figure 13.23 shows the constraint plane g = x +y +z = 1 H = y < 0. Now A, < O and the sign change destroys chopped off by the inequalities x 2 0, y $ 0, z >, 0. What are Example 4. Show that equation (7) has no solution, and the three "endpoints" of this triangle? Find the minimum and choose x, y to make 5x + 6y < - 1000. maximum off = 4x - 2y + 5z on the triangle, by testing f at the endpoints. 32 Minimizef = 2x + 3y + 42subject to g = x + y + z = 1 and x, y, z 2 0. These constraints have multipliers A,> 0, A3 2 0, 26 With an inequality constraint g < k, the multiplier at the I , 2 0. The equations are 2 = A, + i 2 , , and 4 = minimum satisfies A < 0. If k is increased,fmin goes down (since A, + A,. Explain why A, > 0 and A, > 0 and fmin = 2. = dfmin/dk).Explain the reasoning: By increasing k, (more) (fewer) points satisfy the constraints. Therefore (more) (fewer) 33 A wire 4 0 long is used to enclose one or two squares points are available to minimize f: Therefore fmin goes (up) (side x and side y). Maximize the total area x2 + y2 subject to (down). x 2 0 , y$0,4x+4y=40. MIT OpenCourseWare Resource: Calculus Online Textbook Gilbert Strang The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit:
Mathematics, Grade 12. (MAP4C or MTT4G or a mathematics with a similar content.)
TI-84 Graphing Calculator Topics After transferring files between the TI-84 Plus and another calculator, you can then use the sending and/or receiving calculator to transfer the same files to a third calculator, usually without having[more…] The Texas Instrumentswebsite contains over 40 applications that you can download and install on your TI-84 Plus calculator. Most of these application programs are free; those that aren't free are very[more…] GeoMaster on the TI-84 graphing calculator can't find the area of a polygon formed by the intersection of two other polygons because GeoMaster doesn't know that it's there. To get GeoMaster to find the[more…] When you translate a geometric object, you move it in a given direction for a given distance. A vector is used to tell GeoMaster in the TI-84 Plus graphing calculator the direction in which to move the[more…] The Ti-84 Plus calculator's Catalog houses every command and function used by the calculator. However, it's easier to use the keyboard and the menus to access these commands and functions than it is to[more…] One of the benefits of using the TI-84 Plus calculator is that it has a better memory than you do. You can use the previous answer in the next arithmetic expression you want to evaluate, and you can reuse[more…] Most functions housed in the TI-84 Plus calculator's Math menu require that the argument be entered after entering the function. Such functions are easily recognizable by the parenthesis following the[more…] The second step of solving an equation in the TI-84 Plus graphing calculator's Equation Solver is to enter or edit the equation to be solved. The first step is to set the mode. To enter a new equation[more…] To find multiple solutions to an equation in the TI-84 Plus Equation Solver, you first need to find one solution to the equation. Place the cursor anywhere in the line that contains the variable you're[more…] You can create a GeoMaster file in two ways on the TI-84 Plus calculator. If the GeoMaster screen is empty, just start creating your geometric figures. If the GeoMaster screen is not empty, select[more…] Because two points determine a line, segment, ray, or vector, the directions for creating these objects with GeoMaster on the TI-84 Plus graphing calculator are much the same. To construct such objects[more…] A circle is determined by its center and its radius. The GeoMaster DRAW menu contains a plethora of tools that enable you to do mundane things such as construct lines, segments, triangles, polygons, and[more…] A triangle and an arc are determined by three points, so their construction is much the same. The GeoMaster DRAW menu on the TI-84 Plus calculator offers tools that enable you to construct triangles, arcs[more…] You can use the GeoMaster DRAW menu on the TI-84 Plus graphing calculator to create irregular and regular polygons, circles, triangles, and many other basic geometric shapes. To create an irregular polygon[more…] A regular polygon is determined by its center and the number of its sides. You can use the GeoMaster DRAW menu on the TI-84 Plus graphing calculator to create regular and irregular polygons, circles, triangles[more…]
Mathematics for Physics and Physicists Walter Appel "Mathematics for Physics and Physicists is a well-organized resource today's physicists must have to strengthen their math skills and to gain otherwise unattainable insights into their fields of study. Mathematics has always been and is still a precious. . . . One is delighted to see Appel's book maintains a nice balance between rigorous mathematics and physical applications. . . . It will lead potential physicists to embrace mathematics and they will benefit substantially."--Current Engineering Practice ADDITIONAL ENDORSEMENTS: "This nicely organized and pedagogically excellent book not only covers material that students of physics and engineering are conventionally assumed to need at an advanced undergraduate and graduate level but goes beyond that by including some important modern mathematics not usually included in such texts--such as Lebesgue integration, theory of distributions, probability theory, and differential geometry. It treats the material in a rigorous yet friendly manner from a physicist's perspective, and even includes interesting historical asides that convey to a student the idea that mathematics is not a dead subject. It is an excellent choice as a course textbook or as an addition to a physicist's bookshelf."--Elliott Lieb, Princeton University "As happens to many of us who teach a mathematics for physicists course to physics students, Appel--who for years taught such a course at the École Normale Supérieure de Lyon--faced the problem of finding a textbook that is neither too soft or imprecise nor too specialized mathematically. This book is Appel's own elegant way out of this dilemma. Physics students at the graduate level will benefit handsomely."--Michael Kiessling, Rutgers University "Mathematics for Physics and Physicists gives a charming exposition of many important concepts, including topics not covered in standard textbooks. Appel finds an excellent balance between mathematical rigor and physical applications, and the book is interspersed with short biographies of mathematicians and sets of illustrative problems. I wish that kind of book had been available when I was a student."--Andreas Brandhuber, Queen Mary, University of London
Unit specification Aims To give an introduction to the basic ideas of geometry and topology. Brief description This course unit introduces the basic ideas of the geometry of curves and surfaces in Euclidean space, differential forms and elementary topological concepts such as the Euler characteristic. These ideas permeate all modern mathematics and its applications. Intended learning outcomes On successful completion of this module students will have acquired an active knowledge and understanding of the basic concepts of the geometry of curves and surfaces in three-dimensional Euclidean space and will be acquainted with the ways of generalising these concepts to higher dimensions. Future topics requiring this course unit The ideas in this course unit will be developed further in third and fourth level course units in geometry and topology. Syllabus Recollection of lines and planes in R3. Equations in various forms, normal vector to a plane, distance from a point to a plane. Differential forms in R2 and R3. Geometrical meaning of differential forms. Examples: area of parallelogram, volume of parallelipiped.
You will be completing math assignments pertaining to the probability and statistics. Tasks will include watching videos, playing math games online, writing journals, completing discussions, and performing math computation and application skill problems.
Lattices and Ordered Algebraic Structures provides a lucid and concise introduction to the basic results concerning the notion of an order. Although as a whole it is mainly intended for beginning postgraduates, the prerequisities are minimal and selected parts can profitably be used to broaden the horizon of the advanced undergraduate. The treatment is modern, with a slant towards recent developments in the theory of residuated lattices and ordered regular semigroups. This book is a systematic treatment of real algebraic geometry, a subject that has strong interrelation with other areas of mathematics: singularity theory, differential topology, quadratic forms, commutative algebra, model theory, complexity theory etc. The careful and clearly written account covers both basic concepts and up-to-date research topics. It may be used as text for a graduate course. This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods. Moerdijk's lectures offer a detailed introduction to dendroidal sets, which were introduced by himself and Weiss as a foundation for the homotopy theory of operads. The theory of dendroidal sets is based on trees instead of linear orders and has many features analogous to the theory of simplicial sets, but it also reveals new phenomena. The third book of a three-part series, Algebraic, Graphics, and Trigonometric Problem Solving, Second Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and the accompanying practice exercises. Along with the activities and the exercises within the text, MathXL® and MyMathLab® have been enhanced to create a better overall learning experience for the reader. Technology integrated throughout the text helps readers interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops readers' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines. This is a highly self-contained book about algebraic graph theory which iswritten with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. This book constitutes the refereed proceedings of the Third International Conference on Algebraic Informatics, CAI 2009, held in Thessaloniki, Greece, in May 2009. The 16 full papers were carefully reviewed and selected from 25 submissions. The papers cover topics such as algebraic semantics on graph and trees, formal power series, syntactic objects, algebraic picture processing, finite and infinite computations, acceptors and transducers for strings, trees, graphs arrays, etc. decision problems, algebraic characterization of logical theories, process algebra, algebraic algorithms, algebraic coding theory, algebraic aspects of cryptography.
Wisdom is of God Latest News Mathematics The Mathematics Department offers many appropriate levels of courses in Mathematics subjects from Year 8 to 12. It is a major priority in Mathematics to choose the most appropriate level for each student, giving them the best opportunity to maximise their possibility for success at that level. Confidence for any student is essential for success. Opportunity For Success We believe in 'setting the students up for success' wherever possible, not for failure. In Year 8 to 10, Mathematics is compulsory and is taught at three distinct levels - Advanced, Standard and General, and every effort is made to 'place' each student in the appropriate level after studying their previous results. Advanced Mathematics Extension A range of more challenging extension activities are organised for those students in Advanced Mathematics in Year 8 to 10. Students are encouraged to participate in competitions, complete 'Home Sets' of more complicated and challenging Mathematical questions and complete generally harder levels of Mathematics worksheets. Mathletics The College pays for every Mathematics student to be able to access a 'Web-Based' Mathematics revision program called Mathletics. This program caters from the lowest levels of revision in Secondary School, through to the highest academic levels. Your child will be given their ID and password by their Mathematics teacher. This program is a wonderful opportunity for every student to revise, consolidate and develop their mathematical understanding. Any parent can take an opportunity to watch their child 'login' to the program and watch the opportunities that are available to their child, free of charge to every student. Senior School In Year 11 and 12, there exists a wider choice of Mathematics subjects of differing levels than the three levels of Year 8 to 10. The 'lowest' Level 2 course in Mathematics is Stage 1, then Stage 2 (somewhat harder) or Stage 3 (quite academic) and finally, the most able students may choose the Stage 3 'Specialist' course.
Welcome to the learning page of the Mathematics Department at Claverham Community College. The Mathematics Faculty is one of the biggest faculties at the College and boasts being able to assist pupils gaining exceptional results at GCSE. Lessons are structured to engage pupils at every possible opportunity with emphasis on developing personal thinking and learning skills to enable them to be able to use and apply maths in real life problem solving. The resources here are designed to reinforce work carried out in class and often stimulate further the mathematical mind. During KS3 pupils follow the MEP and links to these texts are included here. The topic revision tests and termly tests are taken by all sets based on their level of study. These are part of MEP and form part of APP. All students are expected to record the level at which they are working together with their personal target. This further enhances personal learning giving a clear path for improvement. During KS4 pupils follow a programme of study culminating in an external examination for an EDEXCEL linear GCSE in Mathematics. In order that students make full use of lesson time they are expected to come to each lesson with all equipment including calculator, pair of compasses, and protractor. We recommend the casio calculator which is available from the department at a subsidised rate.
Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions to selected exercises are available from the authors.
The eJMT is indexed by Zentralblatt and Zentralblatt's sister database for Mathematics Education, MathEduc. The eJMT, an international refereed journal, will address technology-based issues in all Mathematical Sciences. The eJMT will deal primarily, but not exclusively, with the use of technologies in mathematics and mathematics education research, pedagogy, and mathematics applications. The eJMT will communicate the most recent applications of technology in pure and applied mathematics and in electronic format, including video and sound. The eJMT will publish the source codes of programs together with submitted papers electronically, so readers can actually continue the experiments. The journal is also a valuable medium for high school mathematics teachers worldwide to discuss effective ways of implementing technology in their teaching. From country to country, the use of technology in high school mathematics varies greatly, and the journal is an excellent place to provide collective worldwide ideas and share their varying experiences. The eJMT has four basic areas of interest: Trend 1: Research in Mathematics and its Applications and Mathematics Education with Technology Trend 2: Instruction in Mathematics and its Applications with Technology Trend 3: Learning in Mathematics and its Applications and Mathematics Education with Technology - Keng Cheng ANG, National Institute of Education, Nanyang Technological University (Singapore), - Matthias KAWSKI, Arizona State University (USA), - Mirek MAJEWSKI, New York Institute of Technology, College of Arts and Sciences (USA).
Performance Task Assessment: Multistep Equations for Algebra 111 pages. A hiker finds himself overexerted after he reaches the summit of Mount Algebra and radios for help. Two hikers converge and then work together to get down the mountain and meet the paramedics. This assessment is an exercise in writing functions from context and using tables, graphs, and equations to provide an extended solution to a problem scenario. It includes both independent and collaborative elements and takes students directly to the heart of drafting meaningful solutions: a solution to a problem illustrates, generalizes, communicates, and verifies the results. An answer is just a number. This lesson is part of a series of problems I created that are focused around graphing linear functions and the use of new tools (tables, graphs, equations) to solve problems in context. Underneath the rich content is an important lesson about presentation and asks: "How important is presentation when drafting solutions?" This assessment is designed to be used after students have a basic understanding of how to solve one-variable equations (including those with variables on both sides). They should also have prior knowledge of coordinate graphing and the concept of functions (but not necessarily function notation). Place it at the end of your unit or combine it with my introduction to standards-based performance tasks and practice tasks (both available separately at my store) for a complete unit on problem solving in context. Also included in this download is a comprehensive grading rubric and a cover page with essential questions, common core standards alignment, student objectives, and teacher notes. Don't forget the free poster. PDF (Acrobat) Document File Be sure that you have an application to open this file type before downloading and/or purchasing. 965.82
Introduction While the focus of these modules in Math and Physics is on conceptual learning, a great deal of the "utility" of introductory courses in calculus and physics is in developing problem-solving skills. Hence any textbook has many traditional examples of problems and many more exercises for the student than are presented within these modules. A proper introduction to problem-solving starts by being reminded that the essence of scientific progress since Galileo is in experimenation and experimentation is founded on quantitative measurement. As an example, consider the pictures below. They show two scenes of a girl walking a cow to pasture while simultaneously holding a kite on a windy day. Which of the two scenes are you more likely to see on a March day in Pennsylvania farmland? Hard as it is to believe, the general principles of physics embodied in Newton's Laws do not, of themselves, preclude either scene from happening. These laws, when coupled with physical intuition as to the masses of objects, do give us a firm indication as to the likelihood of the second scene compared to the first. With that in mind, we can start to look from our qualitative view of how things should move to the techniques for considering quantitative determinations of their motion.
Math Foundations The refresher guide for math skills tested on the GMAT. Kaplan GMAT Math Foundations is the ideal refresher course for the large number of GMAT test ...Show synopsisThe refresher guide for math skills tested on the GMAT. Kaplan GMAT Math Foundations is math concepts is crucial. "Kaplan GMAT Math Foundations" features: Comprehensive coverage of the arithmetic, algebra, and geometry concepts tested by the GMATAn intensive, back-to-basics, tutor-led approach to math reviewHundreds of practice exercises to increase speed and accuracy "Kaplan GMAT Math Foundations" is a great study tool for both test takers who dread the Math section and those whose math skills are not their strength. This guide will give test takers the content review and skill building practice they need to feel confident on test
Contact an Adviser Please note the UW Flexible Option Pilot Program is now closed for registration. For more information sign up for the UW Flexible Option e-Newsletter. Each module in this pilot program is self-paced and competency-based. That means that you study at your own pace and then take assessments (tests) to determine if you have mastered the material. You will begin by taking a pre-assessment of your knowledge. If you already have experience with one or more of these subjects, you may take a practice test to determine if you are ready to go straight to the final assessment. For example, suppose that you are an adult student and you work in an office where you do statistical analysis on a regular basis. However, you've never taken a statistics course. In this case, you can contact a Student Services specialist in Independent Learning and sign up for the pilot Elementary Statistics class. Once registered, you will gain access to a pre-assessment. The pre-assessment helps you understand your level of knowledge of the subject. Based on the results of the pre-assessment, you might choose to study the material further, or, if you are comfortable with the subject, you might choose to take a practice test to formally assess your knowledge. If you do well on the module practice test, then you can take the module exam without having to study the material—since you already know it. If you don't do well on the practice test, then you'll be pointed to materials to help you study. You will not be penalized in any way for testing your knowledge. The goal is your success. The initial pilot program offerings include College Algebra, Elementary Statistics, and a Business Math and Personal Finance Certificate program. The module listings that follow provide detailed information regarding learning outcomes and links to recommended learning materials. Credit Offerings U3600-110 College Algebra (4 modules, 3 undergraduate credits) Provides an introduction to elementary and advanced algebra. Topics include a definition of function; linear and nonlinear functions; graphs, including logarithmic and exponential; theory of polynomial equations; and systems of equations. U3600-117 Elementary Statistics (4 modules, 3 undergraduate credits) Provides a basic understanding and use of statistical concepts and methods to facilitate study and research in other disciplines. The sequence includes measures of central tendency; measures of variability; grouped data; the normal distribution; the central limit theorem; hypothesis testing; estimation; T-distribution; and the Chi-square test. No partial credit is awarded for completing only individual modules in College Algebra and Elementary Statistics. Students must complete the entire module sequence to receive credit for the class. Non-credit Offerings Business Mathematics and Personal Finance Certificate This non-credit program consists of five related course-equivalents which are quantitative in nature and deal with analytical skills. A certificate of completion will be issued to those who show mastery of the competencies in Business Math, Business Statistics, and Personal Finance. Two additional topics, Foundations of Business Math and Preparation for Business Statistics, are available for individuals who would like to brush up on math and analytical basics before attempting the other topics. Required Core Courses Covers basic analysis of data using descriptive statistical measures and charting; calculation of basic probabilities; interpretation of data with confidence intervals and hypothesis testing, including means and proportions of simple populations; comparing means and proportions of two populations; and multiple comparisons (means and proportions of several populations). C216-OF49 Business Math (2 modules, 2 CEUs) Covers foundations of math, including whole numbers, fractions, decimals, and solving for unknowns; and percents and their applications, including percents, discounts, markups and markdowns, payroll, and sales, excise, and property tax.
This Tutorial, intended for mature students, covers the Algebra Topics taught in School and required for College. It makes Algebra easy by carefully explaining the Algebra Rules with examples of how to apply them. Many people have trouble with Algebra because when it was taught in school, they weren\'t ready to absorb the abstract Rules. Now, with maturity, the Algebra Rules are simple to learn and Algebra becomes easy
Ideas obtained from other people, through reading books or papers, by listening to talks, from personal correspondence, or in conversation, must be credited to that person. Even if you substantially extend the idea or apply it to a different situation, you must cite the original idea. Because of the issue of credit, keeping up in one's field is not only the mark of a good mathematician, but of an ethical one. All projects, papers, and talks done for a grade should describe the source and background of the problem, give the names of contributors to the problem, and cite relevant papers. Mathematics is very much a social activity. Major conferences, small seminars, and private professional discussions are the lifeblood of research. You are expected to talk about your work. On the other hand, the quantity, quality, and originality of your research are major components of your grade. Students should be careful not to "steal" each other's work. You should be aware of what others are working on and how it relates to your topic. Faculty have an obligation to do the same. Sometimes one must walk a fine line. Stetson faculty are expected to freely discuss their research interests and ideas with all students in order to help students find interesting projects. They are also sometimes willing to work on a suitable problem posed by the student. A student should not take the idea of one faculty member and propose it to another without the consent of the first faculty member. New results are often disseminated informally, in talks or preprints, so as to keep everyone up to date. New ideas presented in this fashion are the property of the author. If his or her idea is used prior to formal submission, proper acknowledgment is still due. When writing or speaking about mathematics, you should not intentionally suppress the work of others, nor improperly detract from that work. If you discover an error in your own work, you should attempt to correct it or withdraw it. Do not announce results you don't yet have; and once announced, results should be submitted in a timely fashion. All of the above is doubly important if you intend to publish your results in an academic journal. If you discover you have achieved the same result simultaneously with someone else, it may be appropriate to offer or accept joint authorship. The mathematical community does not recognize "independent discovery" based on ignorance of well-known results. The AMS statement also addresses other issues, including that of equal opportunity: "Mathematical ability must be respected wherever it is found, without regard to race, gender, ethnicity, age, sexual orientation, religious or political belief, or disability."
Our users: Great stuff! As I was doing an Internet search on effective math software, I came across the Algebrator Web site. I decided to purchase the software after seeing the online demo and never regretted it. Thanks! Christian Terry, ID. This new version is a vast improvement over the old one. John Davis, TX There are other math tutoring programs and then there is Algebrator. No other software even comes close! Seth Lore, IA07-23: prentice hall algebra 2 answer key download second order differential equations matlab finding highest common factor worksheet math trivia for grade 3 ks3 maths worksheets printable Printable Coordinate Graphs math cheats on radicals rational expression ti 84 program PSSA seventh grade formula chart reciprocal,ratios,fractions worksheets for fourth grade factorial expressions on a ti-89 8th grade math worksheets linear equations estimating multiplication and division mix number in to decimal calculator how to tell if a question is combination or permutation solving simultanious equations matrix ti-89 convert decimal to fraction in java simplifying radicals to decimals tricky maths quiz filling and wrapping lessons solve quadratic imaginary ti 89 worksheets graphing linear data how to figure out algebra ratios step-by-step instructions on how to do graphs, functions, and linear systems
books.google.co.uk - As in previous editions, the focus in INTRODUCTORY ALGEBRA remains on the Aufmann Interactive Method (AIM). Students are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. The role of... Algebra
Of what? Would it be of the second edition, that was published in 1992? Or the first? \| 100-101 was not too flattering: \| \| "The majority of the chapters stop short of describing (and therefore \| providing algorithms and code for) state of the art algorithms." \| \| "In other cases the advise section appears to be badly out of date. The \| section on stiff ordinary differential equations does not mention any \| modern codes or techniques, indeed the majority of algorithms discussed \| and the recommended bibliography are almost entirely pre 1980." That would be appropriate if the review was of the first edition. In any case, the review is obsolete. The third edition was published in 2007.
4.6: The Second Derivative, Curvature, Concavity, and Acceleration (4) 4.7: Approximating Functions with Polynomials (4) Chapter 5: Applications of Derivatives 5.1: Extreme Values of a Function (5) 5.2: Reasoning About Functions: Continuity and Differentiability (4) 5.3: Leading Behaviour and L'Hôpital's Rule (5) 5.4: Newton's Method (4) 5.5: Stability of Discrete-Time Dynamical Systems (4) 5.6: The Logistic Dynamical System and More Complex Dynamics (4) 5.7: Case Study: Panting and Deep Breathing (2) Chapter 6: Integrals and Applications 6.1: Differential Equations (4) 6.2: Antiderivatives (4) 6.3: Definite Integral and Area (5) 6.4: Definite and Indefinite Integrals (6) 6.5: Techniques of Integration: Substitution and Integration by Parts (5) 6.6: Applications (4) 6.7: Improper Integrals (4) Chapter 7: Differential Equations 7.1: Basic Models with Differential Equations (4) 7.2: Equilibria and Display of Autonomous Differential Equations (4) 7.3: Stability of Equilibria (4) 7.4: Separable Differential Equations (6) 7.5: Systems of Differential Equations; Predator-Prey Model (4) 7.6: The Phase Plane (3) 7.7: Solutions in the Phase Plane (3) 7.8: The Dynamics of a Neuron (3) Questions Available within WebAssign Most questions from this textbook are available in WebAssign. The online questions are identical to the textbook questions except for minor wording changes necessary for Web use. Whenever possible, variables, numbers, or words have been randomized so that each student receives a unique version of the question. This list is updated nightly.
Mathematics 2 [ LFSAB1102 ] Functions of several real variables ; vector analysis ; linear algebra ; linear differential equations with constant coefficients ; introduction to data analysis and reasoning in a context of random uncertainty. Study and handling of the above-mentioned concepts for their use in later courses. Training in the domains of rigor and abstraction by studying important proofs in calculus or algebra, and by constructing proofs featuring interaction between several different concepts or notions. Resolution of problems or exercises requiring the use of several mathematical tools. Aims After completing this course, students will be able to: Handle functions of several real variables. Master advanced notions in linear algebra. Conduct mathematical reasoning and write short proofs in a rigorous manner. Understand and use different proof techniques. Deal with problems, exercises and proofs for which not all data is provided explicitly. Interpret a problem, exercise or statement from various points of view (e.g. algebraic point of view or geometric point of view). Model mathematical situations involving random elements. Solve exercises and understand results whose difficulty warrants formal definitions and advanced theorems. Approach theories whose formalism exceeds the framework of intuitive examples and which require abstraction.
Here are some details on the exam: - Its day, time, and location are in the "Final Exam" block at the bottom of the syllabus. - It's comprehensive, with emphasis on sections 7.3-7.5 and chapters 8 and 9. (It won't cover book sections I didn't discuss in lecture.) - It's closed book. - You may bring three sheets of notes (8.5" by 11"), each with writing on both sides. I recommend writing on my posted formula sheets, the last of which is in the "Final Exam" block of the syllabus. - It requires a calculator. - I'll provide tables, which are the same as the exam 2 tables (posted in the week 12 line of the syllabus). - It's multiple choice. - I won't ask you to solve huge calculating problems. One way I can test these kinds of problems is to do what I sometimes do in lecture: solve the problem with software myself, and then display the output with some holes that we figure out by hand (with a calculator). - There isn't a sample final exam. Here are some studying steps I recommend: - Solve the homework and quiz problems as they're assigned. (Include the 9.5 homework and Quiz 9, even though they won't earn you points directly.) - Read the formula sheet. Figure out any formula you don't understand by studying the lecture notes or book. Add notes to the formula sheet as needed. (Some students add a few worked examples.) - Review the lecture notes. - Solve additional odd problems from the book, checking your answers in the back. - Redo old quizzes. (Adding low scores to your quiz record doesn't change your high score, which is already recorded in Learn@UW.)
Description An introduction to mathematics which starts with simple arithmetic and algebra and proceeds through to graphs, logarithms, trigonometry to calculus and imaginary numbers. The author, who is internationally renowned for his innovative teaching methods, offers insights into the pleasures of mathematics that will appeal to readers of all backgroundsMathematicians Delight
Think Through Math - Idaho Math Initiative Resource Think Through Math (TTM) is an online, supplemental math instruction and tutoring program that will help raise student achievement in Idaho by providing students with focused instruction, rigorous math problems, access to live certified teachers and a motivation program with rewards for working on math problems. Think Through Math works with the classroom teacher to tailor the online math instruction to the needs of the individual student. Students can access the program anywhere they get Internet access - whether at home, at school or in a library. Currently, schools in more than 25 states across the country are using Think Through Math as a supplement to instruction and noticing increased student achievement. Please visit to learn more about Think Through Math. The Think Through Math (TTM) computer-assisted program is now available to utilize as curriculum for Algebra 1 credit recovery and district approved Alternate Route for Graduation. The Think Through Math curriculum option can be used to facilitate the instruction of the required core math content, by an Idaho certified, highly qualified teacher of record. The programs are simple to implement and monitor, and students will go through a rigorous set of coursework. The credit recovery pathway will give students a solid foundation in Algebra 1 and will prepare them for success in Geometry and future math courses. The Alternate Route for Graduation pathway will give students a solid foundation in math skills aligned to the 10th grade ISAT blueprint. TTM's Algebra 1 credit recovery program initially enrolls students into a pathway of 62 "Target Lessons". The Alternate Route to Graduation program enrolls student into a pathway of 69 "Target Lessons". The pathways were constructed in conjunction with the Idaho SDE. All lessons for the credit recover program are aligned to the Common Core Standards Algebra 1 course and the Idaho Algebra 1 standards. The Alternate Route for Graduation lessons are aligned to the 10th grade ISAT blueprint. When the Smarter Balanced Assessment blueprint has been finalized, TTM will begin working on aligning the pathway to the new assessment blueprint; to be available when Idaho moves to the new assessment. For both programs, students begin by taking a brief placement test which will determine a student's readiness. Based on these results, additional pre-cursor lessons will be added into each student's pathway. Additionally, TTM monitors student progress and adds additional pre-cursors as needed to fill students' learning gaps. When students are working outside of the classroom, they can get assistance from TTM's live certified teachers. As with all coursework, the teacher is a key component in a student's success. The benefit of Think Through Math is that multiple students can be working toward recovering credit on a very individualized basis. An appropriately certified and qualified teacher, however, is necessary to ensure that students have the support to grasp the concepts being presented. As students work through their pathway, reports on student progress are available through the teacher portal. These easy-to-use reports show how much progress students are making toward completing the Target and pre-cursor lessons on their pathway. The Think Through Math program does not qualify as a stand-alone course such as those offered through IDLA. Any computer-based program must be supervised by a properly certified and qualified "teacher of record" who must actually be present in the classroom throughout the sessions. If Think Through Math is used for math credit recovery or alternate route for graduation, a math assignment code should be used when reporting in ISEE and the teacher of record facilitating the program must hold a math endorsement and be highly qualified in order to grant graduation credit. Please see documents below for in-depth information on both programs, including tools for monitoring student progress and grading. Also attached are documents on standard alignment. For Idaho Education Laws and Rules on High School Credit Requirements for Mathematics, please see Idaho Administrative Code 08.03.02.105.01.d. This code can be accessed online at Please be aware that a credit recovery course must meet the appropriate content standards of the original course. For Idaho Education Laws and Rules on Alternate Route for Graduation, please see Idaho Administrative Code 08.03.02.105.06, at Please be aware that alternate routes to graduation are locally established and must go before the local school board for approval. Once approved at the local level, the alternate route for graduation must be sent to the Idaho Board of Education for review. A copy of each districts' alternate plan is kept on file at the Office of the State Board of Education. For more information on the steps to establish a local alternate route for graduation, please contact Allison McClintick with the Idaho Board of Education, [email protected]. To utilize Think Through Math in either capacity communicated in this email, please contact Marisa Alan, TTM Regional Program Manager, at [email protected]. The winner of the 2013 State Showdown and bragging rights for an entire school year is... IDAHO. The Play-by-Play Action Report: The final week of competition, Idaho won the weekly challenge and was awarded 30,000 points. West Virginia battled back and won the Teamwork Bonus round to earn 10,000 points to their overall state score. But Idaho took the lead by winning the Evening/Weekend Bonus rounds to make their overall state score 125,000! Congratulations to all of the classes and students that competed in this year's contest. Everyone is a winner in the eyes of Think Through Math! MVP State Showdown Classes - The two winners for best performance in each of the bonus categories are: Bonus Bonus round winners were selected in a random drawing from all classes that qualified. A class may only win in one bonus category. The Winning classes will receive the following prizes: a TTM Party Pack, a State Showdown banner to hang in their classroom and TTM t-shirts for students and their teacher(s). A Think Through Math representative will be contacting the classroom teacher to coordinate distribution of the prizes. APANGEA IS NOW THINK THROUGH MATH Apangea, the web-based math intervention system available to all schools in the state of Idaho, is now called Think Through Math. And it's undergone more than just a name change. It now offers additional services and is available from grade 3 through Algebra I. Credit Recovery and Alternate Route Programs will also be available through Think Through Math in the very near future. Think Through Math (TTM) is still available at no cost to Idaho schools and districts. "We're changing our name to reflect who we are - a rigorous program that teaches students how to think mathematically," said Kevin McAliley, CEO of Think Through Math. "Leveraging the best of Apangea and everything we have learned from our partners, Think Through Math is focused on preparing all students for the highly anticipated, rigorous Common Core and STAAR assessments with their emphasis on mathematical thinking." Districts across Idaho can not only access and utilize Think Though Math, but they also have access to onsite and online training and coaching. These services are included in TTM's contract with the Idaho State Department of Education (through the Idaho Math Initiative,) and are also available to districts at no charge. TTM's dedicated team of Idaho-specific program managers is ready to work with districts and educators to ensure that schools are utilizing TTM in a way that will improve student achievement and prepare students for the Common Core State Standards. Common Core Ready Targeting the most critical foundations for algebra, Think Through Math provides rigorous instruction and meaningful practice designed to develop understanding. Our mission is to provide underperforming students with an accelerated path to algebra, as well as to college and career. Think Through Math follows the recommendations of the Common Core State Standards for focused and coherent instruction that provides more time for students to reason with, reflect upon, and practice mathematics. Focused Content Think Through Math is focused on the critical foundations for algebra readiness. To accelerate learning for underperforming students, it is not necessary to reteach every missed skill and concept. This program focuses on the development of skills and concepts that are essential for success at grade-level and always teaches in students' zone of proximal development. Coherent Structure The program emphasizes that there is a structure and coherence to mathematics, which helps students apply their understanding to new topics and allows students to successfully transition from elementary content to middle school, high school, and beyond. Deepen Conceptual Understanding Deep understanding means that concepts are well represented and well connected to other concepts. Think Through Math makes extensive use of models-real world situations, manipulatives, number strips, graphs, and diagrams-to help students see the connections between different topics and develop strategies and reasoning that serve as the foundation for learning more abstract math. A Formula For Success Personalized, Adaptive, Intelligent Underperforming students require intensive, individualized instruction that addresses their unique concerns. Think Through Math is intelligent and adaptive; it collects data based on individual responses and adjusts instruction to meet a student's individual needs. LIVE Teacher Support - Bilingual! As students work independently on the computer at home or school, live bilingual support is only a click away. Students have access to just-in-time support from state-certified teachers precisely when they need it. The TTM teachers are available day and night and have plenty of weekend office hours. Unprecedented Motivation Many students are significantly stressed or fatigued by mathematics. Too many have never experienced success with math and have given up. Think Through Math focuses on motivating students to do more math both during and after school with idaho state department of education 2012 Back-to-school teacher toolkit 19 its uniquely 21st century motivation system-a powerful blend of intrinsic and extrinsic motivators. The system is based on a single idea: reward effort. Again, the TTM program and allAgain, the TTM program and all support services are provided at no cost to Idaho educators; training can be provided on a district or site level, based on need. To set up training or for more information, please contact your regional program manager. Informational Fliers TTM Technical Check To determine if your computer is compatible with TTM's technical specifications, sign into your teacher account and click on the username in the upper right hand corner. You will see a drop down menu where "Tech Check" is an option. Click on "Tech Check" and an automatic check will run. For full TTM technical requirements, please refer to the PDF document below. If you are having further difficulties, please contact your school or district IT support staff.
Part of programme Learning outcomes After completing this course with a passing grade the student should be able to Use, explain and apply the fundamental concepts and problem solving methods of one variable Calculus, especially: - Use the derivative to investigate functions, e.g. sketch graphs and solve extremal value problems - Use Taylor's formula to approximate functions with polynomials to a desired degree of accuracy - Explain the definition of the Riemann integral and account for some of its applications, compute integrals using anti-derivatives, partial integration and change of variables - Solve certain linear ordinary differential equations with constant coefficients and explain how they are used in applications - Compute some elementary limits and use these to study the behavior of a function locally or asymptotically Propose mathematical models for applications that can be described by functions of one variable, discuss relevance and accuracy of such models, and be aware of how mathematical software can be used, for example to plot graphs and solve equations Read and understand mathematical text about functions of one variable and their applications, communicate mathematical reasoning and computations within this field orally as well as in writing in such a way that it is easy to follow For the higher grades the student should also be able to: Deduce some particularly important theorems and formulas Generalize and adapt the methods to fit in new situations Solve problem that require complex computations in several steps Explain the mathematical theory behind the concepts limit, continuity, series
Overview Main description An ideal course text or supplement for the many underprepared students enrolled in the required freshman college math course, this revision of the highly successful outline (more than 348,000 copies sold to date) has been updated to reflect the many recent changes in the curriculum. Based on Schaum's critically acclaimed pedagogy of concise theory illustrated by solved problems, Schaum's Outline of College Mathematics features: Mathematical modeling throughout Modernized graphs Graphing and scientific calculator coverage More than 1,500 fully solved problems Another 1,500 supplementary problems And much more Table of contents Elements of AlgebraFunctionsGraphs of FunctionsLinear Equations Simultaneous Linear EquationsQuadratic Functions and EquationsInequalitiesLocus of an Equation The Straight LineFamilies of Straight Lines The CircleArithmetic and Geometric Progressions Infinite Geometric Series Mathematical Induction The Binomial Theorem Permutations Combinations Probability Determinants of Order Two and ThreeDeterminants of Order nSystems of Linear EquationsIntroduction to Transformational GeometryAngles and Arc LengthTrigonometric Functions of a General AngleTrigonometric Functions of an Acute AngleReduction to Functions of Positive Acute AnglesGraphs of the Trigonometric FunctionsFundamental Trigonometric Relations and IdentitiesTrigonometric Functions of Two AnglesSum, Difference, and Product Trigonometric FormulasOblique TrianglesInverse Trigonometric FunctionsTrigonometric EquationsComplex NumbersThe Conic Sections Transformations of CoordinatePoints in SpaceSimultaneous Quadratic EquationsLogarithmsPower, Exponential, and Logarithmic CurvesPolynomial Equations, Rational RootsIrrational Roots of Polynomial Equations Graphs of PolynomialsParametric EquationsThe Derivative Differentiation of Algebraic Expressions Applications of DerivativesIntegration Infinite SequencesInfinite SeriesPower SeriesPolar CoordinatesIntroduction to the Graphing CalculatorThe Number System of AlgebraMathematical Modeling Author comments Philip Schmidt, Ph.D. was the associate provost of Berea College, with the academic rank of professor of mathematics. He is the author of several Schaum's Outline titles. Back cover copy Perfect for high-school seniors and college freshman Covers the fundamentals of basic college mathematics Reflects the newest curriculum Over 1500 solved problems Use with these courses: College Algebra Trigonometry Discrete Mathematics Pre-Calculus Calculus Introduction to Mathematic Modeling SCHAUM'S OUTLINES OVER 30 MILLION SOLD Master the fundamentals of College Mathematics with Schaum's--the high-performance study guide. It will help you cut study time, hone problem-solving skills, and achieve your personal best on exams and projects!Inside, you will find: Complete coverage of the basics in Algebra, Trigonometry, Discrete Mathematics, and Calculus
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them. This book introduces the basic concepts to... More > help you prepare for higher levels of Algebra. -JwL< Less
Algebra 1 Teachers This is a website created for Algebra Teachers that are adopting Common Core Math! Lesson Plans Take a look around the site! You will find unique algebra lesson plans for every skill. In week 2 you can find a great algebra lesson to help students write equations for functions and actually understand them complete with applets. In week 12, students will be motivated to learn algebra when they take a look at the relationship between their favorite movies and profit. For current thoughts and ideas on everything from lesson plans and updates to how it is going in my room, please take a look at my blog. Assessments Assessments are an area where we seem to all be watching and waiting. Please feel free to sign up for the newsletter to receive my assessments. I am in the process of switching them over to a dropbox so that everybody can access them. This will be ready soon. Performance Tasks No one seems to know for sure what they new performance tasks will look like, but offering our students lots of opportunities to learn and grow with some examples will be a great start. You can find some that I have created and examples from other places around the web on the blog. This is a great place to start! Common Core Although this website is using the Common Core State Standards for math as a beginning point, you will find great lessons for algebra regardless of your curriculum. As you browse, I hope you find what you are looking for! Algebra 1 Teachers is a tool that will help with Algebra 1 scope and sequence using the new Common Core State Standards. Lesson plans, assessments, activities, organization, and even tips on keeping your sanity will be addressed on this website and in my newsletter. While there is controversy surrounding the Common Core State Standards, this site is not an endorsement or a place for that discussion. Algebra is the gate keeper for student success and we must do an outstanding job for the kids future and ours as well. Most of us do want to retire someday :) Implementation As we move into implementation of the Common Core Standards we are seeing an amazing shift in what students must be able to accomplish at the end of Algebra 1. Solving linear equations is no longer taught at the Algebra 1 level, but instead at the middle school level. This is wonderful if they retain the information, but what do we do as the middle school is still shifting to the Common Core? I believe that there are some very definable and measurable ways that we can close the gap between what they are supposed to know and what they do know. Scope and Sequence The math department in my school will be using the Mathematics Common Core Toolbox scope and sequence for Algebra 1 next year. We all understand that this is a working document, and that there will need to be changes due to our population and the rate of implementation of the Math Common Core to the grades under us. We will be starting with the necessities and as holes no longer need to be filled and concepts no longer have to be retaught, we will be adding to the list of covered material and Quadrant D type lesson plans. Please sign up for the newsletter to receive assessments. It is my small attempt to keep them out of the hands of the kids :)
BEGIN:VCALENDAR PRODID:-//Microsoft Corporation//Outlook MIMEDIR//EN VERSION:1.0 BEGIN:VEVENT DTSTART:20101113T163000Z DTEND:20101113T180000Z LOCATION:297 DESCRIPTION;ENCODING=QUOTED-PRINTABLE:ABSTRACT: Although physics faculty are incorporating computers to enhance physics education, computation is often viewed as a black box whose inner workings need not be understood. We propose to open up the computational black box by providing Computational Physics (CP) curricula materials based on a problem-solving paradigm that can be incorporated into existing physics classes, or used in stand-alone CP classes. The curricula materials assume a computational science point of view, where understanding of the applied math and the CS is also important, and usually involve a compiled language in order for the students to get closer to the algorithms. The materials derive from a new CP eTextbook available from Compadre that includes video-based lectures, programs, applets, visualizations and animations. SUMMARY:Physics: Examples in Computational Physics I PRIORITY:3 END:VEVENT END:VCALENDAR
Module NUMBER THEORY Module code: MT444P Credits: 5 Semester: 2 Department: MATHEMATICS AND STATISTICS International: Overview Module Objective: To introduce students to classical analytic Number Theory. Topics to include some of the following: The arithmetic functions and identities. Algebraic and transcendental numbers. Continued fractions (including solving congruence equations by continued fractions, periodic continued fractions, Brounkner's Algorithm and Pell's equation). Approximation of irrationals by rationals (Liouville's theorem and construction of transcendental number). Quadratic residues, Euler's criterion and the Quadratic Reciprocity Law (with proof). Jacobi symbol, its reciprocity law and applications. Distribution of primes. Chebyshev's Theorem.
Description This course includes basic concepts and operations, linear equations and inequalities, word problems, exponents, factoring, simple quadratic equations, and graphing.Students will receive a grade of either "Satisfactory" or "Unsatisfactory."One of the requirements for receiving a "Satisfactory" grade is passing the state mandated Florida College Basic Skills Exit Test. Contact Attendance/Withdrawals Regular, punctual attendance is essential to your success in this course and is therefore mandatory.Missing class, arriving late, or leaving early will a ffect your grade, as discussed in the next section ("Methods of Evaluation"). I f you wish to withdraw from this course, it is your responsibility to go to the Admissions Office and do so officially, to avoid receiving a "U" for the course. If you miss a test or quiz for any reason, you will get a 0 on it.Make-up tests are given in extreme conditions, but only if approved by the instructor before the in-class test is given (do not wait until the next day to request a make-up). Make-up tests may be allowed at the discretion of the instructor.The instructor may ask the student to supply supporting documentation (such as a doctor's note). Method of Evaluation You will receive a course grade of either "S" for "Satisfactory" or "U" for "Unsatisfactory," based on whether you qualify for and successfully complete the Florida College Basic Skills Exit Test.If you qualify for the Exit Test and earn a 23 or better on it, you will receive an "S."If you receive less than a 23 on the Exit Test or do not qualify to take it, you will receive a "U."You must earn an "S" to progress to the next course, MAT 1033, Intermediate Algebra, or MTB 1370, Math for Health Related Professions. In order to qualify to take the Exit Test, you must obtain at least a 70% course average.If, at the end of the term, your course average is less than 70%, then you will not qualify to take the Exit Test, and you will receive a "U" for the course.There will be no exceptions. If you qualify for the Exit Test, you will have two opportunities to take it.The first opportunity (the initial test) will be given on Tuesday, December 7, 4:30 – 6:20 pm.The second opportunity (the retake) will be on Thursday, December 9, 4:30 – 6:20 pm.No other testing dates will be provided.If you take the initial test and do not pass it, you may come back for the retake and try again; if you fail to show up for the retake, you will receive a "U" for the course.If you miss the initial test, then the retake will be your only opportunity to take the Exit Test.Make-ups for the Exit Test (other than the retake) are against LSCC policy. Quizzes will typically consist of 5 problems that are very similar to the homework exercises or will be problems that you copy from your homework.They will usually be announced at the previous class session.You must show your work in all cases in order to receive credit for your answers. If you are not present in the room when the quiz is given you will receive a 0 on that quiz.No make-up quizzes will be given for any reason. Your Class Attendance/Participation Grade is based on 180 points.In order to earn the full 180 points, you must attend every class, arrive on time and stay until class is dismissed, maintain appropriate behavior throughout class, and participate fully in all class work assigned by your instructor.For each class you miss, you will lose 6 points.You may also lose points for arriving late, leaving early, behaving inappropriately, or failing to participate in assigned class work.Your Class Attendance/Participation Grade will be the total number of points you have earned, divided by 180. Your Computer Work Grade is based on 250 problems assigned on the "My Math Lab" Internet site ( work will generally be done outside of class, on your own time. You may access "My Math Lab" on the computers in the LSCC Learning Center, or on your home computer if you have appropriate hardware and an Internet connection.However, you are strongly encouraged to do as much of the work as possible in the Learning Center, since there are tutors on duty who can help you with any difficulties that may arise.Details on accessing and using "My Math Lab" will be provided in class.Your Computer Work Grade will be the total number of assigned problems you have correctly completed, divided by 250. My "MyMathLab CourseID" for the Fall 2010 semester is morrill55556. Notes You will receive a course grade of either "S" for "Satisfactory" or "U" for "Unsatisfactory," based on whether you qualify for and successfully complete the Florida College Basic Skills Exit Test.If you qualify for the Exit Test and earn a 23 or better on it,you will receive an "S."If you receive less than a 23 on the Exit Test or do not qualify to take it, you will receive a "U."You must earn an "S" to progress to the next course, MAT 1033, Intermediate Algebra, or MTB 1370, Math for Health Related Professions.
Description The Rockswold/Krieger algebra series uses relevant applications and visualization to show students why math matters, and gives them a conceptual understanding. It answers the common question "When will I ever use this?" Rockswold teaches students the math in context, rather than including the applications at the end of the presentation. By seamlessly integrating meaningful applications that include real data and supporting visuals (graphs, tables, charts, colors, and diagrams), students are able to see how math impacts their lives as they learn the concepts. The authors believe this approach deepens conceptual understanding and better prepares students for future math courses and life. Table of Contents 1. Introduction to Algebra 1.1 Numbers, Variables, and Expressions 1.2 Fractions 1.3 Exponents and Order of Operations 1.4 Real Numbers and the Number Line 1.5 Addition and Subtraction of Real Numbers 1.6 Multiplication and Division of Real Numbers 1.7 Properties of Real Numbers 1.8 Simplifying and Writing Algebraic Expressions 2. Linear Equations and Inequalities 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities 3. Graphing Equations 3.1 Introduction to Graphing 3.2 Linear Equations in Two Variables 3.3 More Graphing of Lines 3.4 Slope and Rates of Change 3.5 Slope-Intercept Form 3.6 Point-Slope Form 3.7 Introduction to Modeling 4. Systems of Linear Equations In Two Variables 4.1 Solving Systems of Linear Equations Graphically and Numerically 4.2 Solving Systems of Linear Equations by Substitution 4.3 Solving Systems of Linear Equations by Elimination 4.4 Systems of Linear Inequalities 5. Polynomials and Exponents 5.1 Rules for Exponents 5.2 Addition and Subtraction of Polynomials 5.3 Multiplication of Polynomials 5.4 Special Products 5.5 Integer Exponents and the Quotient Rule 5.6 Division of Polynomials 6. Factoring Polynomials and Solving Equations 6.1 Introduction to Factoring 6.2 Factoring Trinomials I (x + bx + c) 6.3 Factoring Trinomials II (ax2 + bx + c) 6.4 Special Types of Factoring 6.5 Summary of Factoring 6.6 Solving Equations by Factoring I (Quadratics) 6.7 Solving Equations by Factoring II (Higher Degree) 7. Rational Expressions 7.1 Introduction to Rational Expressions 7.2 Multiplication and Division of Rational Expressions 7.3 Addition and Subtraction with Like Denominators 7.4 Addition and Subtraction with Unlike Denominators 7.5 Complex Fractions 7.6 Rational Equations and Formulas 7.7 Proportions and Variation 8. Radical Expressions 8.1 Introduction to Radical Expressions 8.2 Multiplication and Division of Radical Expressions 8.3 Addition and Subtraction of Radical Expressions 8.4 Simplifying Radical Expressions 8.5 Equations Involving Radical Expressions 8.6 Higher Roots and Rational Exponents 9. Quadratic Equations 9.1 Parabolas 9.2 Introduction to Quadratic Equations 9.3 Solving by Completing the Square 9.4 The Quadratic Formula 9.5 Complex Solutions 9.6 Introduction to Functions Appendix A: Using the Graphing Calculator Appendix B: Sets Answers to Selected Exercises Glossary Bibliography Photo Credits
Many books in linear algebra focus purely on getting students through exams, but this text explains both the how and the why of linear algebra and enables students to begin thinking like mathematicians. The author demonstrates how different topics (geometry, abstract algebra, numerical analysis, physics) make use of vectors in different ways and how these ways are connected, preparing students for further work in these areas. The book is packed with hundreds of exercises ranging from the routine to the challenging. Sketch solutions of the easier exercises are available online. framework), an algorithmic perspective (study of the generic inference schemes) and a "practical" perspective (formalisms and applications). Researchers in a number of fields including artificial intelligence, operational research,... Follow along in The Manga Guide to Linear Algebra as Reiji takes Misa from the absolute basics of this tricky subject through mind-bending operations like performing linear transformations, calculating determinants, and finding eigenvectors and eigenvalues. With memorable examples like miniature golf games and karate tournaments, Reiji transforms abstract concepts into something concrete, understandable, and even fun.... Löwenheim's theorem reflects a critical point in the history of mathematical logic, for it marks the birth of model theory--that is, the part of logic that concerns the relationship between formal theories and their models. However, while the original proofs of other, comparably significant theorems are well understood, this is not the case with Löwenheim's theorem. For example, the very result that scholars attribute to Löwenheim today is not the one that Skolem--a logician raised in the algebraic tradition, like Löwenheim--appears to have attributed to him. In The Birth of Model Theory ,... This book provides the first English translation of Bezout's masterpiece, the General Theory of Algebraic Equations . It follows, by almost two hundred years, the English translation of his famous mathematics textbooks. Here, Bézout presents his approach to solving systems of polynomial equations in several variables and in great detail. He introduces the revolutionary notion of the "polynomial multiplier," which greatly simplifies the problem of variable elimination by reducing it to a system of linear equations. The major result presented in this work, now known as "Bézout's theorem," is... E 6 , E 7 , and E 8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series... Get ready to master the principles and operations of algebra! Master Math: Algebra is a comprehensive reference guide that explains and clarifies algebraic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced topics that will help prepare you for pre-calculus and calculus, the book helps clarify algebra using step-by-step procedures and solutions, along with examples and applications. A complete table of contents and a comprehensive index enable you to quickly find specific topics, and the... Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. Does it make sense to buy two years of insurance on a car that depreciates as soon as you drive...
Introductory Algebra for Collegesemester undergraduate introductory algebra course. The goal of this text is to provide students with a strong foundation in Basic Algebra skills; to develop students' critical thinking and problem-solving capabilities and prepare students for Intermediate Algebra and some service math courses. Topics are presented in an interesting and inviting format incorporating real world sourced data modeling. A 4-color hardback book w/complete text-specific instructor and student print/enhanced media supplement pac... MOREkage. AMATYC/NCTM Standards of Content and Pedagogy integrated in current, researched, real-world Applications, Technology Boxes, Discover For Yourself Boxes and extensively revised Exercise Sets. Early introduction and heavy emphasis on modeling demonstrates and utilizes the concepts of introductory algebra to help students solve problems as well as develop critical thinking skills. One-page Chapter Projects (which may be assigned as collaborative projects or extended applications) conclude each chapter and include references to related Web sites for further student exploration. The influence of mathematics in fine art and their relationships are explored in applications and chapter openers to help students visualize mathematical concepts and recognize the beauty in math.
RESOURCES Absorb Mathematics Absorb Mathematics is an interactive course written by Kadie Armstrong, a mathematician and an expert in developing interactive online content. It offers a huge amount of interactivity - ranging from simple animations that show hidden concepts, to powerful models that allow flexible experimentation. Absorb Mathematics is divided into units – roughly corresponding to a lesson – so you can follow the structure of the course all the way through or use the units individually when covering a particular topic or concept. Each unit provides an engaging narrative supported by interactive animations, our unique simulations and exercises to ensure concepts have been understood. Try the free sample units in your class. Throughout the years as an engineer, I have needed to research topics on engineering, physics, chemistry, mechanics, mathematics, etc. The Internet has made the job infinitely simpler, with the caveat that you have to be careful of your sources. Anyone can post anything on the Internet without peer review, and errors are rampant. The topics listed below are primarily ones that I have researched and generated custom pages for the content. I welcome visitor review and comments on my material to help ensure accuracy. Click here for an incredible resource from the the U.S.
Shop by Age This Algebra Made Simple book is the perfect opportunity to make algebra fun, relevant, and interesting for students! This book has been designed to help students develop a basic understanding of algebraic concepts covered in most Algebra 1 courses. Algebra Made Simple is packed with a wonderful variety of exciting activities students can complete to learn algebra using a variety of everyday applications. Learning algebra has never been more fun!
Glencoe/McGraw-Hill's Real World Problem-Solving Graphic Novels is a unique resource for teachers to reach all students. This workbook is designed for use in any classroom and with students of any ability. The Graphic Novel covers topics in number sense, algebraic thinking, geometry, measurement, probability, statistics, and reasoning.
Globalshiksha presents LearnNext Jharkhand Board Class 9 CD's for Maths and Science. Included lessons with syllabuses are in audio and visual format, solved examples, practice workout, experiments, tests and many more related to Jharkhand Board Class 9 Maths and Science. This CD also include a various set of visual tools and activities on each Lesson with Examples, Experiments, Summary and workout. With the help of this CD you can understand all concepts well, clear all doubts with ease and can score good in the exams. This CD comes with a useful Exam Preparation like Lesson tests usually 20-30 minutes in duration, which will help you to evaluate the understanding of each lesson, and Model tests usually of 150-180 minutes in duration, which cover the whole subject on the lines of final exam pattern. This package can help you sharpen your preparation for final exams, identify your strengths and weaknesses and know answers to all tests with a thorough explanation, overcome exam fear and get well scores in final exams.
Welcome to ONM 12 Introductory Algebra! This course is designed to give you the basic mathematics skills needed for success in future math courses at UMaine and for many personal and professional situations as well. I hope you find it useful and enjoyable. Course topics include: writing and solving linear equations and inequalities (including fractional equations), graphing equations and inequalities using the Cartesian Coordinate system, solving systems of equations, solving quadratic equations by factoring and by the quadratic formula, and several practical applications. This course is worth 3 credits , but these will not count towards your bachelor's degree. They do count toward financial aid, and all UMaine add/drop and withdrawal policies apply. There is a lot of material in this course, and we have learned that students vary greatly in what they need to learn and how difficult it is for them to learn it. You may need just a quick refresher in some areas of the curriculum, but you may need to learn totally new material in other areas. You will need to take responsibility to put in extra time and/or get extra help on areas of the curriculum that are hard for you. If you need extra help or have any questions, concerns, comments or suggestions, please feel free to contact me. Also, please check out the Onward Math Lab at the times and locations above (TBA). Study groups usually work well in mathematics courses, and you are encouraged to work on homework and to study with other students. In addition, your text is supported by "My Math Lab (MML)," a web-based tutorial package provided by the publisher, which includes video clips, diagnostics and prescriptive practice exercises, tutoring, etc. The MyMathLab program is required for this course. I am looking forward to working with you this semester and hope it will be a productive experience for you. Text and Materials: Introductory Algebra for College Students (4th ed.), Robert Blitzer, Pearson/Prentiss Hall. Complete homework assignments on a timely basis by submitting MML problems on a class-by-class basis and text-based homework at each test (includes checking answers in the back of the book, using text and classnotes as a resources. Use additional resources when needed, such as the Onward Math Lab (peer led), meeting with professor, classmates, etc. Work with classmates practicing new material in class (involves teamwork, communicating in mathematics and frequent, careful checking of answers). Prepare notes for exams (can bring these to first two exams) Make test corrections after each exam. Course Requirements: Homework (10% of your grade): Homework is an important part of this course. Most of you can expect to spend more time on homework than you spend in class! Since mathematics involves learning techniques for solving problems, for most people extensive practice is important. In short, you learn how to do mathematics by doing it. Your homework grade will be based on Textbook and MyMathLab assignments: Text-Based Homework (5% of your grade): A set of assignments will be given out at the beginning of the course. All assigned problems should be completed on the date indicated, unless otherwise instructed by the professor. Homework will be collected. At each exam, the homework for the chapters covered by that test will be collected with the test. You will need to keep a notebook showing complete solutions to these problems, not just answers. Make sure to label the section or page numbers and problem numbers on the homework. I will not be correcting your homework, however. You are expected to correct your own homework using the answers in the back of the text and to ask for help if you do not understand how to do a problem. If you need extra help, please stop in during office hours or call me, or use the Onward Math Lab, work with your Peer Advisor or with other students, use MyMathLab, solutions manuals, CD's, or any other resources which you find helpful. When requesting help, please have your notebook and work on assignments present; it will help to me to understand your thought process so far. At the end of this syllabus, you will find a rubric explaining the grading criteria for Textbook-Based Homework. MyMathLab Homework (5% of your grade): In addition, or each section of the text we cover in class, you will be assigned some homework problems in the MyMathLab(MML) program. These will usually be due one-half hour before the start of class. After this deadline, you will not be able to submit the homework, nor will you be able to make it up. However, 5 of the assignments will be dropped. These assignments are not quizzes; you are encouraged to get help, use all resources, work with others, etc. Exams (50% of your grade): You will be given 4 exams worth 100 points each. These will account for 50% of your overall grade in this course. Dates for the exams are listed on the homework assignment outline. Exams must be taken when scheduled, except in case of illness or emergencies. "Not being ready" or "needing more time to study" would not be excused absences. In the event of an illness or emergency on the date of an exam, you must contact me as soon as possible. If a make-up exam is appropriate, it needs to be taken as soon as possible (usually your first day back). Make-up exams will generally be scheduled in the afternoon or evening. Students will not be able to bring study notes to make-up exams (see below). *Students who take a make-up exam without an excused absence will have their grade for that test based on a maximum score of 70. Test Notes (first two exams only): When preparing for an exam, it is usually helpful to make an outline or a "mind map" of the chapter(s) covered by the test. This could also include notes from class and sample problems done in class and from the text. For the first two exams only, you will be allowed to use one or two pages of your own handwritten study notes during the exam. Calculators cannot be used on any tests or quizzes or classwork. Exam Corrections (5% of your grade): We usually do not have time to go over exam questions in class, but you can learn a great deal from reviewing your tests as soon as you get them back. To this end, you will be expected to review each exam after you get it back and to write corrections for any questions you got wrong on that exam. Corrections should be done on a separate sheet of paper and handed in attached on top of the original exam. These Exam Corrections will count for up to 10 points each if they are completely correct and will account for 5% of your overall grade in this course. You will lose one point for each incorrect or incomplete problem (up to a maximum of 10 points), but you may resubmit corrections if you do lose any points until you get the full 10 points. I encourage you to get help in making your Exam Corrections from me, the Onward Math Lab, other students and any other resources you may have. If you do not do the Exam Corrections, you will lose points towards your final grade in this class—these corrections are not optional! *BONUS: You can get a bonus 5 points on your exam if you hand in the Exam Corrections within one week of the date the exam is returned to the whole class and have all problems properly corrected. This will actually increase your exam grade by 5 points. Quizzes (10% of your grade): Quizzes will be given on days, usually during the last 10 minutes of class. Each quiz counts as 10 points and will account for 10% of your overall grade in this course. Sometimes a quiz may be given to be completed outside of class; in this case, that take-home quiz will be due at the start of the very next class (unless otherwise indicated). Quizzes missed cannot be made up. You will, however, be able to discard your two lowest quiz grades. If a quiz is missed because of absence from class or a take-home quiz is not returned at the beginning of the next class, it will be considered a grade of zero and can be one of the "low" quizzes that are dropped. Classwork (10% of your grade): These will be several opportunities to submit graded classwork, which will account for 10% of your overall grade. This will usually involve working in pairs, with each partner getting the same number of points. You can get either 5, 3, 1 or 0 points each day. If you are absent, you get a zero, and cannot make-up the points. Final Exam (15% of your grade): The final exam will be cumulative; it will cover material from the entire course. The final will account for 20% of your overall grade for this course. The final must be taken on the date scheduled. Attendance: Regular attendance is required. (If you are not in class, you will get a zero for any quizzes or graded classwork for that day.) Students who miss a class are responsible for learning the material covered, any announcements, assignments, notes, etc. Most people find it difficult to make up the work if they miss two or more classes in a row. If you miss three classes during the semester and it appears to impair the quality of your work, I may report this fact to your Dean. If there are special circumstances that affect your attendance, please discuss them with me in my office. Disability Accommodations: If you have a disability for which you may be requesting an accommodation, please let me know as soon as possible. You should also contact Ann Smith, director of Disabilities Support Services (at 121 East Annex—telephone number 581-2319) to appropriately document your needs at UM. Grades: Your overall grade for the semester will be determined as follows: Grading Exams 4 @ 100 points 50% Exam Corrections 4 @ 10 pts. 5% *(+20 possible bonus points) Quizzes 10 points each 10% Graded Classwork 5 pts. each 10% Homework Textbook 5% MML 5% 10% Final Exam 15% Total 100% Plus and minus grades will be used: A = 93 – 100 B- = 80 – 82 D+ = 67 – 69 A- = 90 – 92 C+ = 77 – 79 D = 63 – 66 B+ = 87 – 89 C = 73 -76 D- = 60 – 62 B = 83 – 86 C- = 70 – 72 E = below 60 Academic Integrity: Academic dishonesty includes cheating, plagiarism and all forms of misrepresentation in academic work, and is unacceptable at The University of Maine. As stated in The University of Maine's "Student Handbook," plagiarism (the submission of another's work without appropriate attribution) and cheating are violations of The University of Maine Student conduct Code. An instructor who has probable cause or reason to believe a student has cheated may act upon such evidence, and should report the case to the supervising faculty member or the Department Chair for appropriate action. Cel phones must remain off during class, unless there is some special emergency situation. In this case, please check with me first! Non-Onward Students: Credit in this course counts toward financial aid, and all UMaine add/drop/withdrawal policies apply. Check with your Dean for other questions such as "Do I have to take this course?" or "Does the grade for this course count in my GPA?" The Dean of your college makes these decisions. Homework Grading Criteria (Textbook) Homework assigned in the text will count for 5% of your overall grade. It is due on the day of the Exam which covers those new sections for the first time. Homework should be neat and organized (in the order in which it was assigned), showing section and page numbers and problem numbers. The following criteria will be used to determine grades: √+ Homework is complete, correct, and clearly presented. Work is shown, where appropriate. 100% √ Homework is mostly complete, correct, and clearly presented. Work is mostly shown, where appropriate. 85% √– Homework is not complete, but what is handed in is correct, and clearly presented, with work mostly shown, where appropriate; or homework has several sections where work is not shown or incorrect. 70% √– – A significant amount of the homework is not submitted, but what is handed in is correct, and clearly presented, with work mostly shown, where appropriate; or homework has a significant number of sections where work is not shown or incorrect 60% 0 No homework is submitted; or hardly any problems are done; or problems show the answers only, with little or no work. 0% Late Homework: Homework received after the day of the test, will be reduced one grade. Getting Started with MyMathLab (MML)… For ONM012, MyMathLab(MML) is required. You can purchase it at the UMaine Bookstore (new or used) or from the publisher's website.
MA104 COLLEGE ALGEBRA AND TRIGONOMETRY WITH APPLICATIONS IN SCIENCE AND TECHNOLOGY (4 Cr.) COURSE DESCRIPTION Prerequisite: MA 100 (Passed with C- or better.) or satisfactory score on the Math Placement Exam. A graphing calculator or equivalent software is required. NOTE: This course is designed for students who need college algebra but do not intend to take calculus. Offered: Fall, Winter General Introduction and Goals Continued development of students' abilities to manipulate mathematical statements and solve problems. A study of functions, graphing, equation solving techniques, exponents and logarithms, systems of equations, and elementary trigonometry. Emphasis is on applications in the applied sciences.
Aims and Objectives Syllabus Laplace Transform Theory The (one-sided) Laplace transform and its existence. Use of Laplace transforms in solving simple ODEs with constant coefficients and given boundary conditions. Step functions and their transforms. Laplace transforms of standard functions. Uniqueness of the inverse. Elementary properties - linearity, first and second shifting theorems, change of scale. Transforms of derivatives and integrals and of products with powers of t. Transforms of periodic functions.The limit of F(s) as s->infinity. The initial and final value theorems and their uses. Laplace transforms of some further special functions - the saw-tooth function, the dirac delta function. Theorems relating to inversion. The solution of slightly more complicated ordinary differential equations with given initial or boundary conditions - constant coefficient equations, simultaneous equations, some equations with non-constant coefficients, equations with discontinuous forcing terms. (About 8 lectures) Fourier series: Definition of Fourier series. Calculation of coefficients in easy cases. Examples of whole and half range series over various ranges. Elementary properties. (About 5 lectures) Eigenvalues, eigenvectors and eigenfunctions: Eigenvalues and eigenvectors of matrices. Simple harmonic equation. Eigenvalues and eigenfunctions of the simple-harmonic equation with various boundary conditions. Applications of eigenvalues, eigenvectors and eigenfunctions. (About 5 lectures)
I used it in my FSMQ but I don't think it's necessary for C3, C4 or M1. It's useful for checking you have got the transformations of graphs correct, but it's better to just learn to how to do it on paper. (Original post by matthurry) as in texus TI? Im pretty sure there banned on EDEXCEL? :s What is banned is the "graphical" sort i guess, but i'm not sure whether this includes TI-nspire or TI-84; because these are non-CAS and they are allowed on other equivalent exams like AP, SAT, ACT etc.. Otherwise, I think a calculator is expected cause some questions basically require one. You can pick up a decent Graphical calculator for less than £50, think mine cost me around 25-30 several years ago a sharp one. They are worth getting, just to check your answers more than anything. You can also write notes and things in them which may be helpful in maths and other subjects
MATLAB Tutorial CD: Learning MATLAB Superfast This first book on MATLAB has three defining features: a set of quick start lessons that introduce the basics of MATLAB using a learn-by-doing approach; a set of reference chapters that provide in-depth, freshman-level descriptions of MATLAB operators, commands, important concepts and the Desktop graphical user interface (GUI) tools; and an extensive index that can be used to quickly locate information in the book. The book also includes a CDROM, called M-Tutor, which is an interactive computer-based set of lectures and exercises that guide students through new concepts and syntax with useful aids such as audio, video and interactive exercises. The text focuses on a set of core commands that provide sufficient power, but is manageable for a student to understand. A good working knowledge base in MATLAB can be developed using this book, yet it is compact, concise and very affordable. This book is an ideal supplement to a course that introduces MATLAB to solve problems. The book provides an introduction at a level suitable for freshman students, focusing on a set of core operators and commands, making the learning process much more manageable. This also results in a smaller, less intimidating, and very affordable book. Included with the book is a CDROM, which contains interactive, computer-based lectures and exercises that in combination with this book can be used as part of a self-study program to learn MATLAB. Course Hierarchy: Course names include Introduction to Engineering or courses that have a MATLAB component Quickstart Lessons with Exercises – Learn-by-doing approach provides a fast way for students to begin using MATLAB. Hands-on lessons that initially use only a handful of MATLAB commands and gradually introduce more commands. Each lesson consists of a set of command examples that contain the MATLAB commands, the Command Window response, and a description of the example alongside. At the end of each lesson, there is a set of exercises for the student. The MATLAB solutions to the exercises are provided in a following section. A core set of MATLAB operators and commands are presented at a level suitable for a freshman student. These descriptions use figures in their explanations and examples to demonstrate their use. This approach requires less lecture time to describe and explain how to use operators and commands simplifying presentation for instructor and student. Shorter book restricts the content to a core set of operators, commands and tools results in a much than other more comprehensive books on MATLAB. Shorter presentation is less intimidating for students and provides necessary information so they can begin using MATLAB quickly. Appropriate as a supplement for a first year Problem Solving and Computer Tools course. Extensive indices list many common terms for operators, commands and tools providing an excellent reference source to support problem solving.
Courses Available Mathematics An important aspect of the study of mathematics is the skill of analytical thinking and problem solving. Mathematics can be regarded as the most important and fundamental of all sciences that has a bearing on our day to day life. It gives a specialist knowledge in order that student may use their practical knowledge in their field of specialization which may even include diverse fields such as engineering, sales or finance.
prominent. In some approaches, students are engaged in using the tools of algebra to model situations and problems, while, in others, algebra as an abstract language is stressed. While much controversy surrounds the worth and merit of these different perspectives on the subject, additional debates center on the contribution of calculators and other technology, the structure of lessons, and the role of the teacher. Because curricula have already been developed that represent these different perspectives on the subject and on how it might best be taught, one important initiative of SERP might be to design comparative studies of how these curricula are taught in classrooms and what and how diverse students learn algebra over time. In this initiative, cohorts of students could be followed longitudinally. Studies could gather information about the instruction they receive, exposure to curriculum, information on the teachers, and their use of the curriculum and other tools. This initiative will depend on the development of effective assessments (see Initiative 3). As with elementary mathematics, however, knowing why particular curricular interventions produce particular outcomes will require companion controlled experiments at the level of particular program features to test for causality. This kind of research is necessary not only to advance scientific understanding, but also because it provides critical knowledge for teachers who adapt curricula and allows developers to improve curricula or design alternatives that are responsive to research findings. Simultaneous with this effort, SERP can support curriculum development that extends existing curricula in promising directions. The Algebra Cognitive Tutor, for example, emphasizes highly contextualized problem solving. While many fewer students drop out and students master the material covered more quickly and effectively, the curriculum may not achieve the fluency in symbol manipulation and abstract analysis expected for high-achieving students. The developers suggest that the curriculum could quite easily be strengthened in this respect, and a separate accelerated algebra course is likely to yield even better results for high-achieving students. In studying the set of curricula as they are being implemented, SERP as a third-party entity would be well positioned to identify and support promising areas like this for further development.
Posted by kdawsonon Saturday April 03, 2010 @09:39PM from the we-can-forget-it-for-you-wholesale dept. nwm writes "I am trying to refresh my math skills back to the point that I can take college-level statistics and calculus courses. I took everything through AP calculus in high school, had my butt kicked by college calculus, and dropped out shortly thereafter. Twenty+ years later, I need to take a few math courses to wrap up a degree. I've dug around some and found a few sites with useful information, but I'm hoping the Slashdot crowd can offer some good resources — sites, books, programs, online tutors, etc. I really don't want to have to take a series of algebra-geometry-trig 'pre-college' level courses (each at full cost and each a semester long) just to warm my brain up; I'd much rather find some resources, review, cram, and take the placement test with some confidence. Any suggestions?" If you haven't needed a degree or calculus in 20 years, why bother now? If you're job hunting, your time would be better spent making yourself relevant to current employers or starting a consulting business than trying to match your calc and trig skills with a recent grad and get a degree. A degree is a nice "filter" when hiring new applicants, since it proves that they were able to deal with BS for at least 4 years, however with 20 years of actual job experience, you'll do much better off trying to differentiate yourself from the recent grads than you will if you try to "look better on paper." That said, if you want to do this just because it's "unfinished business" lots of community colleges have entire departments dedicated to getting us old folks "up to speed". Just stop by and talk to someone. Most text books have practice questions for each chapter, and some answers in the back. Why not just work through some of those on your own? Math is the kind of subject that you can only learn by doing problems, so I don't think there's any shortcuts. But I suppose if you work on problems, it's nice to have a teacher to help if you get stuck, but perhaps a reasonable substitute would be forums. I don't know how bad you want this but I can tell you that nothing feels better than finishing something you started even if it comes two decades later. What you're mostly going to find in these replies are codices. Not teaching. Not knowledge. You're going to get information sources. What you do with those sources, that will be the teaching, the learning and the progress. No one's going to help you get your math back but you. You're going to get static nonliving information and it's going to be up to you to bring that alive. Frankly, on your part it's going to require the will of a volcano otherwise I suggest a tutor or precalculus class. This material could conceivably be studied by a student on his or her own, but this seldom works out. Students tend to get stuck on something, and, having no goad to keep them going, they try to get past it with decreasing energy, and ultimately develop mental blocks against going on. Having an organized course prevents this by forcing them to face obstacles like exams and assignments. If you attempt this and get stuck, as is almost inevitable, you could try emailing us and we can try to unstick you. Did you catch that last part? You're going to need help. Whether it's bribing your nerdy friends with cases of beer or Star Wars Galaxy Series Five collectible card packs (cough cough) you are going to need guidance at certain points in time. Don't be afraid to ask those around you or -- and I recommend this only in dire cases -- dressing up like a student and rolling into your local university asking to see the precalc professor for help. Your codex might be Wikipedia [wikipedia.org]. Your codex might be Wolfram's MathWorld [wolfram.com]. My codex sits three feet in front of my face as I type this. My codex (and this is purely personal) Bronshtein et al's Handbook of Mathematics [amazon.com]. The binding is acceptable. The paper is not the greatest. The content is priceless. This is not a teaching device. This is my starting point. If I were you my ending point would be at my college's library pouring over all calculus textbooks. The great thing about this starting point is that I like how it lays out all the starting points leading up to that starting point in case I need to start backwards. Another great thing about this particular resource is that it has nearly everything imaginable and is well organized. The bad thing is that it costs $71.97. I think I paid $60 for mine but either way it's not free like Wikipedia. I don't know where you are comfortable starting from but if I were you I would simply research what your learning institutions pre requisites are and spend your free time now acquiring their books and notes in order to make sure you have them covered. All of my old University of Minnesota syllabuses are online [umn.edu] although I cannot find the Math department equivalent (aside from the registration listings). If you could name your courses, I'd suggest books like The Annotated Turing [theannotatedturing.com] which has been a page turner for me and actually starts with basic set theory to work up to automata. I'm guessing you're aiming for more Multivariable and Diff Eq type stuff. Let us know what the courses are and perhaps more human readable works can be suggested that aren't as laboriously mind numbing as reading a codex would be. In my experience in school, if you are motivated to pass, you will find a way to pass (most of the time). But if you are motivated to learn, passing the class will come as a pleasant side effect. Not knocking your stated intentions, but approach this as a learning experience, a thoroughfare in self-enlightenment, and you will reap the test-score rewards. It's not your fault; it's the structure of the educational system. You are clearly not interested in mathematics, since you just want to cram and pass some test. You don't specify exactly for what you need mathematics, but I'm guessing it's for some other thing, possibly something computer related. It's a big lie that you'll ever use calculus for anything except for specialised degrees (and if you were to use it for anything you personally would want to do in your future, you would already be interested in it). It's also profoundly strange that calculus seems to be pinnacle of mathematical education if you're not going to go on to study something like mathematics itself or physics. To put my frustration another way, why doesn't anybody ever ask similar questions for sculpture, or Schaum's Outlines on Basket Weaving or all the other myriad useless things we humans do for our edification? Why is western society obsessed with mathematics, deluded into thinking it's useful in general, and why are people so stressed over learning this useless and dryly-presented subject? Why aren't you required to achieve a certain level of chess expertise before you can complete a computer science degree? A lot of early computer science was concerned with chess playing, let us not forget! It's pointless. It's pointless to cram for exams about subjects you don't care about in order to satisfy requirements you don't genuinely need. My recommendation is, are you really interested in learning this stuff? If so, just spend hours and hours in your local university library in the math section browsing books you're interested in. If you're not really interested, go grab some Schaum's Outlines or the Complete Idiot's guide or whatever, and use that to pass whatever bureaucratic and pointless requirement your educational institute imposes before you're allowed to study what you really want to study. As a scientist I learned a long time ago not to make general and unsubstantiated claims like "No matter what kind of scientist you plan to be, your knowledge of calculus will be essential." As a practicing molecular geneticist and cell biologist I use statistics quite often. I cannot remember ever having to (directly) use calculus in the last 20 years for any of my research. I really enjoyed all of the calculus (and linear and set theory and...) that I took a long time a ago. When I look back at it what I really got out of all my math classes (and O-Chem too for that matter) was the the knowledge that I could learn anything I really set my mind to - if I have to. Really... No business getting a degree in ANYTHING? That's a rather closed and inappropriate (IMO) view. If he's worked in a field for years that doesn't require he use any algebra how's he supposed to keep up with his skills other than doing algebra problems in his spare time? He never indicated the degree he's completing was heavily math-biased or math-dependent. Stats and Calc may be akin to gen-eds. When you paint such ridiculously broad statements you risk your own image before anyone else's. Still, if you can't even pass calculus then there's something wrong. And that's not even the problem- he's looking for help preparing for the placement test. If he's let his skills deteriorate so far that he forgets algebra, then he has no business getting a degree in anything. "Bombed back to the stone age" is best regarded as just an expression. The iron age is here to stay, no matter how much civilization declines. Even if we forget how to smelt iron ore, there would be billions of tons of refined iron lying around in abandoned machinery, buildings, and such. No matter what kind of scientist you plan to be, your knowledge of calculus will be essential. You'll never use statistics This has to be about the worst piece of advice about a science education I've ever seen. Like anything, it depends. Calculus is extraordinarily useful to someone in physics, but less so in biology. Statistics is insanely important in an experimental science (actually it's insanely important in just about any science I can think of). Hell, statistics should be a mandatory class taught in High School. It's far more applicable to everyday life than trig is. They want you to pass calculus for a reason. No matter what kind of scientist you plan to be, your knowledge of calculus will be essential. You'll never use statistics but you will need to use calculus every day. Are you wooshing me here? Having an understanding of what a derivative or integral of a function is a good insight to have, no doubt. But I would argue that statistics is much more broadly applicable, and extremely important for a clear understanding of scientific discourse and all the 'facts' that the poster will encounter. In reply to the original query, what you're going to need to do is a lot of problems. You need to look at this like getting in shape--you can't do it overnight. I returned to college after about 5 years off and needed to take placement exams myself. Turned out the test allowed using a Ti-89. I cheated myself out of really 'placing' myself by being able to approximate/calculate all the multiple choice answers and placed highly. After a few attempts in the classes I was placed in, in the end, I re-took precal and calculus. I could have avoided that if I had actually done a large volume of problems rather than skimming some books and looking at the answers and deciding that it was 'easy enough'. Never look at the answers of problems until you try them. Once you know the right answer, you convince yourself the problem was easy and that you didn't need to do it. This will fuck you over in the end. Find an approach to doing math that makes it enjoyable for you. One thing that helped me a lot was getting a large whiteboard. I find I enjoy doing math more pacing back in front of a board and whatever else comes along with doing work on a board rather than a piece of lined paper. Chalk would have been better. Lastly, ignore the assholes here who are going to berate you for not knowing what they think is simple, obvious knowledge. Math is rife with 'tricks' and non-intuitive methods to solving problems that come through experience. Someone who had a good experience with math through school and went straight into college is not going to understand your position. Good luck to you, and if you really want this, do problems and problems and more problems. Put on some music you love and shred through a book or two. Get help at local colleges. Bribe a friend to help you study, or just hire a tutor. Otherwise, you're going to end up doing it by taking the classes (as I did). One way or another, you have to do the work. The parent is absolutely right. You need practice. Actually, you need what Anders Ericcson calls 'deliberate practice'. Solve every example in the book as follows: Write down the problem. Close the book and try to solve the problem. If you got it right, go on to the next problem. If you didn't get it, look at how the example is solved. Close the book and try again until you get it right. Repeat until you have solved every example in the text. BTW, Jamie Escalante, [wikipedia.org], just died. He was the real life teacher who proved that you can teach calculus to just about anybody. They made the movie 'Stand and Deliver' about his life. Ability is highly over-rated. Most people can, as Escalante proved, learn math to quite a high level of accomplishment. Most people think math is some magic thing that some people just can't get. They are wrong. Almost everyone is wired to learn math. If you are missing some important skills, go back to the level where you were good and start from there. John Mighton points out that most people discover that they have no math ability the same year they have a bad math teacher.;-) If you want, you can learn math as long as you practice, practice, practice. Why is western society obsessed with mathematics, deluded into thinking it's useful in general, and why are people so stressed over learning this useless and dryly-presented subject? Essentially because: 1) Everyone should learn logic and disciplined thought. Otherwise you'll end up with adults who can't read instruction manuals, have an attention span of 5 year olds and can't see their own mistakes and contradictions due to disorganized thought processes and hubris. Math can have a humbling effect on people. 2) Proper mathematics is used constantly by good electrical engineers, physicists and mathematicians. If you want a good engineer, you have to teach him math from childhood. And since you can't have a grade school for scientists and another one for everyone else, everyone has to learn math. 3) Math greatly contributes to keep idiots out of the sciences, med school and other important professions. You are probably just put off by the title of this post. "Help Me Get My Math Back" is a presumptuous start to be sure, but his actual question is fine. And his actual question is not what you addressed at all. He's just asking if you know of any place that has the information he needs in a format that is convenient to him. Your response is just a depressing and pointless toil at windmills. They could learn those from taking courses in logic. Taking math courses isn't necessary for that. 2) Proper mathematics is used constantly by good electrical engineers, physicists and mathematicians. But very few people out of the general population go into those fields. And I would leave out "because mathematicians use it" as a reason for studying mathematics.:) 3) Math greatly contributes to keep idiots out of the sciences, med school and other important professions. You're right. By becoming math majors, those idiots are kept out of those important professions. Generally, from what I've seen in my own experience as a math professor, the best students do not major in math. Other math professors I've talked with have the same opinion. The best students I've had all went into physics, chemistry, or engineering. The ones who went on to major in math were among the weakest students; they went into math precisely because they weren't good enough for those other fields (and admitted it to me). Economics? Are you kidding? Economics is full of mathematics and mathematical models. In sociology, and psychology some scientists build models for phenomena and those models are sometimes mathematical. I am a CS scientist but I work on building such models (models for human behaviors) and most of those models are mathematical. why I switched majors from CompSci - being in a hurry to get a degree in a science and too much bullshit math I'd never use Wow. Don't hate on it just because you thought "hey I'm 'good with computers' and this major says Computer in it" and got burned by math expectations. If you don't love math, you have no business being in CS. Computer Science is a of field mathematics, not an engineering program where you learn to fix and build computers! For (potential) CS freshmen, I recommend going through the classic SICP video lectures [mit.edu]. It's very appropriate how Sussman says over and over things to the effect of "we don't care how this would actually work, we're studying the theory." If that doesn't excite you, or you think functional programming is stupid and inconvenient, or if you can't follow a word he's saying, for your own good switch to a different major because it only gets harder and more theoretical. That course was given to freshmen. Clearly you weren't a good fit for CS, but that's because you couldn't handle the math, not because it was bullshit you'd never use. It's really alarming to me how hostile a lot of posters are to academia. A bachelor's degree isn't a fast-track to get into a career, it's a period of academic study. You really have no business claiming that you completed four years of post-secondary study without some basic understanding of math- and calculus is really, really basic. I think your view of statistics comes from a misunderstanding of it on a fundamental level (not that this is your fault). Statistics and probability theory are the basis for interpreting any kind of quantitative measurement. Beware: trying to interpret measurements without knowing this stuff is perfectly analogous to the way people used to build large buildings (sometimes successfully) without using any mathematical modeling, before things like Hooke's law were well known. Sure, plenty of buildings would collapse, but some fairly sophisticated buildings were built anyway, by people who would be grossly incompetent by today's standards. When someone suggested my skills were due to a magical innate ability, I'd get ticked off and tell them no, everybody has the innate ability. My skills, in fact, came from many hours of tedious practice, doing the same thing over and over until I got it right. I don't think it has to be one or the other. I've never been able to draw worth shit. I probably could learn if I really wanted to, but even as a kid my skills were mediocre at best. Rational thinking and separating out bullshit from what's real I've always been very good at, even as a kid. I think there most certainly are innate talents. The idea that "anyone can do it" might be true if we all had infinite patience, time, and motivation. We don't of course, so we gravitate towards things which we develop at with less effort. If you work at subject A and get half as far as the average person, but work at subject B and get twice as far.. which one do you think most people will pick? Parts of biology are getting insanely mathematical. Very recently, say the last five to ten years, there has been a large influx of mathematicians into biology. They use stochastic analysis to model various processes such as transmission of genes to offspring and growth of cell populations. A decent fraction of the PhD students in my department (maths) are involved in biology. Unfortunately, the problem with economics is that it has TOO MUCH math in it. Or rather, it has too much math misuse. There should be a large amount of statistics, but little calculus. That's because we're dealing with human beings and their obstinate free will. So much of modern economics is about making assumptions so that you can start applying some math to the problem. But the assumptions are often unwarranted, like micro's assumption of "perfect knowledge" that can only exist in a fantasy land. Yes, you're going to have to do a shitload of math to get a degree in economics. But you shouldn't have to. Economics is not a hard science like physics, and should not be treated as such.
About CUPM The Mathematical Association of America's Committee on the Undergraduate Program in Mathematics (CUPM) is charged with making recommendations to guide mathematics departments in designing curricula for their undergraduate students. CUPM began issuing reports in 1953, updating them at roughly 10-year intervals. The committee began work on CUPM Curriculum Guide 2004 in 1999, culminating in recommendations approved unanimously by CUPM in January 2003. CUPM has held panel discussions, met with focus groups, and solicited position papers from prominent mathematicians. Through its Curriculum Foundations Project, CUPM's subcommittee on Curriculum Renewal Across the First Two Years (CRAFTY) has conducted workshops with participants from a broad range of partner disciplines, resulting in position papers on what these disciplines want their students to learn in the mathematics classes that they take. While earlier CUPM reports have focused primarily on the major, CUPM Guide 2004 makes six broad recommendations for the entire college-level mathematics curriculum. These six recommendations have been endorsed by the Board of Governors of the MAA. Additional recommendations concern specific student audiences: students taking general education and introductory courses, those majoring in partner disciplines and preparing for K-8 teaching, and mathematical sciences majors. In addition to recommendations for all majors, CUPM Guide 2004 addresses the special needs of majors who intend to teach or seek nonacademic employment as well as those preparing for graduate study in the mathematical sciences or in related fields. CUPM-Illustrative Resources (CUPM-IR) is an on-line, searchable document that provides examples of successful programs as well as resources that can be used to learn more about what has been and can be done to address the recommendations made in the CUPM Guide 2004.
Further Trigonometry Problems booklet concentrates on Trigonometry problems in 3D. Before Looking at the Sine rule, Cosine rule and 1/2absinC rule. There are approximately 5 lessons worth of work. Made up of worked examples and exercises. I have used this material for some years together with the power pointed lesson plan. PDF (Acrobat) Document File Be sure that you have an application to open this file type before downloading and/or purchasing. 1025.5 KB \| 023 Teacher of Mathematics for over 20 years. Teaching students from the age of 11 up to the age of 19. I have an excellent past record for students external examination results at all levels taught. Several students have gone on to study Mathematics at Oxford or Cambridge.

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