Table Of ContentSpecial Edition Chapters 1-5
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Karl J. Smith Monty J. Strauss Magdalena D. Toda
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Kendall Hunt
p u b lis h in g com pany
Calculus
Seventh Edition
Special Edition, Chapters 1-5
Karl J. Smith
Santa Rosa Junior College
Monty J. Strauss
Texas Tech University
Magdalena D. Toda
Texas Tech University
Kendall Hunt
p u b lis h in g com pany
Cover image Copyright © Shutterstock.com
The cover image of the Louvre was selected to highlight the melding of the old and the new. The study of
calculus has a rich heritage, full of anecdotes and mountaintop ideas, symbolized by the majestic old building
in Paris known as the Louvre. It was constructed as a fortress in 1190 and went thorough many changes until
becoming the palace of the Sun King, Louis XIV, only to be transformed into a museum by Louis XVI. The
building has constantly undergone change, as does mathematics, and in 1989 a modern Pyramid was added to
the museum. At about that time, calculus was “reinvented” using new technologies to capture the ideas of
calculus. Just as the Louvre blends the old and the new in its structure, this book also blends the old and the
new in its presentation. To symbolize this symbiosis, we use the logo:
THINKING IN
<⅛→NE1U DIRECTION
All images, unless otherwise noted, are © Kendall Hunt Publishing Company.
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Copyright © 1995, 1999, 2002 by Prentice-Hall, Inc.
Special Edition © 2006, 2007 by Prentice-Hall, Inc.
Copyright © 2014, 2018 by Kendall Hunt Publishing Company
PAK ISBN 978-1-5249-7135-9
Text ISBN 978-1-5249-7137-3
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the copyright owner.
Published in the United States of America
C
O N T E N T S
Preface v
1 Functions and Graphs 1
1.1 What Is Calculus? 2
1.2 Preliminaries 15
1.3 Lines in the Plane; Parametric Equations 27
1.4 Functions and Graphs 36
1.5 Inverse Functions; Inverse Trigonometric Functions 51
Chapter 1 Review 62
Ethnomathematics by Marcia Ascher 71
Chapter 1 Group Research Project 72
2 Limits and Continuity 73
2.1 The Limit of a Function 74
2.2 Algebraic Computation of Limits 85
2.3 Continuity 95
2.4 Exponential and Logarithmic Functions 107
Chapter 2 Review 122
Chapter 2 Group Research Project 129
3 Differentiation 131
3.1 An Introduction to the Derivative: Tangents 132
3.2 Techniques of Differentiation 145
3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions 154
3.4 Rates of Change: Modeling Rectilinear Motion 161
3.5 The Chain Rule 173
3.6 Implicit Differentiation 181
3.7 Related Rates and Applications 193
3.8 Linear Approximation and Differentials 202
Chapter 3 Review 215
[>233EEED3 Fermat’s Enigma by Simon Singh 217
Chapter 3 Group Research Project 223
4 Additional Applications of the Derivative 225
4.1 Extreme Values of a Continuous Function 226
4.2 The Mean Value Theorem 237
4.3 Using Derivatives to Sketch the Graph of a Function 244
4.4 Curve Sketching with Asymptotes: Limits Involving Infinity 261
4.5 l,H6pitaΓs Rule 273
4.6 Optimization in the Physical Sciences and Engineering 282
4.7 Optimization in Business, Economics, and the Life Sciences 298
Chapter 4 Review 313
Chapter 4 Group Research Project 322
iii
iv Contents
5 Integration 323
5.1 Antidifferentiation 324
5.2 Area as the Limit of a Sum 336
5.3 Riemann Sums and the Definite Integral 344
5.4 The Fundamental Theorems of Calculus 357
5.5 Integration by Substitution 364
5.6 Introduction to Differential Equations 372
5.7 The Mean Value Theorem for Integrals; Average Value 385
5.8 Numerical Integration: The Trapezoidal Rule and Simpson’s Rule 392
5.9 An Alternative Approach: The Logarithm as an Integral 402
Chapter 5 Review 409
Chapter 5 Group Research Project 417
Cumulative Review Problems—Chapters 1-5 418
Appendices 421
A: Introduction to the Theory of Limits 421
B: Selected Proofs 427
C: Significant Digits 429
D: Short Table of Integrals 434
E: Trigonometry 444
F: Parabolas 452
G: Ellipses 462
H: Hyperbolas 473
I: Determinants 484
J: Answers to Selected Problems 489
Index 515
P R E F A C E
“M
1 rZ athematics is the instrument by which the engineer
tunnels our mountains, bridges our rivers, constructs our
aqueducts, erects our factories and makes them musical by
the busy hum of spindles. Take away the results of the rea
soning of mathematics, and there would go with it nearly
all the material achievements which give convenience and
glory to modem civilization. ”
Edward Brooks. Mental Science and Culture,
Philadelphia: Normal Publishing, 1891, p. 255.
THINKING IN &.
FOR THE STU D EN T C∣→NEUJ DIRECTION
You have enrolled in a calculus course, and you might be thinking, “I’ve finally made
it ... All my life I’ve studied mathematics, and now Γm enrolled in ‘the big one.’
Certainly, calculus is the pinnacle of my studies in mathematics!”
But wait! So you are here,... but do you know why? Can you answer the question,
“What is calculus?” In the first section of this text, we begin by answering this question,
and if you look at college catalogues, you will see that calculus is the first course in
college mathematics, not the last one. This course is a prerequisite for almost all the
other nonremedial mathematics courses.
OK, so now you are here in a calculus course and you are asked to pay a great deal
of money for this book. “Why does this book cost so much, and how could it possibly
be worth all that money?” The value of any purchase is relative to many aspects which
differ from person to person, but ultimately, value must be measured by the extent to
which it changes our lives. The intent and goal of this book is to change your life by
giving you life skills which enhance your problem-solving ability. By the time you finish
this course you will have a book, which at different times, you will have loved, hated,
cursed, and respected. You will have a reference book that should last you a lifetime.
Calculus is a difficult subject, and there is no magic key to success; it will require
hard work. This book should make your calculus journey easier because it builds your
problem-solving skills and helps you form some good study habits. You will need to read
the book, work the examples in the book using your own pencil and paper, and make a
commitment to do your mathematics homework on a daily basis. & Read that last sentence
once again. It is the best hint you will see about building success in mathematics. £
We have written this book so that it will be easy for you to know what is important.
Each chapter begins with a preview and ends with a proficiency examination (generally
30 problems) consisting of concept problems (to help you know what is important)
and practice problems (to help you with a self-test).
Important terms are presented in boldface type.
Important ideas, definitions, and procedures are enclosed in screened boxes:
Definitions , Properties , and Procedures
Common pitfalls, helpful hints and explanations are shown using this font.
& WARNINGS are given to call your attention to common mistakes.
V
vi Preface
• Color is used in a functional way to help you “see” what to do next.
• INTERACTIVE figures allow you to “see” a figure in a dynamic way—essential since
calculus discusses dynamic, not static, processes.
Success in this course is a joint effort by the student, the instructor, and the author.
The student must be willing to attend class and devote time to the course on a daily
basis. There is no substitute for working problems in mathematics.
FOR THE INSTRUCTOR
As the instructor of a calculus course, you must select materials which are relevant to
your students at your school or university. Some aspects of your course are necessitated
by your environment, and others are selected by you or a committee in your department.
We have put together a calculus textbook which we believe will meet your needs, as
well as those of your department. Most importantly, our book will meet the needs of
your students in the twenty-first century.
One of the first questions asked by most instructors: “Does this calculus book do early
or late transcendentals?” The quick answer is that this is an early transcendental book,
but we understand the mathematical reasons why one might want a late transcendental
development. For this reason, we have offered an optional Section 5.9 which defines the
logarithm as an integral.
The writing of a calculus textbook is a monumental task, one that grows out of a love
of teaching and of helping students struggle until we see “the light” in their eyes as he
or she has an “ah-hah” moment. The creation of this text has been a life-long journey for
its authors, and to understand the nature of this book we travel back to 1930 to tell you
of the books which have influenced us as we have developed the materials in this text.
At one time, the defining and almost universally used calculus textbook was
Granville, Smith, and Longley’s calculus book published in 1929.* This 516 page book
contained no color, few illustrations, and even fewer problems. At that time, there were
about a million students enrolled in the American university system, and education was
designed to be for the privileged few.
In 1957, Sputnik created a revolution in American education. Rigor was added to
the teaching of calculus. The most influential calculus textbook of this era was a book
by Thomas' which was initially published in 1952 and became the standard text in the
1960s. This book added rigor and problems, but still offered little pedagogy. By the
end of 1968, Thomas was in its fourth edition, and had grown to 818 pages, which
matched the growth of college enrollments at that time to almost four million students.
Textbooks were still relatively inexpensive, but were colorless and without learning aids
or technology. By 1970, Calculus by Apostol÷ was being used as a rigid and rigorous
alternative to calculus,... correct, but lifeless. Many instructors of that time adopted a
“sink-or-swim” attitude as there seemed to be an unlimited supply of students, each year
growing larger and larger.
By the 1980s, college enrollments had passed the ten million mark and calculus
needed to change to keep up with the times. In 1987 the next generation of calcu
lus books was bom. Stewart’s Calculus^ changed both the content and pedagogy of
calculus. During this period of time, there was a movement to “reform” calculus. In 1986
and 1987, Steve Maurer and Ronald Douglas held two conferences, “Toward a Lean and
Lively Calculus” and “Calculus for a New Century: A Pump not a Filter.” This move
ment gave rise to what has become known as the Harvard Calculus approach and the
*W. A. Granville, P. F. Smith, and W. R. Longley, Elements of the Differential and Integral Calculus. Boston: Ginn
and Company, 1929.
i G. B. Thomas, Calculus and Analytic Geometry, Classic Edition (Reprint of the 1952 edition). Reading: Addison
Wesley, 1983.
$Tom Apostle, Calculus (2 volums): Hoboken, Wiley, 1967
$James Stewart, Calculus. Belmont: Brooks/Cole Publishing Company, a division of Wadsworth, Inc., 1987
Preface vii
publication of a book commonly known as the Hughes-Hallett Calculus^- The reform
calculus movement divided the mathematics community into two “camps” and the teach
ing of calculus went through great changes, most of which have had lasting benefits to
students.
With these books (and we have either learned or taught from all of them) and others
we found those features which are essential and those which are superficial. Over the
years, we have class tested our ideas and with this edition we have brought to you a book
which is as fresh as this century, but preserves all the tried-and-true techniques which
have been found to provide the necessary foundation for future work in mathematics. As
college enrollments soar past the twenty million mark, education must now be designed
for the masses. Regardless of the equipment offered at school, individual students have
access to more and more technology (computers, iPads, smartphones, calculators, and
Facebook, etc.). They use that technology not only to learn, but to communicate with
each other in a cyber community. The calculus book for next generation needs to “work”
in today’s world and with today’s changing technology, but at the same time, not compro
mise the student’s ability to work complicated mathematical problems without relying on
technology. These issues were our guiding principle as we wrote this edition of Calculus.
FEATURES OF THIS BOOK
Some of the distinguishing characteristics of the earlier editions are continued with this
edition:
• It is possible to begin the course with either Chapter 1 or Chapter 2 (where the calculus
topics begin).
• This edition offers an early presentation of transcendental functions: logarithms, expo
nential functions, and trigonometric functions are heavily integrated into all chapters
of the book (especially Chapters 1-5).
• Because of user feedback, this edition has expanded drawing lessons for the student.
We offer practical solutions to students on using engineering paper to draw curves
in two and three dimensions. These drawing lessons include hints on plotting points
in three dimensions, drawing circles, ellipses, hyperbolas, trigonometric curves, and
polar-form curves, as well as planes and lines in three dimensions.
• We have taken the introduction of differential equations seriously. Students in many
allied disciplines need to use differential equations early in their studies and conse
quently cannot wait for a postcalculus course. In this edition, we introduce differential
equations in a natural and reasonable way. Slope fields are introduced as a geometric
view of antidifferentiation in Section 5.1, and then are used to introduce a graphi
cal solution to differential equations in Section 5.6. We consider separable differential
equations in Chapter 5 and first-order linear differential equations in Chapter 7, and
demonstrate the use of both, modeling a variety of applied situations. Exact and homo
geneous differential equations appear in Chapter 14, along with an introduction to
second-order linear equations. The “early and often” approach to differential equations
is intended to illustrate their value in continuous modeling and to provide a solid
foundation for further study.
• We continue to utilize the humanness of mathematics. History is not presented as
additional material to leam. Rather we have placed history into problems that lead the
reader from the development of a concept to actually participating in the discovery
process. The problems are designated as historical Quest problems. The problems are
not designed to be “add-on or challenge problems,” but rather to become an integral
part of the assignment. The level of difficulty of Qwe.s ∙ problems ranges from easy to
difficult.
• This edition correctly reflects the precalculus mathematics being taught at most colleges
and universities. We assume knowledge of the trigonometric functions, and in this
, Deborah Hughes-Hallett, Andrew M. Gleason, et. al., Calculus. Hoboken: John Wiley & Sons, 1994.
viii Preface
edition we introduce ex and lnx in Chapter 2 after we have defined the notion of a
limit. We also assume a knowledge of the conic sections and their graphs.
• Think Tank Problems Thinking about and doing mathematics is different than
solving textbook problems. When thinking about and doing mathematics, a proposition
may be true or false, whereas the typical textbook problem has a nice and concise
answer. In the Think Tank problems our task is to prove the proposition true or to
find a counterexample to disprove the proposition. We believe this form of problem to
be important in preparing the student for future work in not only advanced mathematics
courses, but also for analytically oriented courses.
• EXPLORATION PROBLEMS It has been said that mathematical discovery is
directed toward two major goals—the formulation of proofs and the construction of
counterexamples. Most calculus books focus only on the first goal (the body of proofs
and true statements), but we feel that some attention should be paid to the formulation,
exploring concepts that may prove to be true or for which a counterexample is appro
priate. These exploration problems go beyond the category of counterexample problem
to provide opportunities for innovative thinking.
• Journal Problems In an effort to show that “mathematicians work problems too,”
we have reprinted problems from leading mathematics journals. We have chosen prob
lems that are within reach of the intended audience of this book. If students need help
or hints for these problems, they can search out the original presentation and solution
in the cited journal. In addition, we have included problems from various Putnam
examinations. These problems, which are more challenging, are offered in the supple
mentary problems at the end of various chapters and are provided to give insight into
the type of problems that are asked in mathematical competitions. The Putnam Exam
ination is a national annual examination given under the auspices of the Mathematical
Association of America and is designed to recognize mathematically talented college
and university students.
• Modeling continues as a major theme in this edition. Modeling is discussed in Section,
and then appears in almost every section of the book. These applications are desig
nated Modeling Problems or MODELING EXPERIMENTS. Some authors use the
words “Modeling Problem” to refer to any applied problem. We make a distinction
between modeling problems and application problems by defining a modeling prob
lem as follows. A modeling problem is a problem that requires the reader make
some assumptions about the real world in order to derive or come up with the neces
sary mathematical formula or mathematical information to answer the question. These
problems also include real-world examples of modeling by citing the source of the
book or journal that shows the modeling process.
Much of the difficulty students encounter learning the ideas of calculus can be
attributed to ways students study and learn mathematics in high school, which often
involves stressing rote memorization over insight and understanding. On the other hand,
some reform texts are perceived as spending so much time with the development of insight
and understanding that students are not given enough exposure to important computational
and problem-solving skills in order to perform well in more advanced courses. This text
aims at a middle ground by providing sound development, simulating problems, and
well-developed pedagogy within a framework of a traditional tropic structure. “Think,
then do” is a fair summary of our approach.
TEXT CONTENT
The content of this text adapts itself to either semester or quarter systems, and both
differentiation and integration can be introduced in the first course. We begin calculus
with a minimum of review. Cumulative reviews are offered at locations that fit the
way calculus is taught at most colleges and universities. The first one includes Chapters
1-5, the second includes Chapters 6-8. Furthermore there is a cumulative review for
