Amsco's AP Calculus AB BC: Preparing for the Advanced Placement Exams

405 Pages·2004·3.26 MB·English

by  Maxine Lifshitz

AAMMSSCCOO’’SS
AP Calculus
AB / B C
Preparing for the
Advanced Placement
Examinations
Maxine Lifshitz
Mathematics Department Chairperson
Friends Academy
Locust Valley, New York
with
Martha Green
Mathematics Teacher
Baldwin High School
Baldwin, New York
AMSCO
Amsco School Publications, Inc.
315 Hudson Street, New York, N.Y. 10013
Author Maxine Lifshitz is Chair of the Math Department at Friends Academy in
Locust Valley, New York. She received her A.B. degree from Barnard Col-
lege with Honors in Mathematics and her Ph.D. from New York University
in Mathematics Education. She has been a mathematics consultant for
the College Board and a reader of Advanced Placement Calculus Examina-
tions. Dr. Lifshitz has conducted workshops in applications of the graph-
ing calculator both locally at Calculators Help All Teachers (CHAT) and
Long Island Mathematics (Limaçon) conferences, and nationally at
National Council of Teachers of Mathematics (NCTM) and Teachers
Teaching with Technology (T3) conferences. Dr. Lifshitz has published
articles in Mathematics Teacher and The New York State Mathematics
Teachers Journal.
Collaborator Martha Green has taught mathematics at Baldwin High School for the
past 17 years and is currently a reader for the AP Calculus Examinations.
She received a Bachelor’s degree in Engineering from Hofstra University
and a Masters degree in Secondary Education from Adelphi University.
She instructs a graduate-level class on Teaching AP Calculus through The
Effective Teachers Program of the New York State United Teachers
(NYSUT). She has conducted numerous workshops on using calculators
to enhance the teaching of mathematics and has presented at CHAT,
Limaçon, and regional NCTM conferences. She has previously collaborat-
ed with Dr. Lifshitz to conduct workshops at T3International Conferences.
In 2001, the Nassau County Mathematics Teachers Association named
Martha Green Teacher of the Year.
Reviewers
Steven J. Balasiano Brad Huff
Assistant Principal Supervising Mathematics Headmaster
Canarsie High School University High School
Brooklyn, NY Fresno, CA
Terrence Kent
Mathematics Teacher
Downers Grove High School
Downers Grove, IL
Text design by One Dot Inc.
Composition and Line Art by Nesbitt Graphics, Inc.
Please visit our Web site at: www.amscopub.com
When ordering this book, please specify:
eitherR 781 W orAP CALCULUS AB/BC:
PREPARING FOR THE ADVANCED PLACEMENT EXAMINATIONS.
ISBN 1-56765-562-9
NYC Item 56765-562-8
Copyright © 2004 by Amsco School Publications, Inc.
No part of this book may be reproduced in any form without written permission
from the publisher.
Printed in the United States of America
1 2 3 4 5 6 7 8 9 10 09 08 07 06 05 04 03
To all the AP Calculus teachers, from
the ones who accept the challenge a few
weeks before the course begins to those
who continuously refresh and renew
themselves after years of teaching. And to
Seymour, Alissa, and Mariel, who provide
the base and the encouragement for all
my efforts.
Maxine Lifshitz
To my parents, Robert and Martha
Sweeney, who raised me to believe I
could accomplish anything and who
helped me realize my dreams.
Martha Green

CCOONNTTEENNTTSS
AB/BC Topics
Chapter 1 Introduction 3
About the Book 3
How to Use This Book 4
Prerequisites to AP Calculus 4
The AP Calculus AB and BC Courses 4
The AP Calculus Examinations 5
Factors Leading to Success in AP Calculus 6
Chapter 2 Functions and Their Properties 8
2.1 A Review of Basic Functions 8
2.2 Lines 20
2.3 Properties of Functions 22
2.4 Inverses 28
2.5 Translations and Reflections 31
2.6 Parametric Equations 32
Chapter Assessment 35
Contents v
Chapter 3 Limits and Continuity 39
3.1 Functions and Asymptotes 39
3.2 Evaluating Limits as x Approaches a Finite Number c 43
3.3 Evaluating Limits as x Approaches ;q 46
3.4 Special Limits: lim sin and lim 1(cid:2)cos x 48
xS0 x xS0 x
3.5 Evaluating Limits of a Piecewise-Defined Function 50
3.6 Continuity of a Function 52
Chapter Assessment 54
Chapter 4 The Derivative 58
4.1 The Derivative of a Function 58
4.2 The Average Rate of Change of a Function on an Interval 63
4.3 The Definition of the Derivative 66
4.4 Rules for Derivatives 67
4.5 Recognizing the Form of the Derivative 73
4.6 The Equation of a Tangent Line 78
4.7 Differentiability vs. Continuity 81
4.8 Particle Motion 82
4.9 Motion of a Freely Falling Object 86
4.10 Implicit Differentiation 88
4.11 Related Rates 91
4.12 Derivatives of Parametric Equations 93
Chapter Assessment 94
Chapter 5 Applications of the Derivative 98
5.1 Three Theorems: The Extreme Value Theorem,
5.1 Rolle’s Theorem, and the Mean Value Theorem 98
5.2 Critical Values 102
5.3 Concavity and the Second Derivative 110
5.4 Curve Sketching and the Graphing Calculator 113
5.5 Optimization 118
Chapter Assessment 121
Chapter 6 Techniques and Applications
of Antidifferentiation 125
6.1 Antiderivatives 125
6.2 Area Under a Curve: Approximation by Riemann Sums 131
6.3 The Fundamental Theorem of Calculus 138
6.4 The Accumulation Function: An Application of
6.4 Part Two of the Fundamental Theorem 146
6.5 Integration by the Change of Variable or
6.4 u-Substitution Method 150
6.6 Applications of the Integral: Average Value of a Function 155
6.7 Volumes 157
6.8 The Trapezoidal Rule 168
6.9 Arc Length and Area of a Surface of Revolution 172
Chapter Assessment 176
vi Contents
Chapter 7 Separable Differential Equations and
Slope Fields 180
7.1 Separable Differential Equations 180
7.2 Slope Fields 185
7.3 The Connection Between a Slope Field
7.3 and Its Differential Equation 188
Chapter Assessment 192
BC Topics
Chapter 8 Methods of Integration 199
8.1 Integration by Parts 199
8.2 Integration by the Method of Partial Fractions 202
8.3 Improper Integrals 206
Chapter Assessment 209
Chapter 9 Polynomial Approximations and
Infinite Series 213
9.1 Introduction 213
9.2 Sigma Notation 215
9.3 Derivation of the Taylor Polynomial Formula 217
9.4 Finding New Polynomials from Old 221
9.5 Error Formula for Taylor Polynomial Approximation 225
9.6 Sequences and Series 227
9.7 Power Series 242
Chapter Assessment 248
Chapter 10 Logistic Growth, Euler’s Method, and
Other BC Topics 253
10.1 The Logistic Growth Model 253
10.2 Euler’s Approximation Method 257
10.3 Logarithmic Differentiation 261
10.4 L’Hôpital’s Rule 263
10.5 Polar Curves 266
Chapter Assessment 277
Model Examinations
281
AB Model Examination 1 283
AB Model Examination 2 293
BC Model Examination 1 302
BC Model Examination 2 311
Contents vii
Answer Key
321
Chapter 2 323
Chapter 3 331
Chapter 4 335
Chapter 5 346
Chapter 6 350
Chapter 7 356
Chapter 8 360
Chapter 9 361
Chapter 10 367
Model Examinations 371
AB Model Examination 1 371
AB Model Examination 2 378
BC Model Examination 1 383
BC Model Examination 2 389
Index
395
viii Contents
AABB//BBCC TTooppiiccss
CHAPTER 1 Introduction
CHAPTER 2 Functions and Their Properties
CHAPTER 3 Limits and Continuity
CHAPTER 4 The Derivative
CHAPTER 5 Applications of the Derivative
CHAPTER 6 Techniques and Applications
of Antidifferentiation
CHAPTER 7 Separable Differential
Equations and Slope Fields