W. H. FREEMAN AND COMPANYNew York

CALCULUS for AP*SECOND EDITION

JON ROGAWSKIUniversity of California, Los Angeles

RAY CANNONBaylor University, TX

EARLY TRANSCENDENTALS

ROGAWSKI’S

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Library of Congress Control NumberISBN-13: 978-1-4292-5074-0ISBN-10: 1-4292-5074-7

© 2012 by W. H. Freeman and CompanyAll rights reserved

Printed in the United States of AmericaFirst printing

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To Julie

and

To the AP Teachers

CONTENTSROGAWSKI ’S CALCULUS for AP*

Early Transcendentals

Chapter 1 PRECALCULUS REVIEW 1

1.1 Real Numbers, Functions, and Graphs 11.2 Linear and Quadratic Functions 131.3 The Basic Classes of Functions 211.4 Trigonometric Functions 251.5 Inverse Functions 331.6 Exponential and Logarithmic Functions 431.7 Technology: Calculators and Computers 51

Chapter 2 LIMITS 59

2.1 Limits, Rates of Change, and Tangent Lines 592.2 Limits: A Numerical and Graphical Approach 672.3 Basic Limit Laws 772.4 Limits and Continuity 812.5 Evaluating Limits Algebraically 902.6 Trigonometric Limits 952.7 Limits at Infinity 1002.8 Intermediate Value Theorem 1062.9 The Formal Definition of a Limit 110

Preparing for the AP Exam AP2-1

Chapter 3 DIFFERENTIATION 120

3.1 Definition of the Derivative 1203.2 The Derivative as a Function 1293.3 Product and Quotient Rules 1433.4 Rates of Change 1503.5 Higher Derivatives 1593.6 Trigonometric Functions 1653.7 The Chain Rule 1693.8 Derivatives of Inverse Functions 1783.9 Derivatives of General Exponential and Logarithmic

Functions 1823.10 Implicit Differentiation 1883.11 Related Rates 195

Preparing for the AP Exam AP3-1

Chapter 4 APPLICATIONS OF THE DERIVATIVE 207

4.1 Linear Approximation and Applications 2074.2 Extreme Values 2154.3 The Mean Value Theorem and Monotonicity 2264.4 The Shape of a Graph 2344.5 L’Hôpital’s Rule 2414.6 Graph Sketching and Asymptotes 2484.7 Applied Optimization 2574.8 Newton’s Method 2694.9 Antiderivatives 275

Preparing for the AP Exam AP4-1

Chapter 5 THE INTEGRAL 286

5.1 Approximating and Computing Area 2865.2 The Definite Integral 2995.3 The Fundamental Theorem of Calculus, Part I 3095.4 The Fundamental Theorem of Calculus, Part II 3165.5 Net Change as the Integral of a Rate 3225.6 Substitution Method 3285.7 Further Transcendental Functions 3365.8 Exponential Growth and Decay 341

Preparing for the AP Exam AP5-1

Chapter 6 APPLICATIONS OF THE INTEGRAL 357

6.1 Area Between Two Curves 3576.2 Setting Up Integrals: Volume, Density, Average Value 3656.3 Volumes of Revolution 3756.4 The Method of Cylindrical Shells 3846.5 Work and Energy 391

Preparing for the AP Exam AP6-1

Chapter 7 TECHNIQUES OF INTEGRATION 400

7.1 Integration by Parts 4007.2 Trigonometric Integrals 4057.3 Trigonometric Substitution 4137.4 Integrals Involving Hyperbolic and Inverse Hyperbolic

Functions 420

vi

CONTENTS C A L C U L U S vii

7.5 The Method of Partial Fractions 4267.6 Improper Integrals 4367.7 Probability and Integration 4487.8 Numerical Integration 454

Preparing for the AP Exam AP7-1

Chapter 8 FURTHER APPLICATIONS OF THEINTEGRAL AND TAYLORPOLYNOMIALS 467

8.1 Arc Length and Surface Area 4678.2 Fluid Pressure and Force 4748.3 Center of Mass 4808.4 Taylor Polynomials 488

Preparing for the AP Exam AP8-1

Chapter 9 INTRODUCTION TO DIFFERENTIALEQUATIONS 502

9.1 Solving Differential Equations 5029.2 Models Involving y′ = k(y − b) 5119.3 Graphical and Numerical Methods 5169.4 The Logistic Equation 5249.5 First-Order Linear Equations 528

Preparing for the AP Exam AP9-1

Chapter 10 INFINITE SERIES 537

10.1 Sequences 53710.2 Summing an Infinite Series 54810.3 Convergence of Series with Positive Terms 55910.4 Absolute and Conditional Convergence 56910.5 The Ratio and Root Tests 57510.6 Power Series 57910.7 Taylor Series 591

Preparing for the AP Exam AP10-1

Chapter 11 PARAMETRIC EQUATIONS, POLARCOORDINATES, AND VECTORFUNCTIONS 607

11.1 Parametric Equations 60711.2 Arc Length and Speed 62011.3 Polar Coordinates 62611.4 Area, Arc Length, and Slope in Polar Coordinates 63411.5 Vectors in the Plane 64111.6 Dot Product and the Angle between Two Vectors 65311.7 Calculus of Vector-Valued Functions 660

Preparing for the AP Exam AP11-1

Chapter 12 DIFFERENTIATION IN SEVERALVARIABLES 672

12.1 Functions of Two or More Variables 67212.2 Limits and Continuity in Several Variables 68412.3 Partial Derivatives 69212.4 Differentiability and Tangent Planes 70312.5 The Gradient and Directional Derivatives 71112.6 The Chain Rule 72312.7 Optimization in Several Variables 73112.8 Lagrange Multipliers: Optimizing with a Constraint 745

APPENDICES A1A. The Language of Mathematics A1B. Properties of Real Numbers A8C. Induction and the Binomial Theorem A13D. Additional Proofs A18

ANSWERS TO ODD-NUMBERED EXERCISES A27

ANSWERS TO THE ODD-NUMBERED PREPARING FOR THEAP EXAM QUESTIONS A104

REFERENCES A113

PHOTO CREDITS A116

INDEX I1