**Unformatted text preview: ** TO JULIE –Jon
TO ALEXA AND COLTON –Colin Publisher: Terri Ward
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ABOUT THE AUTHORS
COLIN ADAMS C olin Adams is the Thomas T. Read professor of Mathematics at Williams College,
where he has taught since 1985. Colin received his undergraduate degree from MIT
and his PhD from the University of Wisconsin. His research is in the area of knot theory
and low-dimensional topology. He has held various grants to support his research, and
written numerous research articles.
Colin is the author or co-author of The Knot Book, How to Ace Calculus: The Streetwise
Guide, How to Ace the Rest of Calculus: The Streetwise Guide, Riot at the Calc Exam
and Other Mathematically Bent Stories, Why Knot?, Introduction to Topology: Pure and
Applied, and Zombies & Calculus. He co-wrote and appears in the videos “The Great Pi
vs. E Debate” and “Derivative vs. Integral: the Final Smackdown.”
He is a recipient of the Haimo National Distinguished Teaching Award from the
Mathematical Association of America (MAA) in 1998, an MAA Polya Lecturer for 19982000, a Sigma Xi Distinguished Lecturer for 2000-2002, and the recipient of the Robert
Foster Cherry Teaching Award in 2003.
Colin has two children and one slightly crazy dog, who is great at providing the
entertainment. JON ROGAWSKI A s a successful teacher for more than 30 years, Jon Rogawski listened and learned
much from his own students. These valuable lessons made an impact on his thinking,
his writing, and his shaping of a calculus text.
Jon Rogawski received his undergraduate and master’s degrees in mathematics simultaneously from Yale University, and he earned his PhD in mathematics from Princeton
University, where he studied under Robert Langlands. Before joining the Department of
Mathematics at UCLA in 1986, where he was a full professor, he held teaching and visiting
positions at the Institute for Advanced Study, the University of Bonn, and the University
of Paris at Jussieu and Orsay.
Jon’s areas of interest were number theory, automorphic forms, and harmonic analysis on semisimple groups. He published numerous research articles in leading mathematics journals, including the research monograph Automorphic Representations of Unitary
Groups in Three Variables (Princeton University Press). He was the recipient of a Sloan
Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of
the AMS.
Sadly, Jon Rogawski passed away in September 2011. Jon’s commitment to presenting the beauty of calculus and the important role it plays in students’ understanding of the
wider world is the legacy that lives on in each new edition of Calculus. CONTENTS CALCULUS
Chapter 1 PRECALCULUS REVIEW
1.1
1.2
1.3
1.4
1.5 Real Numbers, Functions, and Graphs
Linear and Quadratic Functions
The Basic Classes of Functions
Trigonometric Functions
Technology: Calculators and Computers
Chapter Review Exercises 1
1
12
19
23
32
36 Chapter 2 LIMITS 39 2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9 39
47
56
59
68
73
78
84
87
94 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
Chapter Review Exercises Chapter 3 DIFFERENTIATION
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9 Definition of the Derivative
The Derivative as a Function
Product and Quotient Rules
Rates of Change
Higher Derivatives
Trigonometric Functions
The Chain Rule
Implicit Differentiation
Related Rates
Chapter Review Exercises 97
97
105
117
123
132
137
141
148
155
162 Chapter 4 APPLICATIONS OF THE DERIVATIVE 167
4.1
4.2
4.3
4.4
4.5
4.6
4.7 Linear Approximation and Applications
Extreme Values
The Mean Value Theorem and Monotonicity
The Shape of a Graph
Graph Sketching and Asymptotes
Applied Optimization
Newton’s Method
Chapter Review Exercises Chapter 5 THE INTEGRAL
5.1
5.2
5.3
iv Approximating and Computing Area
The Definite Integral
The Indefinite Integral 167
174
184
190
197
204
216
221 225
225
237
247 5.4
5.5
5.6
5.7 The Fundamental Theorem of Calculus, Part I
The Fundamental Theorem of Calculus, Part II
Net Change as the Integral of a Rate of Change
Substitution Method
Chapter Review Exercises Chapter 6 APPLICATIONS OF THE INTEGRAL
6.1
6.2
6.3
6.4
6.5 Derivative of f (x) = bx and the Number e
Inverse Functions
Logarithms and Their Derivatives
Exponential Growth and Decay
Compound Interest and Present Value
Models Involving y ′ = k(y − b)
L’Hôpital’s Rule
Inverse Trigonometric Functions
Hyperbolic Functions
Chapter Review Exercises Chapter 8 TECHNIQUES OF INTEGRATION
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9 321
321
329
336
345
351
356
361
368
376
386 391 Integration by Parts
391
Trigonometric Integrals
397
Trigonometric Substitution
405
Integrals Involving Hyperbolic and Inverse Hyperbolic
Functions
411
The Method of Partial Fractions
416
Strategies for Integration
424
Improper Integrals
432
Probability and Integration
443
Numerical Integration
450
Chapter Review Exercises
459 Chapter 9 FURTHER APPLICATIONS OF THE
INTEGRAL AND TAYLOR
POLYNOMIALS
9.1
9.2
9.3
9.4 281 Area Between Two Curves
281
Setting Up Integrals: Volume, Density, Average Value 289
Volumes of Revolution
299
The Method of Cylindrical Shells
307
Work and Energy
313
Chapter Review Exercises
319 Chapter 7 EXPONENTIAL FUNCTIONS
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9 254
259
264
270
277 Arc Length and Surface Area
Fluid Pressure and Force
Center of Mass
Taylor Polynomials
Chapter Review Exercises 463
463
470
476
485
496 CONTENTS Chapter 10 INTRODUCTION TO DIFFERENTIAL
EQUATIONS
499
10.1
10.2
10.3
10.4 Solving Differential Equations
Graphical and Numerical Methods
The Logistic Equation
First-Order Linear Equations
Chapter Review Exercises Chapter 11 INFINITE SERIES 499
507
515
519
525 529 11.1
11.2
11.3
11.4
11.5 Sequences
529
Summing an Infinite Series
539
Convergence of Series with Positive Terms
550
Absolute and Conditional Convergence
559
The Ratio and Root Tests and Strategies for Choosing
Tests
564
11.6 Power Series
569
11.7 Taylor Series
579
Chapter Review Exercises
591 Chapter 12 PARAMETRIC EQUATIONS, POLAR
COORDINATES, AND CONIC
SECTIONS
595
12.1
12.2
12.3
12.4
12.5 Parametric Equations
Arc Length and Speed
Polar Coordinates
Area and Arc Length in Polar Coordinates
Conic Sections
Chapter Review Exercises Chapter 13 VECTOR GEOMETRY
13.1
13.2
13.3
13.4
13.5
13.6
13.7 Vectors in the Plane
Vectors in Three Dimensions
Dot Product and the Angle Between Two Vectors
The Cross Product
Planes in 3-Space
A Survey of Quadric Surfaces
Cylindrical and Spherical Coordinates
Chapter Review Exercises Chapter 14 CALCULUS OF VECTOR-VALUED
FUNCTIONS
14.1
14.2
14.3
14.4
14.5
14.6 Vector-Valued Functions
Calculus of Vector-Valued Functions
Arc Length and Speed
Curvature
Motion in 3-Space
Planetary Motion According to Kepler and Newton
Chapter Review Exercises Chapter 15 DIFFERENTIATION IN SEVERAL
VARIABLES
15.1 Functions of Two or More Variables
15.2 Limits and Continuity in Several Variables 595
606
612
620
625
638 641
641
651
661
669
680
686
694
701 705
705
713
722
727
738
747
753 755
755
766 15.3
15.4
15.5
15.6
15.7
15.8 Partial Derivatives
Differentiability and Tangent Planes
The Gradient and Directional Derivatives
The Chain Rule
Optimization in Several Variables
Lagrange Multipliers: Optimizing with a Constraint
Chapter Review Exercises Chapter 16 MULTIPLE INTEGRATION
16.1
16.2
16.3
16.4 Integration in Two Variables
Double Integrals over More General Regions
Triple Integrals
Integration in Polar, Cylindrical, and Spherical
Coordinates
16.5 Applications of Multiple Integrals
16.6 Change of Variables
Chapter Review Exercises Chapter 17 LINE AND SURFACE INTEGRALS
17.1
17.2
17.3
17.4
17.5 Vector Fields
Line Integrals
Conservative Vector Fields
Parametrized Surfaces and Surface Integrals
Surface Integrals of Vector Fields
Chapter Review Exercises Chapter 18 FUNDAMENTAL THEOREMS OF
VECTOR ANALYSIS
18.1 Green’s Theorem
18.2 Stokes’ Theorem
18.3 Divergence Theorem
Chapter Review Exercises
APPENDICES
A.
The Language of Mathematics
B.
Properties of Real Numbers
C.
Induction and the Binomial Theorem
D.
Additional Proofs
ANSWERS TO ODD-NUMBERED EXERCISES v 773
783
790
803
811
825
834 837
837
848
861
872
882
894
907 911
911
921
935
946
960
970 973
973
987
997
1009
A1
A1
A7
A12
A16
ANS1 REFERENCES R1 INDEX I1 Additional content can be accessed online via LaunchPad:
ADDITIONAL PROOFS
• L’Hôpital’s Rule • Error Bounds for Numerical Integration • Comparison Test for Improper Integrals ADDITIONAL CONTENT
• Second Order Differential Equations • Complex Numbers PREFACE
ABOUT CALCULUS
On Teaching Mathematics
I consider myself very lucky to have a career as a teacher and practitioner of mathematics.
When I was young, I decided I wanted to be a writer. I loved telling stories. But I was also
good at math, and, once in college, it didn’t take me long to become enamored with it. I
loved the fact that success in mathematics does not depend on your presentation skills or
your interpersonal relationships. You are either right or you are wrong and there is little
subjective evaluation involved. And I loved the satisfaction of coming up with a solution.
That intensified when I started solving problems that were open research questions that
had previously remained unsolved.
So, I became a professor of mathematics. And I soon realized that teaching mathematics is about telling a story. The goal is to explain to students in an intriguing manner, at
the right pace, and in as clear a way as possible, how mathematics works and what it can
do for you. I find mathematics immensely beautiful. I want students to feel that way, too. On Writing a Calculus Text
I had always thought I might write a calculus text. But that is a daunting task. These days,
calculus books average over a thousand pages. And I would need to convince myself
that I had something to offer that was different enough from what already appears in the
existing books. Then, I was approached about writing the third edition of Jon Rogawski’s
calculus book. Here was a book for which I already had great respect. Jon’s vision of
what a calculus book should be fit very closely with my own. Jon believed that as math
teachers, how we say it is as important as what we say. Although he insisted on rigor at
all times, he also wanted a book that was written in plain English, a book that could be
read and that would entice students to read further and learn more. Moreover, Jon strived
to create a text in which exposition, graphics, and layout would work together to enhance
all facets of a student’s calculus experience.
In writing his book, Jon paid special attention to certain aspects of the text:
1. Clear, accessible exposition that anticipates and addresses student difficulties.
2. Layout and figures that communicate the flow of ideas.
3. Highlighted features that emphasize concepts and mathematical reasoning: Conceptual
Insight, Graphical Insight, Assumptions Matter, Reminder, and Historical Perspective.
4. A rich collection of examples and exercises of graduated difficulty that teach basic
skills, problem-solving techniques, reinforce conceptual understanding, and motivate calculus through interesting applications. Each section also contains exercises that develop
additional insights and challenge students to further develop their skills. vi Coming into the project of creating the third edition, I was somewhat apprehensive.
Here was an already excellent book that had attained the goals set for it by its author. First
and foremost, I wanted to be sure that I did it no harm. On the other hand, I have been
teaching calculus now for 30 years, and in that time, I have come to some conclusions
about what does and does not work well for students.
As a mathematician, I want to make sure that the theorems, proofs, arguments and
development are correct. There is no place in mathematics for sloppiness of any kind.
As a teacher, I want the material to be accessible. The book should not be written at the
mathematical level of the instructor. Students should be able to use the book to learn the
material, with the help of their instructor. Working from the high standard that Jon set, I
have tried hard to maintain the level of quality of the previous edition while making the
changes that I believe will bring the book to the next level. PREFACE vii Placement of Taylor Polynomials
Taylor polynomials appear in Chapter 9, before infinite series in Chapter 11. The goal
here is to present Taylor polynomials as a natural extension of linear approximation.
When teaching infinite series, the primary focus is on convergence, a topic that many
students find challenging. By the time we have covered the basic convergence tests and
studied the convergence of power series, students are ready to tackle the issues involved
in representing a function by its Taylor series. They can then rely on their previous work
with Taylor polynomials and the error bound from Chapter 9. However, the section on
Taylor polynomials is written so that you can cover this topic together with the materials
on infinite series if this order is preferred. Careful, Precise Development
W. H. Freeman is committed to high quality and precise textbooks and supplements.
From this project’s inception and throughout its development and production, quality and
precision have been given significant priority. We have in place unparalleled procedures
to ensure the accuracy of the text:
•
•
•
•
• Exercises and Examples
Exposition
Figures
Editing
Composition Together, these procedures far exceed prior industry standards to safeguard the quality
and precision of a calculus textbook. New to the Third Edition
There are a variety of changes that have been implemented in this edition. Following are
some of the most important.
MORE FOCUS ON CONCEPTS The emphasis has been shifted to focus less on the memorization of specific formulas, and more on understanding the underlying concepts. Memorization can never be completely avoided, but it is in no way the crux of calculus. Students
will remember how to apply a procedure or technique if they see the logical progression
that generates it. And they then understand the underlying concepts rather than seeing the
topic as a black box in which you insert numbers. Specific examples include:
•
•
•
•
• • (Section 1.2) Removed the general formula for the completion of a square and
instead, emphasized the method so students need not memorize the formula.
(Section 8.2) Changed the methods for evaluating trigonometric integrals to focus
on techniques to apply rather than formulas to memorize.
(Chapter 10) Discouraged the memorization of solutions of specific types of differential equations and instead, encouraged the use of methods of solution.
(Section 13.2) Decreased number of formulas for parametrizing a line from two to
one, as the second can easily be derived from the first.
(Section 13.6) De-emphasized the memorization of the various formulas for quadric
surfaces. Instead, moved the focus to slicing with planes to find curves and using
those to determine the shape of the surface. These methods will be useful regardless
of the type of surface it is.
(Section 15.4) Decreased the number of essential formulas for linear approximation
of functions of two variables from four to two, providing the background to derive
the others from these. CHANGES IN NOTATION There are numerous notational changes. Some were made to
bring the notation more into line with standard usage in mathematics and other fields in
which mathematics is applied. Some were implemented to make it easier for students to
remember the meaning of the notation. Some were made to help make the corresponding
concepts that are represented more transparent. Specific examples include: viii PREFACE
• •
• • (Section 4.5) Presented a new notation for graphing that gives the signs of the first
and second derivative and then simple symbols (slanted up and down arrows and
up and down u’s) to help the student keep track of when the graph is increasing or
decreasing and concave up or concave down over the given interval.
(Section 8.1) Simplified the notation for integration by parts and provided a visual
method for remembering it.
(Chapter 11) Changed names of the various tests for convergence/divergence of
infinite series to evoke the usage of the test and thereby make it easier for students
to remember them.
(Chapters 14–18) Rather than using c(t) for a path, we consistently switched to
the vector-valued function r(t). This also allowed us to replace ds with dr as a
differential, which means there is less likely to be confusion with ds, dS and dS. MORE EXPLANATIONS OF DERIVATIONS Occasionally, in the previous edition, a result
was given and verified, without motivating where the derivation came from. I believe it is
important for students to understand how someone might come up with a particular result,
thereby helping them to picture how they might themselves one day be able to derive
results.
•
•
•
• (Section 9.3) Developed the center of mass formulas by first discussing the onedimensional case of a seesaw.
(Section 15.4) Developed the equation of the tangent plane in a manner that makes
geometric sense.
(Section 15.5) Included a proof of the fact the gradient of a function f of three
variables is orthogonal to the surfaces that are the level sets of f .
(Section 15.8) Gave an intuitive explanation for why the Method of Lagrange
Multipliers works. REORDERING AND ADDING TOPICS There were some specific rearrangements among the
sections and additions. These include:
•
• •
•
•
•
•
• A subsection on piecewise-defined functions has been added to Section 1.3.
The section on indefinite integrals (previously Section 4.8) has been moved from
Chapter 4 (Applications of the Derivative) to Chapter 5 (The Integral). This is a
more natural placement for it.
A new section on choosing from amongst the various methods of integration has
been added to Chapter 8.
A subsection on choosing the appropriate convergence/divergence test has been
added to Section 11.5.
An explanation of how to find indefinite limits using power series has been added
to Section 10.6.
The definitions of divergence and curl have been moved from Chapter 18 to Section
17.1. This allows us to utilize them at an appropriate earlier point in the text.
A list all of the different types of integrals that have been introduced in Chapter 17
has been added to Section 17.5.
A subsection on the Vector Form of Green’s Theorem has been added to Section
18.1. NEW EXAMPLES, FIGURES, AND EXERCISES Numerous examples and accompanying
figures have been added to clarify concepts. A variety of exercises have also been added
throughout the text, particularly wher...

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