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
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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