ROGAWSKI’S CALCULUS for AP* EARLY TRANSCENDENTALS JON ROGAWSKI University of California, Los Angeles RAY CANNON Baylor University, TX W. H. FREEMAN AND COMPANY New York SECOND EDITION Director, BFW High School: Craig Bleyer Executive Editor: Ann Heath Publisher: Ruth Baruth Senior Acquisitions Editor: Terri Ward Development Editor: Tony Palermino Development Editor: Julie Z. Lindstrom, Andrew Sylvester Associate Editor: Katrina Wilhelm Assistant Editor: Dora Figueiredo Editorial Assistant: Tyler Holzer Market Development: Steven Rigolosi Executive Marketing Manager: Cindi Weiss Media Editor: Laura Capuano Assistant Media Editor: Catriona Kaplan Senior Media Acquisitions Editor: Roland Cheyney Photo Editor: Ted Szczepanski Photo Researcher: Julie Tesser Cover and Text Designer: Blake Logan Illustrations: Network Graphics and Techsetters, Inc. Illustration Coordinator: Bill Page Production Coordinator: Paul W. Rohloff Composition: Techsetters, Inc. Printing and Binding: RR Donnelley and Sons Library of Congress Control Number ISBN-13: 978-1-4292-5074-0 ISBN-10: 1-4292-5074-7 © 2012 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 www.whfreeman.com To Julie and To the AP Teachers CONTENTS ROGAWSKI’S CALCULUS for AP* Early Transcendentals Chapter 1 PRECALCULUS REVIEW 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Real Numbers, Functions, and Graphs Linear and Quadratic Functions The Basic Classes of Functions Trigonometric Functions Inverse Functions Exponential and Logarithmic Functions Technology: Calculators and Computers Chapter 2 LIMITS 1 1 13 21 25 33 43 51 Chapter 4 APPLICATIONS OF THE DERIVATIVE 207 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Linear Approximation and Applications Extreme Values The Mean Value Theorem and Monotonicity The Shape of a Graph L’Hôpital’s Rule Graph Sketching and Asymptotes Applied Optimization Newton’s Method Antiderivatives Preparing for the AP Exam 59 Chapter 5 THE INTEGRAL 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 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 Preparing for the AP Exam 59 67 77 81 90 95 100 106 110 AP2-1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Approximating and Computing Area The Deﬁnite Integral The Fundamental Theorem of Calculus, Part I The Fundamental Theorem of Calculus, Part II Net Change as the Integral of a Rate Substitution Method Further Transcendental Functions Exponential Growth and Decay Preparing for the AP Exam Chapter 6 APPLICATIONS OF THE INTEGRAL Chapter 3 DIFFERENTIATION 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 120 Definition of the Derivative 120 The Derivative as a Function 129 Product and Quotient Rules 143 Rates of Change 150 Higher Derivatives 159 Trigonometric Functions 165 The Chain Rule 169 Derivatives of Inverse Functions 178 Derivatives of General Exponential and Logarithmic Functions 182 3.10 Implicit Differentiation 188 3.11 Related Rates 195 Preparing for the AP Exam AP3-1 vi 207 215 226 234 241 248 257 269 275 AP4-1 6.1 6.2 6.3 6.4 6.5 286 299 309 316 322 328 336 341 AP5-1 357 Area Between Two Curves 357 Setting Up Integrals: Volume, Density, Average Value 365 Volumes of Revolution 375 The Method of Cylindrical Shells 384 Work and Energy 391 Preparing for the AP Exam AP6-1 Chapter 7 TECHNIQUES OF INTEGRATION 7.1 7.2 7.3 7.4 286 400 Integration by Parts 400 Trigonometric Integrals 405 Trigonometric Substitution 413 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions 420 C ONTE N TS 7.5 7.6 7.7 7.8 The Method of Partial Fractions Improper Integrals Probability and Integration Numerical Integration Preparing for the AP Exam Chapter 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 8.1 8.2 8.3 8.4 Arc Length and Surface Area Fluid Pressure and Force Center of Mass Taylor Polynomials Preparing for the AP Exam 426 436 448 454 AP7-1 467 467 474 480 488 AP8-1 Chapter 9 INTRODUCTION TO DIFFERENTIAL EQUATIONS 502 9.1 9.2 9.3 9.4 9.5 Solving Differential Equations Models Involving y = k(y − b) Graphical and Numerical Methods The Logistic Equation First-Order Linear Equations Preparing for the AP Exam Chapter 10 INFINITE SERIES 10.1 10.2 10.3 10.4 10.5 10.6 10.7 Sequences Summing an Inﬁnite Series Convergence of Series with Positive Terms Absolute and Conditional Convergence The Ratio and Root Tests Power Series Taylor Series Preparing for the AP Exam 502 511 516 524 528 AP9-1 537 537 548 559 569 575 579 591 AP10-1 CALCULUS vii Chapter 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS 607 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Parametric Equations 607 Arc Length and Speed 620 Polar Coordinates 626 Area, Arc Length, and Slope in Polar Coordinates 634 Vectors in the Plane 641 Dot Product and the Angle between Two Vectors 653 Calculus of Vector-Valued Functions 660 Preparing for the AP Exam AP11-1 Chapter 12 DIFFERENTIATION IN SEVERAL VARIABLES 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 Functions of Two or More Variables Limits and Continuity in Several Variables Partial Derivatives Differentiability and Tangent Planes The Gradient and Directional Derivatives The Chain Rule Optimization in Several Variables Lagrange Multipliers: Optimizing with a Constraint 672 672 684 692 703 711 723 731 745 APPENDICES A. The Language of Mathematics B. Properties of Real Numbers C. Induction and the Binomial Theorem D. Additional Proofs A1 A1 A8 A13 A18 ANSWERS TO ODD-NUMBERED EXERCISES A27 ANSWERS TO THE ODD-NUMBERED PREPARING FOR THE AP EXAM QUESTIONS A104 REFERENCES A113 PHOTO CREDITS A116 INDEX I1