Preface xi course. Sections on algebra, the theoretical basis of calculus, including Rolle’s Theorem and the Mean Value Theorem, induction, conics, l’H ˆ opital’s Rule for using derivatives to evaluate limits of an indeterminate form, and Newton’s method of using derivatives to approximate roots constitute Appendices A, C, D, E, F, and G, respectively. Certain appendices can be transported directly into the course. Others can be used as the basis of independent student projects. This book is a preliminary edition and should be viewed as a work in progress. The exposition and choice and sequencing of topics have evolved over the years and will, I expect, continue to evolve. I welcome instructors’ and students’ comments and suggestions on this edition. I can be contacted at the addresses given below. Robin Gottlieb Department of Mathematics 1 Oxford Street Cambridge, MA 02138 gottlieb@math.harvard.edu Acknowledgments A work in progress incurs many debts. I truly appreciate the good humor that participants have shown while working with an evolving course and text. For its progress to this point I’d like to thank all my students and all my fellow instructors and course assistants for their feedback, cooperation, help, and enthusiasm. They include Kevin Oden, Eric Brussel, Eric Towne, Joseph Harris, Andrew Engelward, Esther Silberstein, Ann Ryu, Peter Gilchrist, Tamara Lefcourt, Luke Hunsberger, Otto Bretscher, Matthew Leerberg, Jason Sunderson, Jeanie Yoon, Dakota Pippins, Ambrose Huang, and Barbara Damianic. Special thanks to Eric Towne, without whose help writing course notes in the academic year 1996-1997 this text would not exist. Special thanks also to Eric Brussel whose support for the project has been invaluable, and Peter Gilchrist whose help this past summer was instrumental in getting this preliminary edition ready. Thanks to Matt Leingang and Oliver Knill for technical assistance, to Janine Clookey and Esther Silberstein for start-up assistance, and to everyone in the Harvard Mathematics department for enabling me to work on this book over these past years. I alsowant to acknowledge the type-setting assistance of Paul Anagnostopoulos, Renata D’Arcangelo, Daniel Larson, Eleanor Williams, and numerous others. For the art, I’d like to acknowledge the work of George Nichols, and also of Ben Stephens and Huan Yang. For their work on solutions, thanks go to Peter Gilchrist, Boris Khentov, Dave Marlow, and Sean Owen and coworkers. My thanks to the team at Addison-Wesley for accepting the assortment of materials they were given and carrying out the Herculean task of turning it into a book, especially to Laurie Rosatone for her encouragement and confidence in the project and Ellen Keohane for her assistance and coordination efforts. It has been a special pleasure to work with Julie LaChance in production; I appreciate her effort and support. Thanks also to Joe Vetere, Caroline Fell, Karen Guardino, Sara Anderson, Michael Boezi, Susan Laferriere, and Barbara Atkinson. And thanks to Elka Block and Frank Purcell, for their comments and suggestions. Finally, I want to thank the following people who reviewed this preliminary edition: Dashan Fan, University of Wisconsin, Milwaukee Baxter Johns, Baylor University Michael Moses, George Washington University xii Preface Peter Philliou, Northeastern University Carol S. Schumacher, Kenyon College Eugene Spiegel, The University of Connecticut Robert Stein, California State University, San Bernardino James A. Walsh, Oberlin College To the Student This text has multiple goals. To begin with, you should learn calculus. Your understanding should be deep; you ought to feel it in your bones. Your understanding should be portable; you ought to be able to take it with you and apply it in a variety of contexts. Mathematicians find mathematics exciting and beautiful, and this book may, I hope, provide you with a window through which to see, appreciate, and even come to share this excitement. In some sense mathematics is a language—a way to communicate. You can think of some of your mathematics work as a language lab. Learning any language requires active practice; it requires drill; it requires expressing your own thoughts in that language. But mathematics is more than simply a language. Mathematics is born from inquiry. New mathematics arises from problem solving and from pushing out the boundaries of what is known. Questioning leads to the expansion of knowledge; it is the heart of academic pursuit. From one question springs a host of other questions. Like a branching road, a single inquiry can lead down multiple paths. A path may meander, may lead to a dead-end, detour into fascinating terrain, or steer a straight course toward your destination. The art of questioning, coupled with some good, all-purpose problem-solving skills, may be more important than any neatly packaged set of facts you have tucked under your arm as you stroll away from your studies at the end of the year. For this reason, the text is not a crisp, neatly packed and ironed set of facts. But because you will want to carry away something you can use for reference in the future, this book will supply some concise summaries of the conclusions reached as a result of the investigations in it. We, the author and your instructors, would like you to leave the course equipped with a toolbox of problem-solving skills and strategies—skills and strategies that you have tried and tested throughout the year. We encourage you to break down the complex problems you tackle into a sequence of simpler pieces that can be put together to construct a solution. We urge you to try out your solutions in simple concrete cases and to use numerical, graphical, and analytic methods to investigate problems. We ask that you think about the answers you get, compare them with what you expect, and decide whether your answers are reasonable. Many students will use the mathematics learned in this text in the context of another discipline: biology, medicine, environmental science, physics, chemistry, economics, or one of the social sciences. Therefore, the text offers quite a bit of mathematical modeling— working in the interface between mathematics and other disciplines. Sometimes modeling is treated as an application of mathematics developed, but frequently practical problems from other disciplines provide the questions that lead to the development of mathematical ideas and tools. To learn mathematics successfully you need to actively involve yourself in your studies and work thoughtfully on problems. To do otherwise would be like trying to learn to be a good swimmer without getting in the water. Of course, you’ll need problems to work on. But you’re in luck; you have a slew of them in front of you. Enjoy! Contents Preface vii PART I Functions: An Introduction 1 CHAPTER 1 Functions Are Lurking Everywhere 1 1.1 Functions Are Everywhere 1 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 1: Calibrating Bottles 4 1.2 What Are Functions? Basic Vocabulary and Notation 5 1.3 Representations of Functions 15 CHAPTER 2 Characterizing Functions and Introducing Rates of Change 49 2.1 Features of a Function: Positive/Negative, Increasing/Decreasing, Continuous/Discontinuous 49 2.2 A Pocketful of Functions: Some Basic Examples 61 2.3 Average Rates of Change 73 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 2: Runners 82 2.4 Reading a Graph to Get Information About a Function 84 2.5 The Real Number System: An Excursion 95 CHAPTER 3 Functions Working Together 101 3.1 Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions 101 3.2 Composition of Functions 108 3.3 Decomposition of Functions 119 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 3: Flipping, Shifting, Shrinking, and Stretching: Exercising Functions 123 3.4 Altered Functions, Altered Graphs: Stretching, Shrinking, Shifting, and Flipping 126 xiii xiv Contents PART II Rates of Change: An Introduction to the Derivative 139 CHAPTER 4 Linearity and Local Linearity 139 4.1 Making Predictions: An Intuitive Approach to Local Linearity 139 4.2 Linear Functions 143 4.3 Modeling and Interpreting the Slope 153 ◆ EXPLORATORY PROBLEM FOR CHAPTER 4: Thomas Wolfe’s Royalties for The Story of a Novel 158 4.4 Applications of Linear Models: Variations on a Theme 159 CHAPTER 5 The Derivative Function 169 5.1 Calculating the Slope of a Curve and Instantaneous Rate of Change 169 5.2 The Derivative Function 187 5.3 Qualitative Interpretation of the Derivative 194 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 5: Running Again 206 5.4 Interpreting the Derivative: Meaning and Notation 208 CHAPTER 6 The Quadratics: A Profile of a Prominent Family of Functions 217 6.1 A Profile of Quadratics from a Calculus Perspective 217 6.2 Quadratics From A Noncalculus Perspective 223 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 6: Tossing Around Quadratics 226 6.3 Quadratics and Their Graphs 231 6.4 The Free Fall of an Apple: A Quadratic Model 237 CHAPTER 7 The Theoretical Backbone: Limits and Continuity 245 7.1 Investigating Limits—Methods of Inquiry and a Definition 245 7.2 Left- and Right-Handed Limits; Sometimes the Approach Is Critical 258 7.3 A Streetwise Approach to Limits 265 7.4 Continuity and the Intermediate and Extreme Value Theorems 270 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 7: Pushing the Limit 275 CHAPTER 8 Fruits of Our Labor: Derivatives and Local Linearity Revisited 279 8.1 Local Linearity and the Derivative 279 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 8: Circles and Spheres 286 8.2 The First and Second Derivatives in Context: Modeling Using Derivatives 288 8.3 Derivatives of Sums, Products, Quotients, and Power Functions 290 Contents xv PART III Exponential, Polynomial, and Rational Functions— with Applications 303 CHAPTER 9 Exponential Functions 303 9.1 Exponential Growth: Growth at a Rate Proportional to Amount 303 9.2 Exponential: The Bare Bones 309 9.3 Applications of the Exponential Function 320 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 9: The Derivative of the Exponential Function 328 9.4 The Derivative of an Exponential Function 334 CHAPTER 10 Optimization 341 10.1 Analysis of Extrema 341 10.2 Concavity and the Second Derivative 356 10.3 Principles in Action 361 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 10: Optimization 365 CHAPTER 11 A Portrait of Polynomials and Rational Functions 373 11.1 A Portrait of Cubics from a Calculus Perspective 373 11.2 Characterizing Polynomials 379 11.3 Polynomial Functions and Their Graphs 391 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 11: Functions and Their Graphs: Tinkering with Polynomials and Rational Functions 404 11.4 Rational Functions and Their Graphs 406 PART IV Inverse Functions: A Case Study of Exponential and Logarithmic Functions 421 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? 421 12.1 What Does It Mean for f and g to Be Inverse Functions? 421 12.2 Finding the Inverse of a Function 429 12.3 Interpreting the Meaning of Inverse Functions 434 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 12: Thinking About the Derivatives of Inverse Functions 437 CHAPTER 13 Logarithmic Functions 439 13.1 The Logarithmic Function Defined 439 13.2 The Properties of Logarithms 444 13.3 Using Logarithms and Exponentiation to Solve Equations 449 xvi Contents ◆ EXPLORATORY PROBLEM FOR CHAPTER 13: Pollution Study 458 13.4 Graphs of Logarithmic Functions: Theme and Variations 462 CHAPTER 14 Differentiating Logarithmic and Exponential Functions 467 14.1 The Derivative of Logarithmic Functions 467 ◆ EXPLORATORY PROBLEM FOR CHAPTER 14: The Derivative of the Natural Logarithm 468 14.2 The Derivative of b x Revisited 473 14.3 Worked Examples Involving Differentiation 476 CHAPTER 15 Take It to the Limit 487 15.1 An Interesting Limit 487 15.2 Introducing Differential Equations 497 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 15: Population Studies 507 PART V Adding Sophistication to Your Differentiation 513 CHAPTER 16 Taking the Derivative of Composite Functions 513 16.1 The Chain Rule 513 16.2 The Derivative of x n where n is any Real Number 521 16.3 Using the Chain Rule 523 ◆ EXPLORATORY PROBLEMS FOR CHAPTER 16: Finding the Best Path 528 CHAPTER 17 Implicit Differentiation and its Applications 535 17.1 Introductory Example 535 17.2 Logarithmic Differentiation 538 17.3 Implicit Differentiation 541 17.4 Implicit Differentiation in Context: Related Rates of Change 550 PART VI An Excursion into Geometric Series 559 CHAPTER 18 Geometric Sums, Geometric Series 559 18.1 Geometric Sums 559 18.2 Infinite Geometric Series 566 18.3 A More General Discussion of Infinite Series 572 18.4 Summation Notation 575 18.5 Applications of Geometric Sums and Series 579 Contents xvii PART VII Trigonometric Functions 593 CHAPTER 19 Trigonometry: Introducing Periodic Functions 593 19.1 The Sine and Cosine Functions: Definitions and Basic Properties 594 19.2 Modifying the Graphs of Sine and Cosine 603 19.3 The Function f(x)= tan x 615 19.4 Angles and Arc Lengths 619 CHAPTER 20 Trigonometry—Circles and Triangles 627 20.1 Right-Triangle Trigonometry: The Definitions 627 20.2 Triangles We Know and Love, and the Information They Give Us 635 20.3 Inverse Trigonometric Functions 645 20.4 Solving Trigonometric Equations 651 20.5 Applying Trigonometry to a General Triangle: The Law of Cosines and the Law of Sines 657 20.6 Trigonometric Identities 667 20.7 A Brief Introduction to Vectors 671 CHAPTER 21 Differentiation of Trigonometric Functions 683 21.1 Investigating the Derivative of sin x Graphically, Numerically, and Using Physical Intuition 683 21.2 Differentiating sin x and cos x 688 21.3 Applications 695 21.4 Derivatives of Inverse Trigonometric Functions 703 21.5 Brief Trigonometry Summary 707 PART VIII Integration: An Introduction 711 CHAPTER 22 Net Change in Amount and Area: Introducing the Definite Integral 711 22.1 Finding Net Change in Amount: Physical and Graphical Interplay 711 22.2 The Definite Integral 725 22.3 The Definite Integral: Qualitative Analysis and Signed Area 731 22.4 Properties of the Definite Integral 738 CHAPTER 23 The Area Function and Its Characteristics 743 23.1 An Introduction to the Area Function x a f(t)dt 743 23.2 Characteristics of the Area Function 747 23.3 The Fundamental Theorem of Calculus 757 xviii Contents CHAPTER 24 The Fundamental Theorem of Calculus 761 24.1 Definite Integrals and the Fundamental Theorem 761 24.2 The Average Value of a Function: An Application of the Definite Integral 775 PART IX Applications and Computation of the Integral 783 CHAPTER 25 Finding Antiderivatives—An Introduction to Indefinite Integration 783 25.1 A List of Basic Antiderivatives 783 25.2 Substitution: The Chain Rule in Reverse 787 25.3 Substitution to Alter the Form of an Integral 798 CHAPTER 26 Numerical Methods of Approximating Definite Integrals 805 26.1 Approximating Sums: L n , R n , T n , and M n 805 26.2 Simpson’s Rule and Error Estimates 820 CHAPTER 27 Applying the Definite Integral: Slice and Conquer 827 27.1 Finding “Mass” When Density Varies 827 27.2 Slicing to Find the Area Between Two Curves 843 CHAPTER 28 More Applications of Integration 853 28.1 Computing Volumes 853 28.2 Arc Length, Work, and Fluid Pressure: Additional Applications of the Definite Integral 865 CHAPTER 29 Computing Integrals 877 29.1 Integration by Parts—The Product Rule in Reverse 877 29.2 Trigonometric Integrals and Trigonometric Substitution 886 29.3 Integration Using Partial Fractions 898 29.4 Improper Integrals 903 PART X Series 919 CHAPTER 30 Series 919 30.1 Approximating a Function by a Polynomial 919 30.2 Error Analysis and Taylor’s Theorem 934 30.3 Taylor Series 941 30.4 Working with Series and Power Series 952 30.5 Convergence Tests 964 Contents xix CHAPTER 31 Differential Equations 983 31.1 Introduction to Modeling with Differential Equations 983 31.2 Solutions to Differential Equations: An Introduction 991 31.3 Qualitative Analysis of Solutions to Autonomous Differential Equations 1002 31.4 Solving Separable First Order Differential Equations 1018 31.5 Systems of Differential Equations 1024 31.6 Second Order Homogeneous Differential Equations with Constant Coefficients 1045 Appendices 1051 APPENDIX A Algebra 1051 A.1 Introduction to Algebra: Expressions and Equations 1051 A.2 Working with Expressions 1056 A.3 Solving Equations 1070 APPENDIX B Geometric Formulas 1085 APPENDIX C The Theoretical Basis of Applications of the Derivative 1087 APPENDIX D Proof by Induction 1095 APPENDIX E Conic Sections 1099 E.1 Characterizing Conics from a Geometric Viewpoint 1100 E.2 Defining Conics Algebraically 1101 E.3 The Practical Importance of Conic Sections 1106 APPENDIX F L’H ˆ opital’s Rule: Using Relative Rates of Change to Evaluate Limits 1111 F.1 Indeterminate Forms 1111 APPENDIX G Newton’s Method: Using Derivatives to Approximate Roots 1121 APPENDIX H Proofs to Accompany Chapter 30, Series 1127 Index 1133 . f(x)= tan x 615 19.4 Angles and Arc Lengths 619 CHAPTER 20 Trigonometry—Circles and Triangles 627 20 .1 Right-Triangle Trigonometry: The Definitions 627 20 .2 Triangles We Know and Love, and the. Case Study of Exponential and Logarithmic Functions 421 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone? 421 12. 1 What Does It Mean for f and g to Be Inverse Functions? 421 12. 2 Finding. Guardino, Sara Anderson, Michael Boezi, Susan Laferriere, and Barbara Atkinson. And thanks to Elka Block and Frank Purcell, for their comments and suggestions. Finally, I want to thank the following