MATH 1910_1920 - TO JULIE Jon TO ALEXA AND COLTON Colin Publisher Terri Ward Developmental Editors Tony Palermino Katrina Wilhelm Marketing Manager

MATH 1910_1920 - TO JULIE Jon TO ALEXA AND COLTON Colin...

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Unformatted text preview: TO JULIE –Jon TO ALEXA AND COLTON –Colin Publisher: Terri Ward Developmental Editors: Tony Palermino, Katrina Wilhelm Marketing Manager: Cara LeClair Market Development Manager: Shannon Howard Executive Media Editor: Laura Judge Associate Editor: Marie Dripchak Editorial Assistant: Victoria Garvey Director of Editing, Design, and Media Production: Tracey Kuehn Managing Editor: Lisa Kinne Project Editor: Kerry O’Shaughnessy Production Manager: Paul Rohloff Cover and Text Designer: Blake Logan Illustration Coordinator: Janice Donnola Illustrations: Network Graphics and Techsetters, Inc. Photo Editors: Eileen Liang, Christine Buese Photo Researcher: Eileen Liang Composition: John Rogosich/Techsetters, Inc. Printing and Binding: RR Donnelley Cover photo: ayzek/Shutterstock Library of Congress Preassigned Control Number: 2014959514 Instructor Complimentary Copy: ISBN-13: 978-1-4641-9379-8 ISBN-10: 1-4641-9379-7 Hardcover: ISBN-13: 978-1-4641-2526-3 ISBN-10: 1-4641-2526-0 Loose-leaf: ISBN-13: 978-1-4641-9380-4 ISBN-10: 1-4641-9380-0 © 2015, 2012, 2008 by W. H. Freeman and Company All rights reserved Printed in the United States of America First printing W. H. Freeman and Company, 41 Madison Avenue, New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England ABOUT THE AUTHORS COLIN ADAMS C olin Adams is the Thomas T. Read professor of Mathematics at Williams College, where he has taught since 1985. Colin received his undergraduate degree from MIT and his PhD from the University of Wisconsin. His research is in the area of knot theory and low-dimensional topology. He has held various grants to support his research, and written numerous research articles. Colin is the author or co-author of The Knot Book, How to Ace Calculus: The Streetwise Guide, How to Ace the Rest of Calculus: The Streetwise Guide, Riot at the Calc Exam and Other Mathematically Bent Stories, Why Knot?, Introduction to Topology: Pure and Applied, and Zombies & Calculus. He co-wrote and appears in the videos “The Great Pi vs. E Debate” and “Derivative vs. Integral: the Final Smackdown.” He is a recipient of the Haimo National Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, an MAA Polya Lecturer for 19982000, a Sigma Xi Distinguished Lecturer for 2000-2002, and the recipient of the Robert Foster Cherry Teaching Award in 2003. Colin has two children and one slightly crazy dog, who is great at providing the entertainment. JON ROGAWSKI A s a successful teacher for more than 30 years, Jon Rogawski listened and learned much from his own students. These valuable lessons made an impact on his thinking, his writing, and his shaping of a calculus text. Jon Rogawski received his undergraduate and master’s degrees in mathematics simultaneously from Yale University, and he earned his PhD in mathematics from Princeton University, where he studied under Robert Langlands. Before joining the Department of Mathematics at UCLA in 1986, where he was a full professor, he held teaching and visiting positions at the Institute for Advanced Study, the University of Bonn, and the University of Paris at Jussieu and Orsay. Jon’s areas of interest were number theory, automorphic forms, and harmonic analysis on semisimple groups. He published numerous research articles in leading mathematics journals, including the research monograph Automorphic Representations of Unitary Groups in Three Variables (Princeton University Press). He was the recipient of a Sloan Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of the AMS. Sadly, Jon Rogawski passed away in September 2011. Jon’s commitment to presenting the beauty of calculus and the important role it plays in students’ understanding of the wider world is the legacy that lives on in each new edition of Calculus. CONTENTS CALCULUS Chapter 1 PRECALCULUS REVIEW 1.1 1.2 1.3 1.4 1.5 Real Numbers, Functions, and Graphs Linear and Quadratic Functions The Basic Classes of Functions Trigonometric Functions Technology: Calculators and Computers Chapter Review Exercises 1 1 12 19 23 32 36 Chapter 2 LIMITS 39 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 39 47 56 59 68 73 78 84 87 94 Limits, Rates of Change, and Tangent Lines Limits: A Numerical and Graphical Approach Basic Limit Laws Limits and Continuity Evaluating Limits Algebraically Trigonometric Limits Limits at Infinity Intermediate Value Theorem The Formal Definition of a Limit Chapter Review Exercises Chapter 3 DIFFERENTIATION 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Definition of the Derivative The Derivative as a Function Product and Quotient Rules Rates of Change Higher Derivatives Trigonometric Functions The Chain Rule Implicit Differentiation Related Rates Chapter Review Exercises 97 97 105 117 123 132 137 141 148 155 162 Chapter 4 APPLICATIONS OF THE DERIVATIVE 167 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Linear Approximation and Applications Extreme Values The Mean Value Theorem and Monotonicity The Shape of a Graph Graph Sketching and Asymptotes Applied Optimization Newton’s Method Chapter Review Exercises Chapter 5 THE INTEGRAL 5.1 5.2 5.3 iv Approximating and Computing Area The Definite Integral The Indefinite Integral 167 174 184 190 197 204 216 221 225 225 237 247 5.4 5.5 5.6 5.7 The Fundamental Theorem of Calculus, Part I The Fundamental Theorem of Calculus, Part II Net Change as the Integral of a Rate of Change Substitution Method Chapter Review Exercises Chapter 6 APPLICATIONS OF THE INTEGRAL 6.1 6.2 6.3 6.4 6.5 Derivative of f (x) = bx and the Number e Inverse Functions Logarithms and Their Derivatives Exponential Growth and Decay Compound Interest and Present Value Models Involving y ′ = k(y − b) L’Hôpital’s Rule Inverse Trigonometric Functions Hyperbolic Functions Chapter Review Exercises Chapter 8 TECHNIQUES OF INTEGRATION 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 321 321 329 336 345 351 356 361 368 376 386 391 Integration by Parts 391 Trigonometric Integrals 397 Trigonometric Substitution 405 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions 411 The Method of Partial Fractions 416 Strategies for Integration 424 Improper Integrals 432 Probability and Integration 443 Numerical Integration 450 Chapter Review Exercises 459 Chapter 9 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 9.1 9.2 9.3 9.4 281 Area Between Two Curves 281 Setting Up Integrals: Volume, Density, Average Value 289 Volumes of Revolution 299 The Method of Cylindrical Shells 307 Work and Energy 313 Chapter Review Exercises 319 Chapter 7 EXPONENTIAL FUNCTIONS 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 254 259 264 270 277 Arc Length and Surface Area Fluid Pressure and Force Center of Mass Taylor Polynomials Chapter Review Exercises 463 463 470 476 485 496 CONTENTS Chapter 10 INTRODUCTION TO DIFFERENTIAL EQUATIONS 499 10.1 10.2 10.3 10.4 Solving Differential Equations Graphical and Numerical Methods The Logistic Equation First-Order Linear Equations Chapter Review Exercises Chapter 11 INFINITE SERIES 499 507 515 519 525 529 11.1 11.2 11.3 11.4 11.5 Sequences 529 Summing an Infinite Series 539 Convergence of Series with Positive Terms 550 Absolute and Conditional Convergence 559 The Ratio and Root Tests and Strategies for Choosing Tests 564 11.6 Power Series 569 11.7 Taylor Series 579 Chapter Review Exercises 591 Chapter 12 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS 595 12.1 12.2 12.3 12.4 12.5 Parametric Equations Arc Length and Speed Polar Coordinates Area and Arc Length in Polar Coordinates Conic Sections Chapter Review Exercises Chapter 13 VECTOR GEOMETRY 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Vectors in the Plane Vectors in Three Dimensions Dot Product and the Angle Between Two Vectors The Cross Product Planes in 3-Space A Survey of Quadric Surfaces Cylindrical and Spherical Coordinates Chapter Review Exercises Chapter 14 CALCULUS OF VECTOR-VALUED FUNCTIONS 14.1 14.2 14.3 14.4 14.5 14.6 Vector-Valued Functions Calculus of Vector-Valued Functions Arc Length and Speed Curvature Motion in 3-Space Planetary Motion According to Kepler and Newton Chapter Review Exercises Chapter 15 DIFFERENTIATION IN SEVERAL VARIABLES 15.1 Functions of Two or More Variables 15.2 Limits and Continuity in Several Variables 595 606 612 620 625 638 641 641 651 661 669 680 686 694 701 705 705 713 722 727 738 747 753 755 755 766 15.3 15.4 15.5 15.6 15.7 15.8 Partial Derivatives Differentiability and Tangent Planes The Gradient and Directional Derivatives The Chain Rule Optimization in Several Variables Lagrange Multipliers: Optimizing with a Constraint Chapter Review Exercises Chapter 16 MULTIPLE INTEGRATION 16.1 16.2 16.3 16.4 Integration in Two Variables Double Integrals over More General Regions Triple Integrals Integration in Polar, Cylindrical, and Spherical Coordinates 16.5 Applications of Multiple Integrals 16.6 Change of Variables Chapter Review Exercises Chapter 17 LINE AND SURFACE INTEGRALS 17.1 17.2 17.3 17.4 17.5 Vector Fields Line Integrals Conservative Vector Fields Parametrized Surfaces and Surface Integrals Surface Integrals of Vector Fields Chapter Review Exercises Chapter 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 18.1 Green’s Theorem 18.2 Stokes’ Theorem 18.3 Divergence Theorem Chapter Review Exercises APPENDICES A. The Language of Mathematics B. Properties of Real Numbers C. Induction and the Binomial Theorem D. Additional Proofs ANSWERS TO ODD-NUMBERED EXERCISES v 773 783 790 803 811 825 834 837 837 848 861 872 882 894 907 911 911 921 935 946 960 970 973 973 987 997 1009 A1 A1 A7 A12 A16 ANS1 REFERENCES R1 INDEX I1 Additional content can be accessed online via LaunchPad: ADDITIONAL PROOFS • L’Hôpital’s Rule • Error Bounds for Numerical Integration • Comparison Test for Improper Integrals ADDITIONAL CONTENT • Second Order Differential Equations • Complex Numbers PREFACE ABOUT CALCULUS On Teaching Mathematics I consider myself very lucky to have a career as a teacher and practitioner of mathematics. When I was young, I decided I wanted to be a writer. I loved telling stories. But I was also good at math, and, once in college, it didn’t take me long to become enamored with it. I loved the fact that success in mathematics does not depend on your presentation skills or your interpersonal relationships. You are either right or you are wrong and there is little subjective evaluation involved. And I loved the satisfaction of coming up with a solution. That intensified when I started solving problems that were open research questions that had previously remained unsolved. So, I became a professor of mathematics. And I soon realized that teaching mathematics is about telling a story. The goal is to explain to students in an intriguing manner, at the right pace, and in as clear a way as possible, how mathematics works and what it can do for you. I find mathematics immensely beautiful. I want students to feel that way, too. On Writing a Calculus Text I had always thought I might write a calculus text. But that is a daunting task. These days, calculus books average over a thousand pages. And I would need to convince myself that I had something to offer that was different enough from what already appears in the existing books. Then, I was approached about writing the third edition of Jon Rogawski’s calculus book. Here was a book for which I already had great respect. Jon’s vision of what a calculus book should be fit very closely with my own. Jon believed that as math teachers, how we say it is as important as what we say. Although he insisted on rigor at all times, he also wanted a book that was written in plain English, a book that could be read and that would entice students to read further and learn more. Moreover, Jon strived to create a text in which exposition, graphics, and layout would work together to enhance all facets of a student’s calculus experience. In writing his book, Jon paid special attention to certain aspects of the text: 1. Clear, accessible exposition that anticipates and addresses student difficulties. 2. Layout and figures that communicate the flow of ideas. 3. Highlighted features that emphasize concepts and mathematical reasoning: Conceptual Insight, Graphical Insight, Assumptions Matter, Reminder, and Historical Perspective. 4. A rich collection of examples and exercises of graduated difficulty that teach basic skills, problem-solving techniques, reinforce conceptual understanding, and motivate calculus through interesting applications. Each section also contains exercises that develop additional insights and challenge students to further develop their skills. vi Coming into the project of creating the third edition, I was somewhat apprehensive. Here was an already excellent book that had attained the goals set for it by its author. First and foremost, I wanted to be sure that I did it no harm. On the other hand, I have been teaching calculus now for 30 years, and in that time, I have come to some conclusions about what does and does not work well for students. As a mathematician, I want to make sure that the theorems, proofs, arguments and development are correct. There is no place in mathematics for sloppiness of any kind. As a teacher, I want the material to be accessible. The book should not be written at the mathematical level of the instructor. Students should be able to use the book to learn the material, with the help of their instructor. Working from the high standard that Jon set, I have tried hard to maintain the level of quality of the previous edition while making the changes that I believe will bring the book to the next level. PREFACE vii Placement of Taylor Polynomials Taylor polynomials appear in Chapter 9, before infinite series in Chapter 11. The goal here is to present Taylor polynomials as a natural extension of linear approximation. When teaching infinite series, the primary focus is on convergence, a topic that many students find challenging. By the time we have covered the basic convergence tests and studied the convergence of power series, students are ready to tackle the issues involved in representing a function by its Taylor series. They can then rely on their previous work with Taylor polynomials and the error bound from Chapter 9. However, the section on Taylor polynomials is written so that you can cover this topic together with the materials on infinite series if this order is preferred. Careful, Precise Development W. H. Freeman is committed to high quality and precise textbooks and supplements. From this project’s inception and throughout its development and production, quality and precision have been given significant priority. We have in place unparalleled procedures to ensure the accuracy of the text: • • • • • Exercises and Examples Exposition Figures Editing Composition Together, these procedures far exceed prior industry standards to safeguard the quality and precision of a calculus textbook. New to the Third Edition There are a variety of changes that have been implemented in this edition. Following are some of the most important. MORE FOCUS ON CONCEPTS The emphasis has been shifted to focus less on the memorization of specific formulas, and more on understanding the underlying concepts. Memorization can never be completely avoided, but it is in no way the crux of calculus. Students will remember how to apply a procedure or technique if they see the logical progression that generates it. And they then understand the underlying concepts rather than seeing the topic as a black box in which you insert numbers. Specific examples include: • • • • • • (Section 1.2) Removed the general formula for the completion of a square and instead, emphasized the method so students need not memorize the formula. (Section 8.2) Changed the methods for evaluating trigonometric integrals to focus on techniques to apply rather than formulas to memorize. (Chapter 10) Discouraged the memorization of solutions of specific types of differential equations and instead, encouraged the use of methods of solution. (Section 13.2) Decreased number of formulas for parametrizing a line from two to one, as the second can easily be derived from the first. (Section 13.6) De-emphasized the memorization of the various formulas for quadric surfaces. Instead, moved the focus to slicing with planes to find curves and using those to determine the shape of the surface. These methods will be useful regardless of the type of surface it is. (Section 15.4) Decreased the number of essential formulas for linear approximation of functions of two variables from four to two, providing the background to derive the others from these. CHANGES IN NOTATION There are numerous notational changes. Some were made to bring the notation more into line with standard usage in mathematics and other fields in which mathematics is applied. Some were implemented to make it easier for students to remember the meaning of the notation. Some were made to help make the corresponding concepts that are represented more transparent. Specific examples include: viii PREFACE • • • • (Section 4.5) Presented a new notation for graphing that gives the signs of the first and second derivative and then simple symbols (slanted up and down arrows and up and down u’s) to help the student keep track of when the graph is increasing or decreasing and concave up or concave down over the given interval. (Section 8.1) Simplified the notation for integration by parts and provided a visual method for remembering it. (Chapter 11) Changed names of the various tests for convergence/divergence of infinite series to evoke the usage of the test and thereby make it easier for students to remember them. (Chapters 14–18) Rather than using c(t) for a path, we consistently switched to the vector-valued function r(t). This also allowed us to replace ds with dr as a differential, which means there is less likely to be confusion with ds, dS and dS. MORE EXPLANATIONS OF DERIVATIONS Occasionally, in the previous edition, a result was given and verified, without motivating where the derivation came from. I believe it is important for students to understand how someone might come up with a particular result, thereby helping them to picture how they might themselves one day be able to derive results. • • • • (Section 9.3) Developed the center of mass formulas by first discussing the onedimensional case of a seesaw. (Section 15.4) Developed the equation of the tangent plane in a manner that makes geometric sense. (Section 15.5) Included a proof of the fact the gradient of a function f of three variables is orthogonal to the surfaces that are the level sets of f . (Section 15.8) Gave an intuitive explanation for why the Method of Lagrange Multipliers works. REORDERING AND ADDING TOPICS There were some specific rearrangements among the sections and additions. These include: • • • • • • • • A subsection on piecewise-defined functions has been added to Section 1.3. The section on indefinite integrals (previously Section 4.8) has been moved from Chapter 4 (Applications of the Derivative) to Chapter 5 (The Integral). This is a more natural placement for it. A new section on choosing from amongst the various methods of integration has been added to Chapter 8. A subsection on choosing the appropriate convergence/divergence test has been added to Section 11.5. An explanation of how to find indefinite limits using power series has been added to Section 10.6. The definitions of divergence and curl have been moved from Chapter 18 to Section 17.1. This allows us to utilize them at an appropriate earlier point in the text. A list all of the different types of integrals that have been introduced in Chapter 17 has been added to Section 17.5. A subsection on the Vector Form of Green’s Theorem has been added to Section 18.1. NEW EXAMPLES, FIGURES, AND EXERCISES Numerous examples and accompanying figures have been added to clarify concepts. A variety of exercises have also been added throughout the text, particularly wher...
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