Preface xi
course. Sections on algebra, the theoretical basis of calculus, including Rolles Theorem and the Mean Value Theorem, induction, conics, lHopitals Rule for using derivatives to evaluate limits of an indeterminate form, and Newtons 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 Id 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.
Ialsowanttoacknowledgethetype-settingassistanceofPaulAnagnostopoulos,Renata DArcangelo, Daniel Larson, Eleanor Williams, and numerous others. For the art, Id 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 condence 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, andBarbaraAtkinson.AndthankstoElkaBlockandFrankPurcell, fortheircommentsand 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; yououghttobeabletotakeitwithyouandapplyitinavarietyofcontexts.Mathematicians nd 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 languagea 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.Questioningleadstotheexpansionofknowledge; itistheheartofacademicpursuit. 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 fascinatingterrain,orsteerastraightcoursetowardyourdestination.Theartofquestioning, 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 strategiesskills and strategies that you have tried andtestedthroughouttheyear.Weencourageyoutobreakdownthecomplexproblemsyou 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.
Tolearnmathematicssuccessfullyyouneedtoactivelyinvolveyourselfinyourstudies 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, youll need problems to work on. But youre 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
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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 Wolfes 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 ProÞle of a Prominent Family of Functions 217 6.1 A Prole 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 LimitsMethods of Inquiry and a Denition 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 FunctionsÑ 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 and 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 Dened 439 13.2 The Properties of Logarithms 444
13.3 Using Logarithms and Exponentiation to Solve Equations 449
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