Table Of ContentStudent Outlines
Part One
Joseph E. Borzellino
Patricia Nelson
S O
TUDENT UTLINES
P O
ART NE
JOSEPH E. BORZELLINO
California Polytechnic State University
PATRICIA NELSON
University of Wisconsin-La Crosse
U C
NIVERSITY ALCULUS
Joel Hass
University of California, Davis
Maurice D. Weir
Naval Postgraduate School
George B. Thomas, Jr.
Massachusetts Institute of Technology
TT
PEARSON
Addison
Wesley
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Reproduced by Pearson Addison-Wesley from electronic files supplied by the author.
Copyright © 2007 Pearson Education, Inc.
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out the prior written permission of the publisher. Printed in the United States of America.
ISBN 0-321-39551-4
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PEARSON
,'" .
Addison
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TABLE OF CONTENTS
Study Notes SN-1
Chapter I Functions SN-1
Chapter 2 Limits and Continuity SN-11
Chapter 3 Differentiation SN-19
Chapter 4 Applications of Derivatives SN-29
Chapter 5 Integration SN-35
Chapter 6 Applications of Definite Integrals SN-40
Chapter 7 Techniques of Integration SN-46
Chapter 8 Infinite Sequences and Series SN-54
Chapter 9 Polar Coordinates and Conics SN-65
Appendix A Brief Table of Integrals SN-70
1 Functions 1
1.1 Functions and Their Graphs I
1.2 Combining Functions; Shifting andS caling Graphs 5
1.3 Trigonometric Functions 8
1.4 Exponential Functions 10
1.5 Inverse Functions and Logarithms 12
1.6 Graphing with Calculators and Computers 14
Self-Test 15
Solutions to Self-Test 16
2 Limits and Continuity 18
2.1 Rates of Change and Tangents to Curves 18
2.2 Limit of a Function and Limit Laws 18
2.3 The Precise Definition of a Limit 21
2.4 One-Sided Limits and Limits at Infinity 23
2.5 Infinite Limits and Vertical Asymptotes 25
2.6 Continuity 26
2.7 Tangents and Derivatives at a Point 27
Self-Test 29
Solutions to Self-Test 29
3 Differentiation 31
3.1 The Derivative as a Function 31
3.2 Differentiation Rules for Polynomials, Exponentials, Products and Quotients 33
3.3 The Derivative as a Rate of Change 36
3.4 Derivatives of Trigonometric Functions 38
3.5 The Chain Rule and Parametric Equations 39
3.6 Implicit Differentiation 43
3.7 Derivatives of Inverse Functions and Logarithms 44
3.8 Inverse Trigonometric Functions 48
3.9 Related Rates 49
3.10 Linearization and Differentials 51
3.11 Hyperbolic Functions 52
Self-Test 56
Solutions to Self-Test 57
4 Applications of Derivatives 62
4.1 Extreme Values of Functions 62
4.2 The Mean Value Theorem 63
4.3 Monotonic Functions and the First Derivative Test 64
4.4 Concavity and Curve Sketching 65
4.5 Applied Optimization Problems 69
4.6 Indeterminate Forms and L,H6pitaΓs Rule 72
4.7 Newton’s Method 73
4.8 Antiderivatives 73
Self-Test 76
Solutions to Self-Test 76
5 Integration 80
5.1 Estimating with Finite Sums 80
5.2 Sigma Notation and Limits of Finite Sums 81
5.3 The Definite Integral 82
5.4 The Fundamental Theorem of Calculus 85
5.5 Indefinite Integrals and the Substitution Rule 87
5.6 Substitution and Area Between Curves 89
5.7 The Logarithm Defined as an Integral 92
Self-Test 96
Solutions to Self-Test 96
6 Applications of Definite Integrals 99
6.1 Volumes by Slicing and Rotation About an Axis 99
6.2 Volumes by Cylindrical Shells 101
6.3 Lengths of Plane Curves 103
6.4 Areas of Surfaces of Revolution and the Theorems of Pappus 104
6.5 Exponential Change and Separable Differential Equations 105
6.6 Work 106
6.7 Moments and Centers of Mass 108
Self-Test 110
Solutions to Self-Test 111
7 Techniques of Integration 114
7.1 Integration by Parts 114
7.2 Trigonmetric Integrals 116
7.3 Trigonmetric Substitutions 117
7.4 Integration of Rational Functions by Partial Fractions 119
7.5 Integral Tables and Computer Algebra Systems 122
7.6 Numerical Integration 122
7.7 Improper Integrals 124
Self-Test 127
Solutions to Self-Test 127
8 Infinite Sequences and Series 132
8.1 Sequences 132
8.2 Infinite Series 135
8.3 The Integral Test 137
8.4 Comparison Tests 138
8.5 The Ratio and Root Tests 139
8.6 Alternating Series, Absolute and Conditional Convergence 139
8.7 Power Series 143
8.8 Taylor and Maclaurin Series 145
8.9 Convergence of Taylor Series 146
8.10 The Binomial Series 150
Self-Test 151
Solutions to Self-Test 152
9 Polar Coordinates and Conics 155
9.1 Polar Coordinates 155
9.2 Graphing in Polar Coordinates 157
9.3 Areas and Lengths in Polar Coordinates 158
9.4 Conic Sections 160
9.5 Conics in Polar Coordinates 163
9.6 Conics and Parametric Equations; the Cycloid 166
Self-Test 167
Solutions to Self-Test 167
PREFACE TO THE STUDENT
This manual has been designed especially for you, the student. It conforms to University
Calculus, by Joel Hass, Maurice D. Weir, and George B. Thomas Jr. It is intended as a self-study
workbook to assist you in mastering the basic ideas of calculus. Although this manual was written to
conform to the University Calculus, it can be used to accompany any standard calculus textbook and
course.
ORGANIZATION AND LEARNING OBJECTIVES
The study manual is organized section by section to correspond with University Calculus. For
each section we specify its main ideas by stating appropriate learning OBJECTIVES. Each objective
states a particular task for you to perform in order to master that objective. Usually the task requires
you to solve a certain type of problem related to the discussion in the text; sometimes the task requires
you to demonstrate proficiency with certain key terms or concepts. Each objective is highly specific
and is followed by one or more examples which illustrate its requirements.
Correct answers to each example are supplied at the bottom of the page. Some examples are
partially worked out and require you to formulate intermediate steps. Others are broken down into a
sequence of steps to guide you through the procedures and techniques associated with its solution.
Many others are questions similar to the exercises found in the text; however, each example has been
carefully selected not to repeat examples or problems in University Calculus so that you retain the
full array of the text problems for practice and further learning.
SELF-TESTS
At the end of each chapter there is a SELF-TEST. Each test is followed by complete solutions to
all of the test problems. The test problems cover the objectives and are similar in scope and difficulty
to the examples in this manual and the examples and problems in University Calculus. The test should
be useful in preparing for class examinations.
HOW TO USE THIS STUDY GUIDE
We recommend that this manual be used in the following ways:
1. Read the textbook: Carefully read the section of University Calculus assigned to you by your
calculus instructor.
2. Study the learning objectives and examples: Read each objective and work through the associ
ated example(s) in the corresponding section of this manual. You should conceal the answers to
the examples at the bottom of the page. Work with pencil and scratch paper as you work through
each solution, writing any intermediate results in the blanks provided.
3. Check your answers: After you have arrived at your solution, and any blanks for a given problem
are filled in, compare your answers with the correct answers at the bottom of the page. If you
have difficulty or do not fully understand the given answers, review the material in the textbook
or consult with your instructor.
4. Do the chapter self-test: After you complete a chapter in University Calculus, review the objec
tives for that chapter in this manual. Then take the chapter self-test and compare your solutions
with those provided. Problems in the self-test sometimes bring together several ideas from the
chapter.
5. Compare your notes with the STUDY NOTES written to accompany University Calculus. You
may find a new way of looking at a concept or an approach to a topic which differs from what
your instructor presents in class. Seeing mathematical concepts from different angles is often
highly useful.
GUIDANCE FROM YOUR INSTRUCTOR
We caution you that the learning objectives given in this manual by no means exhaust all the
possible objectives that could be written for a careful study of calculus; we have tried to identify the
main ones. However, your instructor may have additional requirements. For instance, he or she may
want you to be able to prove certain theorems or derive results in the text. We have not stated objectives
of this sort. Also, your instructor may consider some objectives far more important than others and not
require that you master some objectives at all. Moreover, your instructor may require critical thinking
and writing exercises that ask you to explain the reasoning behind your answers. It is imperative that
you find out specifically what your instructor considers essential, and study accordingly. This manual
should be helpful to you both in identifying the tasks and successfully mastering them. The problems
assigned by your instructor should help you discover those concepts and applications of calculus that
your instructor wishes to stress.
