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calculus
eighth edition

James Stewart
M c Master University
and
University of Toronto

Australia • Brazil • Mexico • Singapore • United Kingdom • United States
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Calculus, Eighth Edition
James Stewart

© 2016, 2012 Cengage Learning

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Library of Congress Control Number: 2015937035
Student Edition:
ISBN: 978-1-285-74062-1
Loose-leaf Edition:
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Print Number: 03     Print Year: 2015
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Contents
Preface  xi
To the Student  xxiii
Calculators, Computers, and other graphing devices  xxiv
Diagnostic tests  xxvi

A Preview of Calculus   1

1









1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8





Four Ways to Represent a Function  10
Mathematical Models: A Catalog of Essential Functions  23
New Functions from Old Functions  36
The Tangent and Velocity Problems  45
The Limit of a Function  50
Calculating Limits Using the Limit Laws  62
The Precise Definition of a Limit  72
Continuity 82
Review 94

Principles of Problem Solving 98

2




 95






2.1

Derivatives and Rates of Change  106
 Writing Project  •  Early Methods for Finding Tangents  117
2.2 The Derivative as a Function  117
2.3 Differentiation Formulas  130
 Applied Project  •  Building a Better Roller Coaster  144
2.4 Derivatives of Trigonometric Functions  144
2.5 The Chain Rule  152
 Applied Project  •  Where Should a Pilot Start Descent?  161
2.6 Implicit Differentiation  161
 Laboratory Project  •  Families of Implicit Curves  168

iii
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

iv

Contents






2.7
2.8
2.9

Rates of Change in the Natural and Social Sciences  169
Related Rates  181
Linear Approximations and Differentials  188
 Laboratory Project  •  Taylor Polynomials  194

Review 195



Problems Plus  200

3













3.1

Maximum and Minimum Values  204
 Applied Project  •  The Calculus of Rainbows  213
3.2 The Mean Value Theorem  215
3.3 How Derivatives Affect the Shape of a Graph  221
3.4 Limits at Infinity; Horizontal Asymptotes  231
3.5 Summary of Curve Sketching  244
3.6 Graphing with Calculus and Calculators  251
3.7 Optimization Problems  258
 Applied Project  •  The Shape of a Can  270
 Applied Project  •  Planes and Birds: Minimizing Energy   271
3.8 Newton’s Method  272
3.9 Antiderivatives 278

Review 285



Problems Plus  289

4








4.1
4.2

Areas and Distances  294
The Definite Integral  306
 Discovery Project  • Area Functions 319
4.3 The Fundamental Theorem of Calculus  320
4.4 Indefinite Integrals and the Net Change Theorem  330
 Writing Project  •  Newton, Leibniz, and the Invention of Calculus  339
4.5 The Substitution Rule  340

Review 348



Problems Plus  352

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Contents
v

5








5.1

Areas Between Curves  356
 Applied Project  •  The Gini Index  364
5.2 Volumes 366
5.3 Volumes by Cylindrical Shells  377
5.4 Work 383
5.5 Average Value of a Function  389
 Applied Project  •  Calculus and Baseball  392

Review 393



Problems Plus  395

6


6.1



Instructors may cover either Sections 6.2–6.4 or Sections 6.2*–6.4*. See the Preface.

Inverse Functions  400

6.2

Exponential Functions and
Their Derivatives  408

6.2* The Natural Logarithmic
Function 438

6.3

Logarithmic
Functions 421

6.3* The Natural Exponential
Function 447

6.4

Derivatives of Logarithmic
Functions 428

6.4* General Logarithmic and
Exponential Functions  455




6.5

Exponential Growth and Decay  466
 Applied Project  •  Controlling Red Blood Cell Loss During Surgery  473




6.6

Inverse Trigonometric Functions  474
 Applied Project  •  Where to Sit at the Movies  483



6.7




6.8

Hyperbolic Functions  484
Indeterminate Forms and l’Hospital’s Rule  491
 Writing Project  •  The Origins of l’Hospital’s Rule  503

Review 503



Problems Plus  508

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

vi

Contents

7










7.1
7.2
7.3
7.4
7.5
7.6

Integration by Parts  512
Trigonometric Integrals  519
Trigonometric Substitution  526
Integration of Rational Functions by Partial Fractions  533
Strategy for Integration  543
Integration Using Tables and Computer Algebra Systems  548
 Discovery Project  •  Patterns in Integrals  553
7.7 Approximate Integration  554
7.8 Improper Integrals  567

Review 577



Problems Plus  580

8









8.1

Arc Length 584
 Discovery Project  •  Arc Length Contest  590
8.2 Area of a Surface of Revolution  591
 Discovery Project  •  Rotating on a Slant  597
8.3 Applications to Physics and Engineering  598
 Discovery Project  •  Complementary Coffee Cups  608
8.4 Applications to Economics and Biology  609
8.5 Probability 613

Review 621



Problems Plus  623

9








9.1
9.2
9.3

Modeling with Differential Equations  626
Direction Fields and Euler’s Method  631
Separable Equations  639
 Applied Project  •  How Fast Does a Tank Drain?  648
 Applied Project  •  Which Is Faster, Going Up or Coming Down?  649
9.4 Models for Population Growth  650
9.5 Linear Equations  660

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Contents
vii



9.6

Predator-Prey Systems  667

Review 674



Problems Plus  677

10










10.1 Curves Defined by Parametric Equations  680
 Laboratory Project  •  Running Circles Around Circles  688
10.2 Calculus with Parametric Curves  689
 Laboratory Project  • Bézier Curves 697
10.3 Polar Coordinates  698
 Laboratory Project  •  Families of Polar Curves  708
10.4 Areas and Lengths in Polar Coordinates  709
10.5 Conic Sections  714
10.6 Conic Sections in Polar Coordinates  722

Review 729



Problems Plus  732

11
















11.1 Sequences 734
 Laboratory Project  • Logistic Sequences 747
11.2 Series 747
11.3 The Integral Test and Estimates of Sums  759
11.4 The Comparison Tests  767
11.5 Alternating Series  772
11.6 Absolute Convergence and the Ratio and Root Tests  777
11.7 Strategy for Testing Series  784
11.8 Power Series  786
11.9 Representations of Functions as Power Series  792
11.10 Taylor and Maclaurin Series  799
 Laboratory Project  •  An Elusive Limit  813
 Writing Project  •  How Newton Discovered the Binomial Series  813
11.11 Applications of Taylor Polynomials  814
 Applied Project  •  Radiation from the Stars  823





Review 824

Problems Plus  827

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

viii

Contents

12









12.1 Three-Dimensional Coordinate Systems 832
12.2 Vectors 838
12.3 The Dot Product  847
12.4 The Cross Product  854
 Discovery Project  •  The Geometry of a Tetrahedron  863
12.5 Equations of Lines and Planes  863
 Laboratory Project  •  Putting 3D in Perspective  873
12.6 Cylinders and Quadric Surfaces  874





7et1206un03
04/21/10
MasterID: 01462

Review 881

Problems Plus  884

13






13.1 Vector Functions and Space Curves 888
13.2 Derivatives and Integrals of Vector Functions  895
13.3 Arc Length and Curvature  901
13.4 Motion in Space: Velocity and Acceleration  910
 Applied Project  • Kepler’s Laws 920





Review 921

Problems Plus  924

14











14.1 Functions of Several Variables  928
14.2 Limits and Continuity  943
14.3 Partial Derivatives  951
14.4 Tangent Planes and Linear Approximations  967
 Applied Project  •  The Speedo LZR Racer  976
14.5 The Chain Rule  977
14.6 Directional Derivatives and the Gradient Vector  986
14.7 Maximum and Minimum Values  999
 Applied Project  •  Designing a Dumpster  1010
 Discovery Project  •  Quadratic Approximations and Critical Points  1010

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Contents
ix





14.8 Lagrange Multipliers  1011
 Applied Project  • Rocket Science 1019
 Applied Project  • Hydro-Turbine Optimization 1020





Review 1021

Problems Plus  1025

15













15.1 Double Integrals over Rectangles  1028
15.2 Double Integrals over General Regions  1041
15.3 Double Integrals in Polar Coordinates  1050
15.4 Applications of Double Integrals  1056
15.5 Surface Area  1066
15.6 Triple Integrals  1069
 Discovery Project  •  Volumes of Hyperspheres  1080
15.7 Triple Integrals in Cylindrical Coordinates  1080
 Discovery Project  •  The Intersection of Three Cylinders   1084
15.8 Triple Integrals in Spherical Coordinates  1085
 Applied Project  • Roller Derby 1092
15.9 Change of Variables in Multiple Integrals  1092





Review 1101

Problems Plus  1105

16










16.1 Vector Fields  1108
16.2 Line Integrals  1115
16.3 The Fundamental Theorem for Line Integrals  1127
16.4 Green’s Theorem  1136
16.5 Curl and Divergence  1143
16.6 Parametric Surfaces and Their Areas  1151
16.7 Surface Integrals  1162
16.8 Stokes’ Theorem  1174
 Writing Project  •  Three Men and Two Theorems  1180

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

x

Contents




16.9 The Divergence Theorem  1181
16.10 Summary 1187





Review 1188

Problems Plus  1191

17





17.1
17.2
17.3
17.4

Second-Order Linear Equations  1194
Nonhomogeneous Linear Equations  1200
Applications of Second-Order Differential Equations  1208
Series Solutions  1216



Review 1221










Numbers, Inequalities, and Absolute Values  A2
Coordinate Geometry and Lines  A10
Graphs of Second-Degree Equations  A16
Trigonometry A24
Sigma Notation  A34
Proofs of Theorems  A39
Complex Numbers  A48
Answers to Odd-Numbered Exercises  A57

A
B
C
D
E
F
G
H

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface
A great discovery solves a great problem but there is a grain of discovery in the
solution of any problem. Your problem may be modest; but if it challenges your
curiosity and brings into play your inventive faculties, and if you solve it by your
own means, you may experience the tension and enjoy the triumph of discovery.
g e o r g e p o lya

The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried
to write a book that assists students in discovering calculus—both for its practical power
and its surprising beauty. In this edition, as in the first seven editions, I aim to convey
to the student a sense of the utility of calculus and develop technical competence, but I
also strive to give some appreciation for the intrinsic beauty of the subject. Newton
undoubtedly experienced a sense of triumph when he made his great discoveries. I want
students to share some of that excitement.
The emphasis is on understanding concepts. I think that nearly everybody agrees that
this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation:
Focus on conceptual understanding.
I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and
graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently, the Rule of Three has been expanded
to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as
well.
In writing the eighth edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book
contains elements of reform, but within the context of a traditional curriculum.

I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions.
Calculus: Early Transcendentals, Eighth Edition, is similar to the present textbook
except that the exponential, logarithmic, and inverse trigonometric functions are
covered in the first semester.
● Essential Calculus, Second Edition, is a much briefer book (840 pages), though it
contains almost all of the topics in Calculus, Eighth Edition. The relative brevity is
achieved through briefer exposition of some topics and putting some features on the
website.


xi
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xii

Preface



Essential Calculus: Early Transcendentals, Second Edition, resembles Essential
Calculus, but the exponential, logarithmic, and inverse trigonometric functions are
covered in Chapter 3.
Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is
woven throughout the book instead of being treated in separate chapters.



Calculus: Early Vectors introduces vectors and vector functions in the first semester
and integrates them throughout the book. It is suitable for students taking engineering and physics courses concurrently with calculus.



Brief Applied Calculus is intended for students in business, the social sciences, and
the life sciences.



Biocalculus: Calculus for the Life Sciences is intended to show students in the life
sciences how calculus relates to biology.



Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all
the content of Biocalculus: Calculus for the Life Sciences as well as three additional chapters covering probability and statistics.



The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I’ve incorporated into this edition:










The data in examples and exercises have been updated to be more timely.
New examples have been added (see Examples 5.1.5, 11.2.5, and 14.3.3, for
instance). And the solutions to some of the existing examples have been amplified.
Three new projects have been added: The project Planes and Birds: Minimizing
Energy (page 271) asks how birds can minimize power and energy by flapping their
wings versus gliding. The project Controlling Red Blood Cell Loss During Surgery
(page 473) describes the ANH procedure, in which blood is extracted from the
patient before an operation and is replaced by saline solution. This dilutes the
patient’s blood so that fewer red blood cells are lost during bleeding and the
extracted blood is returned to the patient after surgery. In the project The Speedo
LZR Racer (page 976) it is explained that this suit reduces drag in the water and, as
a result, many swimming records were broken. Students are asked why a small
decrease in drag can have a big effect on performance.
I have streamlined Chapter 15 (Multiple Integrals) by combining the first two sections so that iterated integrals are treated earlier.
More than 20% of the exercises in each chapter are new. Here are some of my
favorites: 2.1.61, 2.2.34–36, 3.3.30, 3.3.54, 3.7.39, 3.7.67, 4.1.19–20, 4.2.67–68,
4.4.63, 5.1.51, 6.2.79, 6.7.54, 6.8.90, 8.1.39, 12.5.81, 12.6.29–30, 14.6.65–66.
In addition, there are some good new Problems Plus. (See Problems 10–12 on
page 201, Problem 10 on page 290, Problems 14–15 on pages 353–54, and Problem 8 on page 1026.)

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface
xiii

Conceptual Exercises
The most important way to foster conceptual understanding is through the problems
that we assign. To that end I have devised various types of problems. Some exercise sets
begin with requests to explain the meanings of the basic concepts of the section. (See, for
instance, the first few exercises in Sections 1.5, 1.8, 11.2, 14.2, and 14.3.) Similarly, all
the review sections begin with a Concept Check and a True-False Quiz. Other exercises
test conceptual understanding through graphs or tables (see Exercises 2.1.17, 2.2.33–36,
2.2.45–50, 9.1.11–13, 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–38,
14.1.41–44, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.6–8, 16.1.11–18, 16.2.17–18, and
16.3.1–2).
Another type of exercise uses verbal description to test conceptual understanding (see
Exercises 1.8.10, 2.2.64, 3.3.57–58, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.7.25,
3.4.33–34, and 9.4.4).

Graded Exercise Sets
Each exercise set is carefully graded, progressing from basic conceptual exercises and
skill-development problems to more challenging problems involving applications and
proofs.

Real-World Data
My assistants and I spent a great deal of time looking in libraries, contacting companies
and government agencies, and searching the Internet for interesting real-world data to
introduce, motivate, and illustrate the concepts of calculus. As a result, many of the
examples and exercises deal with functions defined by such numerical data or graphs.
See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake),
Exercise 2.2.33 (unemployment rates), Exercise 4.1.16 (velocity of the space shuttle
Endeavour), and Figure 4 in Section 4.4 (San Francisco power consumption). Functions
of two variables are illustrated by a table of values of the wind-chill index as a function
of air temperature and wind speed (Example 14.1.2). Partial derivatives are introduced
in Section 14.3 by examining a column in a table of values of the heat index (perceived
air temperature) as a function of the actual temperature and the relative humidity. This
example is pursued further in connection with linear approximations (Example 14.4.3).
Directional derivatives are introduced in Section 14.6 by using a temperature contour
map to estimate the rate of change of temperature at Reno in the direction of Las Vegas.
Double integrals are used to estimate the average snowfall in Colorado on December
20–21, 2006 (Example 15.1.9). Vector fields are introduced in Section 16.1 by depictions
of actual velocity vector fields showing San Francisco Bay wind patterns.

Projects
One way of involving students and making them active learners is to have them work
(perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve
applications that are designed to appeal to the imagination of students. The project after
Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height
or to fall back to its original height. (The answer might surprise you.) The project after
Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of
a rocket so as to minimize the total mass while enabling the rocket to reach a desired
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xiv

Preface

velocity. Laboratory Projects involve technology; the one following Section 10.2 shows
how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projects ask students to compare present-day methods with those of the founders of
calculus—Fermat’s method for finding tangents, for instance. Suggested references are
supplied. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 7.6). Others explore
aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6),
and intersections of three cylinders (after Section 15.7). Additional projects can be found
in the Instructor’s Guide (see, for instance, Group Exercise 4.1: Position from Samples).

Problem Solving
Students usually have difficulties with problems for which there is no single well-defined
procedure for obtaining the answer. I think nobody has improved very much on George
Polya’s four-stage problem-solving strategy and, accordingly, I have included a version
of his problem-solving principles following Chapter 1. They are applied, both explicitly
and implicitly, throughout the book. After the other chapters I have placed sections called
Problems Plus, which feature examples of how to tackle challenging calculus problems.
In selecting the varied problems for these sections I kept in mind the following advice
from David Hilbert: “A mathematical problem should be difficult in order to entice us,
yet not inaccessible lest it mock our efforts.” When I put these challenging problems on
assignments and tests I grade them in a different way. Here I reward a student significantly for ideas toward a solution and for recognizing which problem-solving principles
are relevant.

Dual Treatment of Exponential and Logarithmic Functions
There are two possible ways of treating the exponential and logarithmic functions and
each method has its passionate advocates. Because one often finds advocates of both
approaches teaching the same course, I include full treatments of both methods. In Sections 6.2, 6.3, and 6.4 the exponential function is defined first, followed by the logarithmic function as its inverse. (Students have seen these functions introduced this way since
high school.) In the alternative approach, presented in Sections 6.2*, 6.3*, and 6.4*, the
logarithm is defined as an integral and the exponential function is its inverse. This latter
method is, of course, less intuitive but more elegant. You can use whichever treatment
you prefer.
If the first approach is taken, then much of Chapter 6 can be covered before Chapters 4 and 5, if desired. To accommodate this choice of presentation there are specially
identified problems involving integrals of exponential and logarithmic functions at the
end of the appropriate sections of Chapters 4 and 5. This order of presentation allows a
faster-paced course to teach the transcendental functions and the definite integral in the
first semester of the course.
For instructors who would like to go even further in this direction I have prepared an
alternate edition of this book, called Calculus: Early Transcendentals, Eighth Edition,
in which the exponential and logarithmic functions are introduced in the first chapter.
Their limits and derivatives are found in the second and third chapters at the same time
as polynomials and the other elementary functions.

Tools for Enriching Calculus
TEC is a companion to the text and is intended to enrich and complement its contents.
(It is now accessible in the eBook via CourseMate and Enhanced WebAssign. Selected
Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey
Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface
xv

approach. In sections of the book where technology is particularly appropriate, marginal
icons direct students to TEC Modules that provide a laboratory environment in which
they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises.
Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration,
to assigning specific exercises from those included with each Module, or to creating
additional exercises, labs, and projects that make use of the Visuals and Modules.
TEC also includes Homework Hints for representative exercises (usually odd-numbered) in every section of the text, indicated by printing the exercise number in red.
These hints are usually presented in the form of questions and try to imitate an effective
teaching assistant by functioning as a silent tutor. They are constructed so as not to reveal
any more of the actual solution than is minimally necessary to make further progress.

Enhanced WebAssign
Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends
on ease of use, grading precision, and reliability. With the Eighth Edition we have been
working with the calculus community and WebAssign to develop an online homework
system. Up to 70% of the exercises in each section are assignable as online homework,
including free response, multiple choice, and multi-part formats.
The system also includes Active Examples, in which students are guided in step-bystep tutorials through text examples, with links to the textbook and to video solutions.

Website
Visit CengageBrain.com or stewartcalculus.com for these additional materials:


Homework Hints



Algebra Review



Lies My Calculator and Computer Told Me



History of Mathematics, with links to the better historical websites





Additional Topics (complete with exercise sets): Fourier Series, Formulas for the
Remainder Term in Taylor Series, Rotation of Axes
Archived Problems (drill exercises that appeared in previous editions, together with
their solutions)



Challenge Problems (some from the Problems Plus sections from prior editions)



Links, for particular topics, to outside Web resources



Selected Visuals and Modules from Tools for Enriching Calculus (TEC)

Diagnostic Tests

The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry.

A Preview of Calculus

This is an overview of the subject and includes a list of questions to motivate the study
of calculus.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xvi

Preface

1  Functions and Limits

From the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the
standard functions from these four points of view. The material on limits is motivated
by a prior discussion of the tangent and velocity problems. Limits are treated from
descriptive, graphical, numerical, and algebraic points of view. Section 1.7, on the precise
epsilon-delta defintion of a limit, is an optional section.

2 Derivatives

The material on derivatives is covered in two sections in order to give students more
time to get used to the idea of a derivative as a function. The examples and exercises
explore the meanings of derivatives in various contexts. Higher derivatives are introduced in Section 2.2.

3 Applications of Differentiation

The basic facts concerning extreme values and shapes of curves are deduced from the
Mean Value Theorem. Graphing with technology emphasizes the interaction between
calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to raise your
head 42° to see the top of a rainbow.

4 Integrals

The area problem and the distance problem serve to motivate the definite integral, with
sigma notation introduced as needed. (Full coverage of sigma notation is provided in
Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables.

5 Applications of Integration

Here I present the applications of integration—area, volume, work, average value—that
can reasonably be done without specialized techniques of integration. General methods
are emphasized. The goal is for students to be able to divide a quantity into small pieces,
estimate with Riemann sums, and recognize the limit as an integral.

6 Inverse Functions:

As discussed more fully on page xiv, only one of the two treatments of these functions
need be covered. Exponential growth and decay are covered in this chapter.

7 Techniques of Integration

All the standard methods are covered but, of course, the real challenge is to be able to
recognize which technique is best used in a given situation. Accordingly, in Section 7.5,
I present a strategy for integration. The use of computer algebra systems is discussed in
Section 7.6.

8 Further Applications
of Integration

Here are the applications of integration—arc length and surface area—for which it is
useful to have available all the techniques of integration, as well as applications to biology, economics, and physics (hydrostatic force and centers of mass). I have also
included a section on probability. There are more applications here than can realistically
be covered in a given course. Instructors should select applications suitable for their
students and for which they themselves have enthusiasm.

9 Differential Equations

Modeling is the theme that unifies this introductory treatment of differential equations.
Direction fields and Euler’s method are studied before separable and linear equations are
solved explicitly, so that qualitative, numerical, and analytic approaches are given equal

Exponential, Logarithmic, and
Inverse Trigonometric Functions

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface
xvii

consideration. These methods are applied to the exponential, logistic, and other models
for population growth. The first four or five sections of this chapter serve as a good
introduction to first-order differential equations. An optional final section uses predatorprey models to illustrate systems of differential equations.
10  Parametric Equations
and Polar Coordinates

This chapter introduces parametric and polar curves and applies the methods of calculus
to them. Parametric curves are well suited to laboratory projects; the two presented here
involve families of curves and Bézier curves. A brief treatment of conic sections in polar
coordinates prepares the way for Kepler’s Laws in Chapter 13.

11  Infinite Sequences and Series

The convergence tests have intuitive justifications (see page 759) as well as formal
proofs. Numerical estimates of sums of series are based on which test was used to prove
convergence. The emphasis is on Taylor series and polynomials and their applications
to physics. Error estimates include those from graphing devices.

12  Vectors and the
Geometry of Space

The material on three-dimensional analytic geometry and vectors is divided into two
chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and
surfaces.

13  Vector Functions

This chapter covers vector-valued functions, their derivatives and integrals, the length
and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler’s laws.

14  Partial Derivatives

Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific
column in a table of values of the heat index (perceived air temperature) as a function
of the actual temperature and the relative humidity.

15  Multiple Integrals

Contour maps and the Midpoint Rule are used to estimate the average snowfall and
average temperature in given regions. Double and triple integrals are used to compute
probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of
intersections of three cylinders. Cylindrical and spherical coordinates are introduced in
the context of evaluating triple integrals.

16  Vector Calculus

Vector fields are introduced through pictures of velocity fields showing San Francisco
Bay wind patterns. The similarities among the Fundamental Theorem for line integrals,
Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.

17 Second-Order
Differential Equations

Since first-order differential equations are covered in Chapter 9, this final chapter deals
with second-order linear differential equations, their application to vibrating springs and
electric circuits, and series solutions.

Calculus, Eighth Edition, is supported by a complete set of ancillaries developed
under my direction. Each piece has been designed to enhance student understanding
and to facilitate creative instruction. The tables on pages xxi–xxii describe each of these
ancillaries.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xviii

Preface

The preparation of this and previous editions has involved much time spent reading the
reasoned (but sometimes contradictory) advice from a large number of astute reviewers.
I greatly appreciate the time they spent to understand my motivation for the approach
taken. I have learned something from each of them.

Eighth Edition Reviewers
Jay Abramson, Arizona State University
Adam Bowers, University of California San Diego
Neena Chopra, The Pennsylvania State University
Edward Dobson, Mississippi State University
Isaac Goldbring, University of Illinois at Chicago
Lea Jenkins, Clemson University
Rebecca Wahl, Butler University

Technology Reviewers
Maria Andersen, Muskegon Community College
Eric Aurand, Eastfield College
Joy Becker, University of Wisconsin–Stout
Przemyslaw Bogacki, Old Dominion University
Amy Elizabeth Bowman, University of Alabama
in Huntsville
Monica Brown, University of Missouri–St. Louis
Roxanne Byrne, University of Colorado at Denver and
Health Sciences Center
Teri Christiansen, University of Missouri–Columbia
Bobby Dale Daniel, Lamar University
Jennifer Daniel, Lamar University
Andras Domokos, California State University, Sacramento
Timothy Flaherty, Carnegie Mellon University
Lee Gibson, University of Louisville
Jane Golden, Hillsborough Community College
Semion Gutman, University of Oklahoma
Diane Hoffoss, University of San Diego
Lorraine Hughes, Mississippi State University
Jay Jahangiri, Kent State University
John Jernigan, Community College of Philadelphia

Brian Karasek, South Mountain Community College
Jason Kozinski, University of Florida
Carole Krueger, The University of Texas at Arlington
Ken Kubota, University of Kentucky
John Mitchell, Clark College
Donald Paul, Tulsa Community College
Chad Pierson, University of Minnesota, Duluth
Lanita Presson, University of Alabama in Huntsville
Karin Reinhold, State University of New York at Albany
Thomas Riedel, University of Louisville
Christopher Schroeder, Morehead State University
Angela Sharp, University of Minnesota, Duluth
Patricia Shaw, Mississippi State University
Carl Spitznagel, John Carroll University
Mohammad Tabanjeh, Virginia State University
Capt. Koichi Takagi, United States Naval Academy
Lorna TenEyck, Chemeketa Community College
Roger Werbylo, Pima Community College
David Williams, Clayton State University
Zhuan Ye, Northern Illinois University

Previous Edition Reviewers
B. D. Aggarwala, University of Calgary
John Alberghini, Manchester Community College
Michael Albert, Carnegie-Mellon University
Daniel Anderson, University of Iowa
Amy Austin, Texas A&M University
Donna J. Bailey, Northeast Missouri State University
Wayne Barber, Chemeketa Community College
Marilyn Belkin, Villanova University
Neil Berger, University of Illinois, Chicago
David Berman, University of New Orleans
Anthony J. Bevelacqua, University of North Dakota
Richard Biggs, University of Western Ontario

Robert Blumenthal, Oglethorpe University
Martina Bode, Northwestern University
Barbara Bohannon, Hofstra University
Jay Bourland, Colorado State University
Philip L. Bowers, Florida State University
Amy Elizabeth Bowman, University of Alabama in Huntsville
Stephen W. Brady, Wichita State University
Michael Breen, Tennessee Technological University
Robert N. Bryan, University of Western Ontario
David Buchthal, University of Akron
Jenna Carpenter, Louisiana Tech University
Jorge Cassio, Miami-Dade Community College

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface
xix

Jack Ceder, University of California, Santa Barbara
Scott Chapman, Trinity University
Zhen-Qing Chen, University of Washington—Seattle
James Choike, Oklahoma State University
Barbara Cortzen, DePaul University
Carl Cowen, Purdue University
Philip S. Crooke, Vanderbilt University
Charles N. Curtis, Missouri Southern State College
Daniel Cyphert, Armstrong State College
Robert Dahlin
M. Hilary Davies, University of Alaska Anchorage
Gregory J. Davis, University of Wisconsin–Green Bay
Elias Deeba, University of Houston–Downtown
Daniel DiMaria, Suffolk Community College
Seymour Ditor, University of Western Ontario
Greg Dresden, Washington and Lee University
Daniel Drucker, Wayne State University
Kenn Dunn, Dalhousie University
Dennis Dunninger, Michigan State University
Bruce Edwards, University of Florida
David Ellis, San Francisco State University
John Ellison, Grove City College
Martin Erickson, Truman State University
Garret Etgen, University of Houston
Theodore G. Faticoni, Fordham University
Laurene V. Fausett, Georgia Southern University
Norman Feldman, Sonoma State University
Le Baron O. Ferguson, University of California—Riverside
Newman Fisher, San Francisco State University
José D. Flores, The University of South Dakota
William Francis, Michigan Technological University
James T. Franklin, Valencia Community College, East
Stanley Friedlander, Bronx Community College
Patrick Gallagher, Columbia University–New York
Paul Garrett, University of Minnesota–Minneapolis
Frederick Gass, Miami University of Ohio
Bruce Gilligan, University of Regina
Matthias K. Gobbert, University of Maryland, Baltimore County
Gerald Goff, Oklahoma State University
Stuart Goldenberg, California Polytechnic State University
John A. Graham, Buckingham Browne & Nichols School
Richard Grassl, University of New Mexico
Michael Gregory, University of North Dakota
Charles Groetsch, University of Cincinnati
Paul Triantafilos Hadavas, Armstrong Atlantic State University
Salim M. Haïdar, Grand Valley State University
D. W. Hall, Michigan State University
Robert L. Hall, University of Wisconsin–Milwaukee
Howard B. Hamilton, California State University, Sacramento
Darel Hardy, Colorado State University
Shari Harris, John Wood Community College
Gary W. Harrison, College of Charleston
Melvin Hausner, New York University/Courant Institute
Curtis Herink, Mercer University
Russell Herman, University of North Carolina at Wilmington
Allen Hesse, Rochester Community College

Randall R. Holmes, Auburn University
James F. Hurley, University of Connecticut
Amer Iqbal, University of Washington—Seattle
Matthew A. Isom, Arizona State University
Gerald Janusz, University of Illinois at Urbana-Champaign
John H. Jenkins, Embry-Riddle Aeronautical University,
Prescott Campus
Clement Jeske, University of Wisconsin, Platteville
Carl Jockusch, University of Illinois at Urbana-Champaign
Jan E. H. Johansson, University of Vermont
Jerry Johnson, Oklahoma State University
Zsuzsanna M. Kadas, St. Michael’s College
Nets Katz, Indiana University Bloomington
Matt Kaufman
Matthias Kawski, Arizona State University
Frederick W. Keene, Pasadena City College
Robert L. Kelley, University of Miami
Akhtar Khan, Rochester Institute of Technology
Marianne Korten, Kansas State University
Virgil Kowalik, Texas A&I University
Kevin Kreider, University of Akron
Leonard Krop, DePaul University
Mark Krusemeyer, Carleton College
John C. Lawlor, University of Vermont
Christopher C. Leary, State University of New York at Geneseo
David Leeming, University of Victoria
Sam Lesseig, Northeast Missouri State University
Phil Locke, University of Maine
Joyce Longman, Villanova University
Joan McCarter, Arizona State University
Phil McCartney, Northern Kentucky University
Igor Malyshev, San Jose State University
Larry Mansfield, Queens College
Mary Martin, Colgate University
Nathaniel F. G. Martin, University of Virginia
Gerald Y. Matsumoto, American River College
James McKinney, California State Polytechnic University, Pomona
Tom Metzger, University of Pittsburgh
Richard Millspaugh, University of North Dakota
Lon H. Mitchell, Virginia Commonwealth University
Michael Montaño, Riverside Community College
Teri Jo Murphy, University of Oklahoma
Martin Nakashima, California State Polytechnic University,
Pomona
Ho Kuen Ng, San Jose State University
Richard Nowakowski, Dalhousie University
Hussain S. Nur, California State University, Fresno
Norma Ortiz-Robinson, Virginia Commonwealth University
Wayne N. Palmer, Utica College
Vincent Panico, University of the Pacific
F. J. Papp, University of Michigan–Dearborn
Mike Penna, Indiana University–Purdue University Indianapolis
Mark Pinsky, Northwestern University
Lothar Redlin, The Pennsylvania State University
Joel W. Robbin, University of Wisconsin–Madison
Lila Roberts, Georgia College and State University

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xx

Preface

E. Arthur Robinson, Jr., The George Washington University
Richard Rockwell, Pacific Union College
Rob Root, Lafayette College
Richard Ruedemann, Arizona State University
David Ryeburn, Simon Fraser University
Richard St. Andre, Central Michigan University
Ricardo Salinas, San Antonio College
Robert Schmidt, South Dakota State University
Eric Schreiner, Western Michigan University
Mihr J. Shah, Kent State University–Trumbull
Qin Sheng, Baylor University
Theodore Shifrin, University of Georgia
Wayne Skrapek, University of Saskatchewan
Larry Small, Los Angeles Pierce College
Teresa Morgan Smith, Blinn College
William Smith, University of North Carolina
Donald W. Solomon, University of Wisconsin–Milwaukee
Edward Spitznagel, Washington University
Joseph Stampfli, Indiana University
Kristin Stoley, Blinn College

M. B. Tavakoli, Chaffey College
Magdalena Toda, Texas Tech University
Ruth Trygstad, Salt Lake Community College
Paul Xavier Uhlig, St. Mary’s University, San Antonio
Stan Ver Nooy, University of Oregon
Andrei Verona, California State University–Los Angeles
Klaus Volpert, Villanova University
Russell C. Walker, Carnegie Mellon University
William L. Walton, McCallie School
Peiyong Wang, Wayne State University
Jack Weiner, University of Guelph
Alan Weinstein, University of California, Berkeley
Theodore W. Wilcox, Rochester Institute of Technology
Steven Willard, University of Alberta
Robert Wilson, University of Wisconsin–Madison
Jerome Wolbert, University of Michigan–Ann Arbor
Dennis H. Wortman, University of Massachusetts, Boston
Mary Wright, Southern Illinois University–Carbondale
Paul M. Wright, Austin Community College
Xian Wu, University of South Carolina

In addition, I would like to thank R. B. Burckel, Bruce Colletti, David Behrman, John
Dersch, Gove Effinger, Bill Emerson, Dan Kalman, Quyan Khan, Alfonso Gracia-Saz,
Allan MacIsaac, Tami Martin, Monica Nitsche, Lamia Raffo, Norton Starr, and Jim Trefzger for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from
their calculus texts; COMAP for permission to use project material; George Bergman,
David Bleecker, Dan Clegg, Victor Kaftal, Anthony Lam, Jamie Lawson, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker
for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay,
Larry Riddle, Philip Straffin, and Klaus Volpert for ideas for projects; Dan Anderson,
Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises
and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in
proofreading; Andy Bulman-Fleming, Lothar Redlin, Gina Sanders, and Saleem Watson
for additional proofreading; and Jeff Cole and Dan Clegg for their careful preparation
and proofreading of the answer manuscript.
In addition, I thank those who have contributed to past editions: Ed Barbeau, Jordan Bell, George Bergman, Fred Brauer, Andy Bulman-Fleming, Bob Burton, David
Cusick, Tom DiCiccio, Garret Etgen, Chris Fisher, Leon Gerber, Stuart Goldenberg,
Arnold Good, Gene Hecht, Harvey Keynes, E. L. Koh, Zdislav Kovarik, Kevin Kreider,
Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Mary Pugh, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Dusty Sabo, Doug Shaw, Dan Silver,
Simon Smith, Norton Starr, Saleem Watson, Alan Weinstein, and Gail Wolkowicz.
I also thank Kathi Townes, Stephanie Kuhns, Kristina Elliott, and Kira Abdallah of
TECHarts for their production services and the following Cengage Learning staff:
Cheryll Linthicum, content project manager; Stacy Green, senior content developer;
Samantha Lugtu, associate content developer; Stephanie Kreuz, product assistant; Lynh
Pham, media developer; Ryan Ahern, marketing manager; and Vernon Boes, art director.
They have all done an outstanding job.
I have been very fortunate to have worked with some of the best mathematics editors
in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth,
Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, Richard Stratton, Liz Covello, and now Neha
Taleja. All of them have contributed greatly to the success of this book.
james stewart
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Instructor’s Guide
by Douglas Shaw
ISBN 978-1-305-27178-4
Each section of the text is discussed from several viewpoints.
The Instructor’s Guide contains suggested time to allot, points
to stress, text discussion topics, core materials for lecture,
workshop/discussion suggestions, group work exercises in
a form suitable for handout, and suggested homework
assignments.
Complete Solutions Manual
Single Variable
By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker
ISBN 978-1-305-27610-9
Multivariable
By Dan Clegg and Barbara Frank
ISBN 978-1-305-27611-6
Includes worked-out solutions to all exercises in the text.
Printed Test Bank
By William Steven Harmon
ISBN 978-1-305-27180-7
Contains text-specific multiple-choice and free response test
items.
Cengage Learning Testing Powered by Cognero
(login.cengage.com)
This flexible online system allows you to author, edit, and
manage test bank content from multiple Cengage Learning
solutions; create multiple test versions in an instant; and
deliver tests from your LMS, your classroom, or wherever you
want.

TEC TOOLS FOR ENRICHING™ CALCULUS
By James Stewart, Harvey Keynes, Dan Clegg, and developer
Hubert Hohn
Tools for Enriching Calculus (TEC) functions as both a
powerful tool for instructors and as a tutorial environment
in which students can explore and review selected topics. The
Flash simulation modules in TEC include instructions, written
and audio explanations of the concepts, and exercises. TEC
is accessible in the eBook via CourseMate and Enhanced
WebAssign. Selected Visuals and Modules are available at
www.stewartcalculus.com.
  Enhanced WebAssign®
www.webassign.net
Printed Access Code: ISBN 978-1-285-85826-5
Instant Access Code ISBN: 978-1-285-85825-8
Exclusively from Cengage Learning, Enhanced WebAssign
offers an extensive online program for Stewart’s Calculus
to encourage the practice that is so critical for concept
mastery. The meticulously crafted pedagogy and exercises
in our proven texts become even more effective in Enhanced
WebAssign, supplemented by multimedia tutorial support and
immediate feedback as students complete their assignments.
Key features include:
n  T
housands of homework problems that match your textbook’s end-of-section exercises
  Opportunities for students to review prerequisite skills and
content both at the start of the course and at the beginning
of each section

n

  Read It eBook pages, Watch It videos, Master It tutorials,
and Chat About It links

n

  A customizable Cengage YouBook with highlighting, notetaking, and search features, as well as links to multimedia
resources

n

  Personal Study Plans (based on diagnostic quizzing) that
identify chapter topics that students will need to master

n

  A WebAssign Answer Evaluator that recognizes and accepts
equivalent mathematical responses in the same way an
instructor grades

n

Stewart Website
www.stewartcalculus.com
Contents: Homework Hints  n  Algebra Review  n Additional
Topics  n  Drill exercises  n  Challenge Problems  n Web
Links  n  History of Mathematics  n  Tools for Enriching
Calculus (TEC)

■ Electronic items  ■ Printed items

  A Show My Work feature that gives instructors the option
of seeing students’ detailed solutions

n

  Visualizing Calculus Animations, Lecture Videos, and more

n

(Table continues on page xxii)

xxi
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Cengage Customizable YouBook
YouBook is an eBook that is both interactive and customizable. Containing all the content from Stewart’s Calculus,
YouBook features a text edit tool that allows instructors to
modify the textbook narrative as needed. With YouBook,
instructors can quickly reorder entire sections and chapters
or hide any content they don’t teach to create an eBook that
perfectly matches their syllabus. Instructors can further
customize the text by adding instructor-created or YouTube
video links. Additional media assets include animated figures,
video clips, highlighting and note-taking features, and more.
YouBook is available within Enhanced WebAssign.
CourseMate
CourseMate is a perfect self-study tool for students, and
requires no set up from instructors. CourseMate brings course
concepts to life with interactive learning, study, and exam
preparation tools that support the printed textbook. CourseMate for Stewart’s Calculus includes an interactive eBook,
Tools for Enriching Calculus, videos, quizzes, flashcards,
and more. For instructors, CourseMate includes Engagement
Tracker, a first-of-its-kind tool that monitors student
engagement.
CengageBrain.com
To access additional course materials, please visit
www.cengagebrain.com. At the CengageBrain.com home
page, search for the ISBN of your title (from the back cover of
your book) using the search box at the top of the page. This
will take you to the product page where these resources can
be found.

Student Solutions Manual
Single Variable
By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker
ISBN 978-1-305-27181-4
Multivariable
By Dan Clegg and Barbara Frank
ISBN 978-1-305-27182-1
Provides completely worked-out solutions to all oddnumbered exercises in the text, giving students a chance to

check their answer and ensure they took the correct steps
to arrive at the answer. The Student Solutions Manual
can be ordered or accessed online as an eBook at
www.cengagebrain.com by searching the ISBN.
Study Guide
Single Variable
By Richard St. Andre
ISBN 978-1-305-27913-1
Multivariable
By Richard St. Andre
ISBN 978-1-305-27184-5
For each section of the text, the Study Guide provides students
with a brief introduction, a short list of concepts to master,
and summary and focus questions with explained answers.
The Study Guide also contains self-tests with exam-style
questions. The Study Guide can be ordered or accessed online
as an eBook at www.cengagebrain.com by searching the
ISBN.
A Companion to Calculus
By Dennis Ebersole, Doris Schattschneider, Alicia Sevilla,
and Kay Somers
ISBN 978-0-495-01124-8
Written to improve algebra and problem-solving skills of
students taking a calculus course, every chapter in this
companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to
understand and solve calculus problems related to that topic.
It is designed for calculus courses that integrate the review of
precalculus concepts or for individual use. Order a copy of
the text or access the eBook online at www.cengagebrain.com
by searching the ISBN.
Linear Algebra for Calculus
by Konrad J. Heuvers, William P. Francis, John H. Kuisti,
Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner
ISBN 978-0-534-25248-9
This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic
ideas of linear algebra. Order a copy of the text or access
the eBook online at www.cengagebrain.com by searching the
ISBN.

■ Electronic items  ■ Printed items

xxii
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

To the Student
Reading a calculus textbook is different from reading a
newspaper or a novel, or even a physics book. Don’t be discouraged if you have to read a passage more than once
in order to understand it. You should have pencil and paper
and calculator at hand to sketch a diagram or make a
calculation.
Some students start by trying their homework problems
and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section
of the text before attempting the exercises. In particular, you
should look at the definitions to see the exact meanings of
the terms. And before you read each example, I suggest that
you cover up the solution and try solving the problem yourself. You’ll get a lot more from looking at the solution if
you do so.
Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected, step-by-step fashion with explanatory sentences—
not just a string of disconnected equations or formulas.
The answers to the odd-numbered exercises appear at the
back of the book, in Appendix H. Some exercises ask for a
verbal explanation or interpretation or description. In such
cases there is no single correct way of expressing the
answer, so don’t worry that you haven’t found the definitive
answer. In addition, there are often several different forms
in which to express a numerical or algebraic answer, so if
your answer differs from mine, don’t immediately assume
you’re wrong. For example, if the answer given in the back
of the book is s2 2 1 and you obtain 1y (1 1 s2 ), then
you’re right and rationalizing the denominator will show
that the answers are equivalent.
The icon ; indicates an exercise that definitely requires
the use of either a graphing calculator or a computer with
graphing software. But that doesn’t mean that graphing
devices can’t be used to check your work on the other exercises as well. The symbol CAS is reserved for problems in

which the full resources of a computer algebra system (like
Maple, Mathematica, or the TI-89) are required.
You will also encounter the symbol |, which warns you
against committing an error. I have placed this symbol in
the margin in situations where I have observed that a large
proportion of my students tend to make the same mistake.
Tools for Enriching Calculus, which is a companion to
this text, is referred to by means of the symbol TEC and can
be accessed in the eBook via Enhanced WebAssign and
CourseMate (selected Visuals and Modules are available at
www.stewartcalculus.com). It directs you to modules in
which you can explore aspects of calculus for which the
computer is particularly useful.
You will notice that some exercise numbers are printed
in red: 5. This indicates that Homework Hints are available
for the exercise. These hints can be found on stewartcalculus.com as well as Enhanced WebAssign and CourseMate.
The homework hints ask you questions that allow you to
make progress toward a solution without actually giving
you the answer. You need to pursue each hint in an active
manner with pencil and paper to work out the details. If a
particular hint doesn’t enable you to solve the problem, you
can click to reveal the next hint.
I recommend that you keep this book for reference purposes after you finish the course. Because you will likely
forget some of the specific details of calculus, the book will
serve as a useful reminder when you need to use calculus in
subsequent courses. And, because this book contains more
material than can be covered in any one course, it can also
serve as a valuable resource for a working scientist or
engineer.
Calculus is an exciting subject, justly considered to be
one of the greatest achievements of the human intellect. I
hope you will discover that it is not only useful but also
intrinsically beautiful.
james stewart

xxiii
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Calculators, Computers, and
Other Graphing Devices

xxiv

© Dan Clegg

You can also use computer software such
as Graphing Calculator by Pacific Tech
(www.pacifict.com) to perform many of these
functions, as well as apps for phones and
tablets, like Quick Graph (Colombiamug) or
Math-Studio (Pomegranate Apps). Similar
functionality is available using a web interface
at WolframAlpha.com.

© Dan Clegg

© Dan Clegg

Advances in technology continue to bring a wider variety of tools for
doing mathematics. Handheld calculators are becoming more powerful, as are software programs and Internet resources. In addition,
many mathematical applications have been released for smartphones
and tablets such as the iPad.
Some exercises in this text are marked with a graphing icon ; ,
which indicates that the use of some technology is required. Often this
means that we intend for a graphing device to be used in drawing the
graph of a function or equation. You might also need technology to
find the zeros of a graph or the points of intersection of two graphs.
In some cases we will use a calculating device to solve an equation or
evaluate a definite integral numerically. Many scientific and graphing
calculators have these features built in, such as the Texas Instruments
TI-84 or TI-Nspire CX. Similar calculators are made by Hewlett Packard, Casio, and Sharp.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

The CAS icon is reserved for problems in which the full resources of
a computer algebra system (CAS) are required. A CAS is capable of
doing mathematics (like solving equations, computing derivatives or
integrals) symbolically rather than just numerically.
Examples of well-established computer algebra systems are the computer software packages Maple and Mathematica. The WolframAlpha
website uses the Mathematica engine to provide CAS functionality
via the Web.
Many handheld graphing calculators have CAS capabilities, such
as the TI-89 and TI-Nspire CX CAS from Texas Instruments. Some
tablet and smartphone apps also provide these capabilities, such as the
previously mentioned MathStudio.

© Dan Clegg

© Dan Clegg

© Dan Clegg

In general, when we use the term “calculator” in this book, we mean
the use of any of the resources we have mentioned.

xxv
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Diagnostic Tests
Success in calculus depends to a large extent on knowledge of the mathematics that
precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas.
After taking each test you can check your answers against the given answers and, if
necessary, refresh your skills by referring to the review materials that are provided.

A
1. Evaluate each expression without using a calculator.
(a)
s23d4 (b)
234 (c)
324

SD

22

5 23
2
(d)
(e)
(f)
16 23y4
5 21
3
2.
Simplify each expression. Write your answer without negative exponents.



(a) s200 2 s32

s3a 3b 3 ds4ab 2 d 2
(b)

S

D

22

3x 3y2 y 3
(c)
x 2 y21y2
3. Expand and simplify.
sx 1 3ds4x 2 5d
(a) 3sx 1 6d 1 4s2x 2 5d (b)
(c) ssa 1 sb dssa 2 sb d (d)
s2x 1 3d2
(e) sx 1 2d3
4. Factor each expression.
(a)
4x 2 2 25 (b)
2x 2 1 5x 2 12
3
2
(c)
x 2 3x 2 4x 1 12 (d)
x 4 1 27x
3y2
1y2
21y2
(e)
3x 2 9x 1 6x (f)
x 3 y 2 4xy

5. Simplify the rational expression.
x 2 1 3x 1 2
2x 2 2 x 2 1
x13
(b)
?
2
x 2x22
x2 2 9
2x 1 1
y
x
2
2
x
x11
x
y
(c) 2
2
(d)
x 24
x12
1
1
2
y
x

(a)

xxvi
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Diagnostic Tests

xxvii

6. Rationalize the expression and simplify.
s10
s4 1 h 2 2
(a)
(b)
h
s5 2 2
7. Rewrite by completing the square.
(a)
x 2 1 x 1 1 (b)
2x 2 2 12x 1 11
8. Solve the equation. (Find only the real solutions.)
2x
2x 2 1
(a)
x 1 5 − 14 2 12 x (b)

x11
x
(c)
x 2 2 x 2 12 − 0 (d)
2x 2 1 4x 1 1 − 0

|

|

(e)
x 4 2 3x 2 1 2 − 0 (f)
3 x 2 4 − 10
(g)
2xs4 2 xd21y2 2 3 s4 2 x − 0

9.
Solve each inequality. Write your answer using interval notation.
(a)
24 , 5 2 3x < 17 (b)
x 2 , 2x 1 8
(c)
xsx 2 1dsx 1 2d . 0 (d)
x24 ,3
2x 2 3
(e)
<1
x11

|

|

10. State whether each equation is true or false.
(a)
s p 1 qd2 − p 2 1 q 2 (b)
sab − sa sb
1 1 TC
(c)
−11T
sa 2 1 b 2 − a 1 b (d)
C
1
1
1
1yx
1
− 2 (f)

(e)
x2y
x
y
ayx 2 byx
a2b

answers to diagnostic test a: algebra
1. (a) 81

1
(b) 281 (c)
81


(d) 25

1
(e) 94 (f)
8

1
6. (a) 5s2 1 2s10 (b)
s4 1 h 1 2

x
2. (a) 6s2 (b)
48a 5b7 (c)
9y7

7. (a) s x 1 12 d 1 34

3. (a) 11x 2 2 (b)
4x 2 1 7x 2 15
(c) a 2 b (d)
4x 2 1 12x 1 9
(e) x 3 1 6x 2 1 12x 1 8

8. (a) 6

4. (a) s2x 2 5ds2x 1 5d (b)
s2x 2 3dsx 1 4d
(c) sx 2 3dsx 2 2dsx 1 2d (d)
xsx 1 3dsx 2 2 3x 1 9d
21y2
(e) 3x sx 2 1dsx 2 2d (f)
xysx 2 2dsx 1 2d
x12
x21
(b)
x22
x23
1
(c)
(d)
2sx 1 yd
x22
5. (a)

2

(b) ­
2sx 2 3d2 2 7
(b)
1 (c)
23, 4

2 22
(d) 21 6 12 s2 (e)
61, 6s2 (f)
3, 3

(g) 12
5

9. (a) f24, 3d (b)
s22, 4d
(c) s22, 0d ø s1, `d (d)
s1, 7d
(e) s21, 4g
10. (a) False
(d) False

(b) True
(e) False

(c) False
(f) True

If you had difficulty with these problems, you may wish to consult the
Review of Algebra on the website www.stewartcalculus.com.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xxviii

Diagnostic Tests

B
1. Find an equation for the line that passes through the point s2, 25d and

(a) has slope 23

(b) is parallel to the x-axis

(c) is parallel to the y-axis

(d) is parallel to the line 2x 2 4y − 3

2. Find an equation for the circle that has center s21, 4d and passes through the point s3, 22d.

3. Find the center and radius of the circle with equation x 2 1 y 2 2 6x 1 10y 1 9 − 0.

4. Let As27, 4d and Bs5, 212d be points in the plane.

(a) Find the slope of the line that contains A and B.

(b) Find an equation of the line that passes through A and B. What are the intercepts?

(c) Find the midpoint of the segment AB.

(d) Find the length of the segment AB.

(e) Find an equation of the perpendicular bisector of AB.

(f) Find an equation of the circle for which AB is a diameter.

5. Sketch the region in the xy-plane defined by the equation or inequalities.

| |

| |

21 < y < 3 (b)
x , 4 and y , 2
(a)
(c)
y , 1 2 12 x (d)
y > x2 2 1
(e)
x 2 1 y 2 , 4 (f)
9x 2 1 16y 2 − 144

answers to diagnostic test b: analytic geometry
1. (a) y − 23x 1 1 (b)
y − 25
(c) x −

2 (d)
y − 12 x 2 6

5.
(a)

y

(b)

3

2. sx 1 1d2 1 s y 2 4d2 − 52

x

_1

4. (a) 234
(b) 4x 1 3y 1 16 − 0; x-intercept 24, y-intercept 2 16
3
(c) s21, 24d
(d) 20
(e) 3x 2 4y − 13
(f) sx 1 1d2 1 s y 1 4d2 − 100

(d)

_4

1

1
4x

0

(e)

y
2

_1

y

0

y=1- 2 x
2

x

_2

y

0

(c)

2

0

3. Center s3, 25d, radius 5

y

1

x

(f )
≈+¥=4

0

y=≈-1

2

x

y
3

0

4 x

If you had difficulty with these problems,
you may wish to consult
6et-dtba05a-f
5.20.06
the review of analytic geometry in Appendixes
B and C.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Diagnostic Tests

xxix

C
y

1. The graph of a function f is given at the left.

(a) State the value of f s21d.

(b) Estimate the value of f s2d.

(c) For what values of x is f sxd − 2?
1

(d) Estimate the values of x such that f sxd − 0.
0
x
1

(e) State the domain and range of f.


2. If f sxd − x 3, evaluate the difference quotient

f s2 1 hd 2 f s2d
and simplify your answer.
h


3. Find the domain of the function.
Figure For Problem 1

3
2x 1 1
x
s
(a)
f sxd − 2
(b)
tsxd − 2
(c)
hsxd − s4 2 x 1 sx 2 2 1
x 1x22
x 11


4. How are graphs of the functions obtained from the graph of f ?
(a)
y − 2f sxd (b)
y − 2 f sxd 2 1 (c)
y − f sx 2 3d 1 2

5. Without using a calculator, make a rough sketch of the graph.
(a)
y − x 3 (b)
y − sx 1 1d3 (c)
y − sx 2 2d3 1 3
2
(d)
y − 4 2 x (e)
y − sx (f)
y − 2 sx
(g)
y − 22 x (h)
y − 1 1 x21

H

1 2 x 2 if x < 0

6. Let f sxd −
2x 1 1 if x . 0


(a) Evaluate f s22d and f s1d.

(b) Sketch the graph of f.


7. If f sxd − x 2 1 2x 2 1 and tsxd − 2x 2 3, find each of the following functions.
(a)
f 8 t (b)
t 8 f (c)
t8t8t

answers to diagnostic test C: functions
1. (a) 22
(b) 2.8

(c) 23, 1 (d)
22.5, 0.3
(e) f23, 3g, f22, 3g

5. (a)

0

4. (a) Reflect about the x-axis
(b) Stretch vertically by a factor of 2, then shift 1 unit
downward
(c) Shift 3 units to the right and 2 units upward

(d)

(g)

1

x

_1

(e)

2

x

(2, 3)
x

0

1

x

1

x

x

0

(f)

y

0

(h)

y

y

0

1

y
1

0
_1

y

1

y
4

0

(c)

y

1

2. 12 1 6h 1 h 2
3. (a) s2`, 22d ø s22, 1d ø s1, `d
(b) s2`, `d
(c) s2`, 21g ø f1, 4g

(b)

y

1

x

0

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

x

xxx

Diagnostic Tests

6. (a) 23, 3 (b)

7. (a) s f 8 tdsxd − 4x 2 2 8x 1 2

y

(b) s t 8 f dsxd − 2x 2 1 4x 2 5

1
_1

0

x

(c) s t 8 t 8 tdsxd − 8x 2 21

If you had difficulty
with these problems, you should look at sections 1.1–1.3 of this book.
4c3DTCax06b
10/30/08

D
1. Convert from degrees to radians.
(a)
3008 (b)
2188

2. Convert from radians to degrees.
(a)
5 y6 (b)
2

3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle
of 308.

4. Find the exact values.
(a)
tans y3d (b)
sins7 y6d (c)
secs5 y3d


5. Express the lengths a and b in the figure in terms of .
24
a

6. If sin x − 13 and sec y − 54, where x and y lie between 0 and y2, evaluate sinsx 1 yd.
¨

7. Prove the identities.
b
2 tan x
(a)
tan sin 1 cos − sec (b) 2 − sin 2x
1 1 tan x
Figure For Problem 5


8. Find all values of x such that sin 2x − sin x and 0 < x < 2 .

9. Sketch the graph of the function y − 1 1 sin 2x without using a calculator.

answers to diagnostic test D: trigonometry
1. (a) 5 y3 (b)
2 y10
2. (a) 1508 (b)
3608y < 114.68
3. 2 cm

1
6. 15
s4 1 6 s2 d

8. 0, y3, , 5 y3, 2
y
2

9.

4. (a) s3 (b)
221 (c)
2
5. (a) 24 sin (b)
24 cos



0

π

x

4c3DTDax09

If you had difficulty with these problems, you should look
at Appendix D of this book.
10/30/08

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

A Preview of Calculus
By the time you finish this course, you will be able to calculate the length of the curve used to design the Gateway Arch
in St. Louis, determine where a pilot should start descent
for a smooth landing, compute the force on a baseball bat
when it strikes the ball, and measure the amount of light
sensed by the human eye as the pupil changes size.

calculus is fundamentally different from the mathematics that you have studied previously: calculus is less static and more dynamic. It is concerned with change and motion; it deals
with quantities that approach other quantities. For that reason it may be useful to have an overview
of the subject before beginning its intensive study. Here we give a glimpse of some of the main
ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety
of problems.

1
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

2

a preview of calculus

The Area Problem



The origins of calculus go back at least 2500 years to the ancient Greeks, who found
areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles.
It is a much more difficult problem to find the area of a curved figure. The Greek
method of exhaustion was to inscribe polygons in the figure and circumscribe polygons
about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.

A∞

A™





A=A¡+A™+A£+A¢+A∞

FIGURE 1





A∞







A¡™



FIGURE 2

Let An be the area of the inscribed polygon with n sides. As n increases, it appears that
An becomes closer and closer to the area of the circle. We say that the area of the circle
is the limit of the areas of the inscribed polygons, and we write

TEC  In the Preview Visual, you
can see how areas of inscribed and
circumscribed polygons approximate
the area of a circle.

y

A − lim An
nl`

The Greeks themselves did not use limits explicitly. However, by indirect reasoning,
Eudoxus (fifth century bc) used exhaustion to prove the familiar formula for the area of
a circle: A − r 2.
We will use a similar idea in Chapter 4 to find areas of regions of the type shown in
Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4),
let the width of the rectangles decrease, and then calculate A as the limit of these sums
of areas of rectangles.
y

y

(1, 1)

y

(1, 1)

(1, 1)

(1, 1)

y=≈
A
0

1

x

0

1
4

1
2

3
4

1

x

0

1

x

0

1
n

1

x

FIGURE 3

The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 4 for finding areas will also enable
us to compute the volume of a solid, the length of a curve, the force of water against a
dam, the mass and center of gravity of a rod, and the work done in pumping water out
of a tank.

The Tangent Problem
Consider the problem of trying to find an equation of the tangent line t to a curve with
equation y − f sxd at a given point P. (We will give a precise definition of a tangent line in

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.



a preview of calculus
y

Chapter 1. For now you can think of it as a line that touches the curve at P as in Figure 5.)
Since we know that the point P lies on the tangent line, we can find the equation of t if we
know its slope m. The problem is that we need two points to compute the slope and we
know only one point, P, on t. To get around the problem we first find an approximation
to m by taking a nearby point Q on the curve and computing the slope mPQ of the secant
line PQ. From Figure 6 we see that

t
y=ƒ
P

0

x

FIGURE 5  
The tangent line at P
y

1

mPQ −

m − lim mPQ
Q lP

Q { x, ƒ}

ƒ-f(a)

P { a, f(a)}

and we say that m is the limit of mPQ as Q approaches P along the curve. Because x
approaches a as Q approaches P, we could also use Equation 1 to write

x-a

a

x

x

FIGURE 6  
The secant line at PQ
y

f sxd 2 f sad

x2a

Now imagine that Q moves along the curve toward P as in Figure 7. You can see that
the secant line rotates and approaches the tangent line as its limiting position. This means
that the slope mPQ of the secant line becomes closer and closer to the slope m of the tangent line. We write

t

0

3

t
Q
P

0

FIGURE 7  
Secant lines approaching the
tangent line

x

2

f sxd 2 f sad

x2a

m − lim

xla

Specific examples of this procedure will be given in Chapter 1.
The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main
ideas behind differential calculus are due to the French mathematician Pierre Fermat (1601–1665) and were developed by the English mathematicians John Wallis
(1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the German mathematician Gottfried Leibniz (1646–1716).
The two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems
in a sense that will be described in Chapter 4.

Velocity
When we look at the speedometer of a car and read that the car is traveling at 48 miyh,
what does that information indicate to us? We know that if the velocity remains constant,
then after an hour we will have traveled 48 mi. But if the velocity of the car varies, what
does it mean to say that the velocity at a given instant is 48 miyh?
In order to analyze this question, let’s examine the motion of a car that travels along a
straight road and assume that we can measure the distance traveled by the car (in feet) at
l-second intervals as in the following chart:
t − Time elapsed ssd

0

1

2

3

4

5

d − Distance sftd

0

2

9

24

42

71

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4

a preview of calculus

As a first step toward finding the velocity after 2 seconds have elapsed, we find the average velocity during the time interval 2 < t < 4:
change in position
time elapsed

average velocity −

42 2 9
422



− 16.5 ftys
Similarly, the average velocity in the time interval 2 < t < 3 is
average velocity −

24 2 9
− 15 ftys
322

We have the feeling that the velocity at the instant t − 2 can’t be much different from the
average velocity during a short time interval starting at t − 2. So let’s imagine that the distance traveled has been measured at 0.l-second time intervals as in the following chart:
t

2.0

2.1

2.2

2.3

2.4

2.5

d

9.00

10.02

11.16

12.45

13.96

15.80

Then we can compute, for instance, the average velocity over the time interval f2, 2.5g:
average velocity −

15.80 2 9.00
− 13.6 ftys
2.5 2 2

The results of such calculations are shown in the following chart:
Time interval

f2, 3g

f2, 2.5g

f2, 2.4g

f2, 2.3g

f2, 2.2g

f2, 2.1g

Average velocity sftysd

15.0

13.6

12.4

11.5

10.8

10.2

The average velocities over successively smaller intervals appear to be getting closer to
a number near 10, and so we expect that the velocity at exactly t − 2 is about 10 ftys. In
Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value
of the average velocities over smaller and smaller time intervals.
In Figure 8 we show a graphical representation of the motion of the car by plotting the
distance traveled as a function of time. If we write d − f std, then f std is the number of
feet traveled after t seconds. The average velocity in the time interval f2, tg is

d

Q { t, f(t)}

average velocity −

which is the same as the slope of the secant line PQ in Figure 8. The velocity v when
t − 2 is the limiting value of this average velocity as t approaches 2; that is,

20
10
0

change in position
f std 2 f s2d

time elapsed
t22

P { 2, f(2)}
1

2

FIGURE 8

3

4

5

t

v − lim

tl2

f std 2 f s2d
t22

and we recognize from Equation 2 that this is the same as the slope of the tangent line
to the curve at P.

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a preview of calculus

5

Thus, when we solve the tangent problem in differential calculus, we are also solving
problems concerning velocities. The same techniques also enable us to solve problems
involving rates of change in all of the natural and social sciences.

The Limit of a Sequence
In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now
known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning
space and time that were held in his day. Zeno’s second paradox concerns a race between
the Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as follows, that Achilles could never pass the tortoise: Suppose that Achil­les starts at position
a 1 and the tortoise starts at position t1. (See Figure 9.) When Achilles reaches the point
a 2 − t1, the tortoise is farther ahead at position t2. When Achilles reaches a 3 − t2, the tortoise is at t3. This process continues indefinitely and so it appears that the tortoise will
always be ahead! But this defies common sense.

Achilles

FIGURE 9



tortoise

a™





a∞

...



t™





...

One way of explaining this paradox is with the idea of a sequence. The successive positions of Achilles sa 1, a 2 , a 3 , . . .d or the successive positions of the tortoise st1, t2 , t3 , . . .d
form what is known as a sequence.
In general, a sequence ha nj is a set of numbers written in a definite order. For instance,
the sequence

h1, 12 , 13 , 14 , 15 , . . . j
can be described by giving the following formula for the nth term:
an −
a¢ a £

a™

0

We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the
terms of the sequence a n − 1yn are becoming closer and closer to 0 as n increases. In
fact, we can find terms as small as we please by making n large enough. We say that the
limit of the sequence is 0, and we indicate this by writing


1

(a)
1

lim

nl`

1 2 3 4 5 6 7 8

(b)

FIGURE 10

1
n

n

1
−0
n

In general, the notation
lim a n − L

nl`

is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large.

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6

a preview of calculus

The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number. For instance, if
a 1 − 3.1
a 2 − 3.14
a 3 − 3.141
a 4 − 3.1415
a 5 − 3.14159
a 6 − 3.141592
a 7 − 3.1415926
f


then

lim a n −

nl`

The terms in this sequence are rational approximations to .
Let’s return to Zeno’s paradox. The successive positions of Achilles and the tortoise
form sequences ha nj and htn j, where a n , tn for all n. It can be shown that both sequences
have the same limit:
lim a n − p − lim tn

nl`

nl`

It is precisely at this point p that Achilles overtakes the tortoise.

The Sum of a Series
Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man
standing in a room cannot walk to the wall. In order to do so, he would first have to
go half the distance, then half the remaining distance, and then again half of what still
remains. This process can always be continued and can never be ended.” (See Figure 11.)

1
2

FIGURE 11

1
4

1
8

1
16

Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances
as follows:
3

1−

1
1
1
1
1
1 1 1
1 ∙∙∙ 1 n 1 ∙∙∙
2
4
8
16
2

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a preview of calculus

7

Zeno was arguing that it doesn’t make sense to add infinitely many numbers together.
But there are other situations in which we implicitly use infinite sums. For instance, in
decimal notation, the symbol 0.3 − 0.3333 . . . means
3
3
3
3
1
1
1
1 ∙∙∙
10
100
1000
10,000
and so, in some sense, it must be true that
3
3
3
3
1
1
1
1
1 ∙∙∙ −
10
100
1000
10,000
3
More generally, if dn denotes the nth digit in the decimal representation of a number, then
0.d1 d2 d3 d4 . . . −

d1
d2
d3
dn
1 2 1 3 1 ∙∙∙ 1 n 1 ∙∙∙
10
10
10
10

Therefore some infinite sums, or infinite series as they are called, have a meaning. But
we must define carefully what the sum of an infinite series is.
Returning to the series in Equation 3, we denote by sn the sum of the first n terms of
the series. Thus
s1 − 12 − 0.5
s2 − 12 1 14 − 0.75
s3 − 12 1 14 1 18 − 0.875
1
s4 − 12 1 14 1 18 1 16
− 0.9375
1
1
s5 − 12 1 14 1 18 1 16
1 32
− 0.96875
1
1
1
s6 − 12 1 14 1 18 1 16
1 32
1 64
− 0.984375
1
1
1
1
s7 − 12 1 14 1 18 1 16
1 32
1 64
1 128
− 0.9921875



f
1
s10 − 12 1 14 1 ∙ ∙ ∙ 1 1024
< 0.99902344



f
s16 −

1
1
1
1 1 ∙ ∙ ∙ 1 16 < 0.99998474
2
4
2

Observe that as we add more and more terms, the partial sums become closer and closer
to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently
many terms of the series), we can make the partial sum sn as close as we please to the number 1. It therefore seems reasonable to say that the sum of the infinite series is 1 and to
write
1
1
1
1
1 1 1 ∙∙∙ 1 n 1 ∙∙∙ − 1
2
4
8
2
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8

a preview of calculus

In other words, the reason the sum of the series is 1 is that
lim sn − 1

nl`

In Chapter 11 we will discuss these ideas further. We will then use Newton’s idea of
combining infinite series with differential and integral calculus.

Summary
We have seen that the concept of a limit arises in trying to find the area of a region, the
slope of a tangent to a curve, the velocity of a car, or the sum of an infinite series. In
each case the common theme is the calculation of a quantity as the limit of other, easily
calculated quantities. It is this basic idea of a limit that sets calculus apart from other
areas of mathematics. In fact, we could define calculus as the part of mathematics that
deals with limits.
After Sir Isaac Newton invented his version of calculus, he used it to explain the
motion of the planets around the sun. Today calculus is used in calculating the orbits of
satellites and spacecraft, in predicting population sizes, in estimating how fast oil prices
rise or fall, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas. We will explore
some of these uses of calculus in this book.
In order to convey a sense of the power of the subject, we end this preview with a list
of some of the questions that you will be able to answer using calculus:
rays from sun

138°
rays from sun

observer

42°



1. How can we explain the fact, illustrated in Figure 12, that the angle of elevation
from an observer up to the highest point in a rainbow is 42°? (See page 213.)



2. How can we explain the shapes of cans on supermarket shelves? (See page 270.)



3. Where is the best place to sit in a movie theater? (See page 483.)



4. How can we design a roller coaster for a smooth ride? (See page 144.)



5. How far away from an airport should a pilot start descent? (See page 161.)



6. How can we fit curves together to design shapes to represent letters on a laser
printer? (See page 697.)



7. How can we estimate the number of workers that were needed to build the Great
Pyramid of Khufu in ancient Egypt? (See page 388.)



8. Where should an infielder position himself to catch a baseball thrown by an
outfielder and relay it to home plate? (See page 392.)



9. Does a ball thrown upward take longer to reach its maximum height or to fall
back to its original height? (See page 649.)

FIGURE 12

10. How can we explain the fact that planets and satellites move in elliptical orbits?
(See page 916.)
11. How can we distribute water flow among turbines at a hydroelectric station so
as to maximize the total energy production? (See page 1020.)
12. If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which
of them reaches the bottom first? (See page 1092.)

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1
Often a graph is the best
way to represent a function
because it conveys so much
information at a glance.
Shown is a graph of the
vertical ground acceleration
created by the 2011
earthquake near Tohoku,
Japan. The earthquake
had a magnitude of 9.0 on
the Richter scale and was
so powerful that it moved
northern Japan 8 feet closer
to North America.

Functions and Limits

Pictura Collectus/Alamy

(cm/s@)
2000
1000
0

time

_1000
_2000
0

50

100

150

200

Seismological Society of America

The fundamental objects that we deal with in calculus are functions. We stress that a
function can be represented in different ways: by an equation, in a table, by a graph, or in words.
We look at the main types of functions that occur in calculus and describe the process of using
these functions as mathematical models of real-world phenomena.

In A Preview of Calculus (page 1) we saw how the idea of a limit underlies the various
branches of calculus. It is therefore appropriate to begin our study of calculus by investigating
limits of functions and their properties.

9
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10

Chapter 1  Functions and Limits

Year

Population
(millions)

1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010

1650
1750
1860
2070
2300
2560
3040
3710
4450
5280
6080
6870

Functions arise whenever one quantity depends on another. Consider the following four
situations.
A. The area A of a circle depends on the radius r of the circle. The rule that connects r
and A is given by the equation A − r 2. With each positive number r there is associated one value of A, and we say that A is a function of r.
B. The human population of the world P depends on the time t. The table gives estimates of the world population Pstd at time t, for certain years. For instance,
Ps1950d < 2,560,000,000
But for each value of the time t there is a corresponding value of P, and we say that
P is a function of t.
C. The cost C of mailing an envelope depends on its weight w. Although there is no
simple formula that connects w and C, the post office has a rule for determining C
when w is known.
D. The vertical acceleration a of the ground as measured by a seismograph during an
earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by
seismic activity during the Northridge earthquake that shook Los Angeles in 1994.
For a given value of t, the graph provides a corresponding value of a.
a

{cm/s@}

100
50

5

FIGURE 1
Vertical ground acceleration
during the Northridge earthquake

10

15

20

25

30

t (seconds)

_50
Calif. Dept. of Mines and Geology

Each of these examples describes a rule whereby, given a number (r, t, w, or t), another
number (A, P, C, or a) is assigned. In each case we say that the second number is a
function of the first number.
A function f is a rule that assigns to each element x in a set D exactly one
element, called f sxd, in a set E.
We usually consider functions for which the sets D and E are sets of real numbers.
The set D is called the domain of the function. The number f sxd is the value of f at x
and is read “ f of x.” The range of f is the set of all possible values of f sxd as x varies
throughout the domain. A symbol that represents an arbitrary number in the domain of a
function f is called an independent variable. A symbol that represents a number in the
range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable.

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11

Section  1.1   Four Ways to Represent a Function

x
(input)

f

ƒ
(output)

FIGURE 2

Machine diagram for a function f

x

ƒ
a

f(a)

f

D

It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of
the function f, then when x enters the machine, it’s accepted as an input and the machine
produces an output f sxd according to the rule of the function. Thus we can think of the
domain as the set of all possible inputs and the range as the set of all possible outputs.
The preprogrammed functions in a calculator are good examples of a function as a
machine. For example, the square root key on your calculator computes such a function.
You press the key labeled s (or s x ) and enter the input x. If x , 0, then x is not in the
domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x > 0, then an approximation to s x will appear in the display. Thus the
s x key on your calculator is not quite the same as the exact mathematical function f
defined by f sxd − s x .
Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow
connects an element of D to an element of E. The arrow indicates that f sxd is associated
with x, f sad is associated with a, and so on.
The most common method for visualizing a function is its graph. If f is a function
with domain D, then its graph is the set of ordered pairs

|

hsx, f sxdd x [ Dj

E

(Notice that these are input-output pairs.) In other words, the graph of f consists of all
points sx, yd in the coordinate plane such that y − f sxd and x is in the domain of f.
The graph of a function f gives us a useful picture of the behavior or “life history”
of a function. Since the y-coordinate of any point sx, yd on the graph is y − f sxd, we can
read the value of f sxd from the graph as being the height of the graph above the point x
(see Figure 4). The graph of f also allows us to picture the domain of f on the x-axis and
its range on the y-axis as in Figure 5.

FIGURE 3

Arrow diagram for f

y

y

{ x, ƒ}
range

ƒ
f (2)

f (1)
0

1

2

x

x

FIGURE 4
y

0

domain

x

FIGURE 5

Example 1   The graph of a function f is shown in Figure 6.
(a)  Find the values of f s1d and f s5d.
(b)  What are the domain and range of f ?

1
0

y ƒ(x)

Solution

1

x

FIGURE 6
The notation for intervals is given in
Appendix A.

(a)  We see from Figure 6 that the point s1, 3d lies on the graph of f, so the value of f
at 1 is f s1d − 3. (In other words, the point on the graph that lies above x − 1 is 3 units
above the x-axis.)
When x − 5, the graph lies about 0.7 units below the x-axis, so we estimate that
f s5d < 20.7.
(b)  We see that f sxd is defined when 0 < x < 7, so the domain of f is the closed interval f0, 7g. Notice that f takes on all values from 22 to 4, so the range of f is


|

hy 22 < y < 4j − f22, 4g

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12

Chapter 1  Functions and Limits
y

Example 2   Sketch the graph and find the domain and range of each function.
(a)  fsxd − 2x 2 1 (b) 
tsxd − x 2
Solution

y=2x-1
0
-1

x

1
2

FIGURE 7
y

(2, 4)

y=≈
(_1, 1)

(a)  The equation of the graph is y − 2x 2 1, and we recognize this as being the equation of a line with slope 2 and y-intercept 21. (Recall the slope-intercept form of the
equation of a line: y − mx 1 b. See Appendix B.) This enables us to sketch a portion
of the graph of f in Figure 7. The expression 2x 2 1 is defined for all real numbers, so
the domain of f is the set of all real numbers, which we denote by R. The graph shows
that the range is also R.
(b) Since ts2d − 2 2 − 4 and ts21d − s21d2 − 1, we could plot the points s2, 4d and
s21, 1d, together with a few other points on the graph, and join them to produce the
graph (Figure 8). The equation of the graph is y − x 2, which represents a parabola (see
Appendix C). The domain of t is R. The range of t consists of all values of tsxd, that is,
all numbers of the form x 2. But x 2 > 0 for all numbers x and any positive number y is a
square. So the range of t is hy y > 0j − f0, `d. This can also be seen from Figure 8. ■

|

1
0

1

x

Example 3  If f sxd − 2x 2 2 5x 1 1 and h ± 0, evaluate

f sa 1 hd 2 f sad
.
h

Solution  We first evaluate f sa 1 hd by replacing x by a 1 h in the expression for f sxd:

FIGURE 8

f sa 1 hd − 2sa 1 hd2 2 5sa 1 hd 1 1
− 2sa 2 1 2ah 1 h 2 d 2 5sa 1 hd 1 1
− 2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1
The expression

Then we substitute into the given expression and simplify:
f sa 1 hd 2 f sad
s2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1d 2 s2a 2 2 5a 1 1d

h
h

f sa 1 hd 2 f sad
h
in Example 3 is called a difference
quotient and occurs frequently in
calculus. As we will see in Chapter
2, it represents the average rate of
change of f sxd between x − a and
x − a 1 h.



2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1 2 2a 2 1 5a 2 1
h



4ah 1 2h 2 2 5h
− 4a 1 2h 2 5
h



Representations of Functions
There are four possible ways to represent a function:
  verbally
●  numerically
●  visually
●  algebraically  


(by a description in words)
(by a table of values)
(by a graph)
(by an explicit formula)

If a single function can be represented in all four ways, it’s often useful to go from one
representation to another to gain additional insight into the function. (In Example 2, for
instance, we started with algebraic formulas and then obtained the graphs.) But certain
functions are described more naturally by one method than by another. With this in mind,
let’s reexamine the four situations that we considered at the beginning of this section.

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13

Section  1.1   Four Ways to Represent a Function

A. The most useful representation of the area of a circle as a function of its radius is
probably the algebraic formula Asrd − r 2, though it is possible to compile a table
of values or to sketch a graph (half a parabola). Because a circle has to have a positive radius, the domain is hr r . 0j − s0, `d, and the range is also s0, `d.
B. We are given a description of the function in words: Pstd is the human population of
the world at time t. Let’s measure t so that t − 0 corresponds to the year 1900. The
table of values of world population provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It
too is a useful representation; the graph allows us to absorb all the data at once. What
about a formula? Of course, it’s impossible to devise an explicit formula that gives
the exact human population Pstd at any time t. But it is possible to find an expression
for a function that approximates Pstd. In fact, using methods explained in Section
1.2, we obtain the approximation

|

t
(years
since 1900)

Population
(millions)

0
10
20
30
40
50
60
70
80
90
100
110

1650
1750
1860
2070
2300
2560
3040
3710
4450
5280
6080
6870

Pstd < f std − s1.43653 3 10 9 d s1.01395d t
Figure 10 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit
formula that approximates the behavior of our given function. We will see, however,
that the ideas of calculus can be applied to a table of values; an explicit formula is
not necessary.

P

P

5x10'

0

5x10'

20

40
60
80
Years since 1900

FIGURE 9

100

120

t

0

20

40
60
80
Years since 1900

100

120

t

FIGURE 10

A function defined by a table of
values is called a tabular function.
w (ounces)

Cswd (dollars)

0,w<1
1,w<2
2,w<3
3,w<4
4,w<5




0.98
1.19
1.40
1.61
1.82




The function P is typical of the functions that arise whenever we attempt to apply
calculus to the real world. We start with a verbal description of a function. Then we
may be able to construct a table of values of the function, perhaps from instrument
readings in a scientific experiment. Even though we don’t have complete knowledge
of the values of the function, we will see throughout the book that it is still possible
to perform the operations of calculus on such a function.
C. Again the function is described in words: Let Cswd be the cost of mailing a large envelope with weight w. The rule that the US Postal Service used as of 2015 is as follows:
The cost is 98 cents for up to 1 oz, plus 21 cents for each additional ounce (or less)
up to 13 oz. The table of values shown in the margin is the most convenient representation for this function, though it is possible to sketch a graph (see Example 10).
D. The graph shown in Figure 1 is the most natural representation of the vertical acceleration function astd. It’s true that a table of values could be compiled, and it is
even possible to devise an approximate formula. But everything a geologist needs to

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14

Chapter 1  Functions and Limits

know— amplitudes and patterns — can be seen easily from the graph. (The same is
true for the patterns seen in electrocardiograms of heart patients and polygraphs for
lie-detection.)
In the next example we sketch the graph of a function that is defined verbally.
T

Example 4   When you turn on a hot-water faucet, the temperature T of the water
depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.

0

SOLUTION  The initial temperature of the running water is close to room temperature
because the water has been sitting in the pipes. When the water from the hot-water tank
starts flowing from the faucet, T increases quickly. In the next phase, T is constant at
the tempera­ture of the heated water in the tank. When the tank is drained, T decreases
to the temperature of the water supply. This enables us to make the rough sketch of T
as a function of t in Figure 11.


t

FIGURE 11

In the following example we start with a verbal description of a function in a physical
situation and obtain an explicit algebraic formula. The ability to do this is a useful skill
in solving calculus problems that ask for the maximum or minimum values of quantities.

Example 5   A rectangular storage container with an open top has a volume of 10 m3.
The length of its base is twice its width. Material for the base costs $10 per square
meter; material for the sides costs $6 per square meter. Express the cost of materials as
a function of the width of the base.

h
w

SOLUTION  We draw a diagram as in Figure 12 and introduce notation by letting w and
2w be the width and length of the base, respectively, and h be the height.
The area of the base is s2wdw − 2w 2, so the cost, in dollars, of the material for the
base is 10s2w 2 d. Two of the sides have area wh and the other two have area 2wh, so the
cost of the material for the sides is 6f2swhd 1 2s2whdg. The total cost is therefore

C − 10s2w 2 d 1 6f2swhd 1 2s2whdg − 20 w 2 1 36 wh

2w

FIGURE 12

To express C as a function of w alone, we need to eliminate h and we do so by using
the fact that the volume is 10 m3. Thus
w s2wdh − 10

which gives

PS   In setting up applied functions as
in Example 5, it may be useful to review
the principles of problem solving as
discussed on page 98, particularly
Step 1: Understand the Problem.

10
5
2 −
2w
w2

h−

Substituting this into the expression for C, we have

S D

C − 20w 2 1 36w

5

w

2

− 20w 2 1

180
w

Therefore the equation
Cswd − 20w 2 1

180
w

    w . 0

expresses C as a function of w.

Example 6   Find the domain of each function.
(a)  f sxd − sx 1 2          (b) tsxd −



1
x2 2 x

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Section  1.1   Four Ways to Represent a Function

Domain Convention
If a function is given by a formula
and the domain is not stated explicitly, the convention is that the domain
is the set of all numbers for which
the formula makes sense and defines
a real number.

15

SOLUTION

(a)  Because the square root of a negative number is not defined (as a real number),
the domain of f consists of all values of x such that x 1 2 > 0. This is equivalent to
x > 22, so the domain is the interval f22, `d.
(b) Since
1
1
tsxd − 2

x 2x
xsx 2 1d
and division by 0 is not allowed, we see that tsxd is not defined when x − 0 or x − 1.
Thus the domain of t is

|

hx x ± 0, x ± 1j
which could also be written in interval notation as
s2`, 0d ø s0, 1d ø s1, `d


y

The graph of a function is a curve in the xy-plane. But the question arises: which
curves in the xy-plane are graphs of functions? This is answered by the following test.

x=a
(a, b)

0

The Vertical Line Test  A curve in the xy-plane is the graph of a function of x if
and only if no vertical line intersects the curve more than once.
x

a

(a) This curve represents a function.
y
(a, c)

x=a

(a, b)
0



a

x

(b) This curve doesn’t represent
a function.

FIGURE 13

The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each
vertical line x − a intersects a curve only once, at sa, bd, then exactly one function value
is defined by f sad − b. But if a line x − a intersects the curve twice, at sa, bd and sa, cd,
then the curve can’t represent a function because a function can’t assign two different
values to a.
For example, the parabola x − y 2 2 2 shown in Figure 14(a) is not the graph of a
function of x because, as you can see, there are vertical lines that intersect the parabola
twice. The parabola, however, does contain the graphs of two functions of x. Notice
that the equation x − y 2 2 2 implies y 2 − x 1 2, so y − 6sx 1 2 . Thus the upper
and lower halves of the parabola are the graphs of the functions f sxd − s x 1 2 [from
Example 6(a)] and tsxd − 2s x 1 2 . [See Figures 14(b) and (c).]
We observe that if we reverse the roles of x and y, then the equation x − hsyd − y 2 2 2
does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h.
y

(_2, 0)

FIGURE 14

0

y

x

_2 0

(a) x=¥-2

(b) y=œ„„„„
x+2

y

x

_2

0

x

(c) y=_ œ„„„„
x+2

Piecewise Defined Functions
The functions in the following four examples are defined by different formulas in dif­
ferent parts of their domains. Such functions are called piecewise defined functions.

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16

Chapter 1  Functions and Limits

Example 7  A function f is defined by
f sxd −

H

1 2 x if x < 21
x2
if x . 21

Evaluate f s22d, f s21d, and f s0d and sketch the graph.
Solution  Remember that a function is a rule. For this particular function the rule is
the following: First look at the value of the input x. If it happens that x < 21, then the
value of f sxd is 1 2 x. On the other hand, if x . 21, then the value of f sxd is x 2.

Since 22 < 21, we have f s22d − 1 2 s22d − 3.
Since 21 < 21, we have f s21d − 1 2 s21d − 2.

y

Since 0 . 21, we have f s0d − 0 2 − 0.

1

_1

0

1

x

FIGURE 15

How do we draw the graph of f ? We observe that if x < 21, then f sxd − 1 2 x,
so the part of the graph of f that lies to the left of the vertical line x − 21 must coincide with the line y − 1 2 x, which has slope 21 and y-intercept 1. If x . 21,
then f sxd − x 2, so the part of the graph of f that lies to the right of the line x − 21
must coincide with the graph of y − x 2, which is a parabola. This enables us to sketch
the graph in Figure 15. The solid dot indicates that the point s21, 2d is included on the
graph; the open dot indicates that the point s21, 1d is excluded from the graph.

The next example of a piecewise defined function is the absolute value function.
Recall that the absolute value of a number a, denoted by a , is the distance from a to 0
on the real number line. Distances are always positive or 0, so we have

| |

For a more extensive review of
absolute values, see Appendix A.

| a | > 0    for every number a
For example,

| 3 | − 3   | 23 | − 3   | 0 | − 0   | s2 2 1 | − s2 2 1   | 3 2 | − 2 3
In general, we have

| a | − a   if
| a | − 2a  if

a>0
a,0

(Remember that if a is negative, then 2a is positive.)

Example 8
  Sketch the graph of the absolute value function f sxd − | x |.

y

SOLUTION  From the preceding discussion we know that

y=| x |

|x| −
0

FIGURE 16

x

H

x
if x > 0
2x if x , 0

Using the same method as in Example 7, we see that the graph of f coincides with the
line y − x to the right of the y-axis and coincides with the line y − 2x to the left of the
y-axis (see Figure 16).


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Section  1.1   Four Ways to Represent a Function

17

y

Example 9   Find a formula for the function f graphed in Figure 17.

1

SOLUTION  The line through s0, 0d and s1, 1d has slope m − 1 and y-intercept b − 0,
so its equation is y − x. Thus, for the part of the graph of f that joins s0, 0d to s1, 1d,
we have

0

x

1

f sxd − x    if 0 < x < 1

FIGURE 17

The line through s1, 1d and s2, 0d has slope m − 21, so its point-slope form is

Point-slope form of the equation of
a line:
y 2 y1 − msx 2 x 1 d

y 2 0 − s21dsx 2 2d    or    y − 2 2 x
So we have

See Appendix B.

f sxd − 2 2 x    if 1 , x < 2

We also see that the graph of f coincides with the x-axis for x . 2. Putting this information together, we have the following three-piece formula for f :

H

x
if 0 < x < 1
f sxd − 2 2 x if 1 , x < 2
0
if x . 2



Example 10   In Example C at the beginning of this section we considered the cost
Cswd of mailing a large envelope with weight w. In effect, this is a piecewise defined
function because, from the table of values on page 13, we have

C
1.50
1.00

Cswd −
0.50





0

1

2

3

4

5

figure 18

w

0.98
1.19
1.40
1.61




if
if
if
if

0,w<1
1,w<2
2,w<3
3,w<4


The
graph is shown in Figure 18. You can see why functions similar to this one are
called step functions—they jump from one value to the next. Such functions will be

studied in Chapter 2.

Symmetry
If a function f satisfies f s2xd − f sxd for every number x in its domain, then f is called
an even function. For instance, the function f sxd − x 2 is even because

y

f(_x)

f s2xd − s2xd2 − x 2 − f sxd

ƒ
_x

0

FIGURE 19  
An even function

x

x

The geometric significance of an even function is that its graph is symmetric with respect
to the y-axis (see Figure 19). This means that if we have plotted the graph of f for x > 0,
we obtain the entire graph simply by reflecting this portion about the y-axis.
If f satisfies f s2xd − 2f sxd for every number x in its domain, then f is called an odd
function. For example, the function f sxd − x 3 is odd because
f s2xd − s2xd3 − 2x 3 − 2f sxd

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.


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