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REFERENCE PAGES

Cut here and keep for reference

ALGEBRA

G E O M E T RY

ARITHMETIC OPERATIONS

GEOMETRIC FORMULAS

a
c
ad ϩ bc
ϩ ෇
b
d
bd
a
b
a
d
ad
෇ ϫ ෇
c
b
c
bc
d

a͑b ϩ c͒ ෇ ab ϩ ac
a
c
aϩc
෇ ϩ
b
b
b

Formulas for area A, circumference C, and volume V:
Triangle

Circle

Sector of Circle

A ෇ 12 bh
෇ 12 ab sin ␪

A ෇ ␲r 2
C ෇ 2␲ r

A ෇ 12 r 2␪
s ෇ r ␪ ͑␪ in radians͒

a

EXPONENTS AND RADICALS
xm
෇ x mϪn

xn
1
xϪn ෇ n
x

x m x n ෇ x mϩn
͑x ͒ ෇ x
m n

mn

ͩͪ

n

x
y

͑xy͒n ෇ x n y n

෇

ͱ
n

n
n
n
xy ෇ s
xs

y
s

r

s

¨

b

r

xn
yn

Sphere
V ෇ 43 ␲ r 3
A ෇ 4␲ r 2

n
n
x m͞n ෇ s
x m ෇ (s
x )m

n
x 1͞n ෇ s
x

r

h

¨

n
x
x
s
෇ n
y
sy

Cylinder
V ෇ ␲ r 2h

Cone
V ෇ 13 ␲ r 2h

r
r

h

h

FACTORING SPECIAL POLYNOMIALS

r

x 2 Ϫ y 2 ෇ ͑x ϩ y͒͑x Ϫ y͒
x 3 ϩ y 3 ෇ ͑x ϩ y͒͑x 2 Ϫ xy ϩ y 2͒
x 3 Ϫ y 3 ෇ ͑x Ϫ y͒͑x 2 ϩ xy ϩ y 2͒

DISTANCE AND MIDPOINT FORMULAS
BINOMIAL THEOREM

Distance between P1͑x1, y1͒ and P2͑x 2, y2͒:

͑x ϩ y͒2 ෇ x 2 ϩ 2xy ϩ y 2

͑x Ϫ y͒2 ෇ x 2 Ϫ 2xy ϩ y 2

d ෇ s͑x 2 Ϫ x1͒2 ϩ ͑ y2 Ϫ y1͒2

͑x ϩ y͒3 ෇ x 3 ϩ 3x 2 y ϩ 3xy 2 ϩ y 3
͑x Ϫ y͒3 ෇ x 3 Ϫ 3x 2 y ϩ 3xy 2 Ϫ y 3
͑x ϩ y͒n ෇ x n ϩ nx nϪ1y ϩ
ϩ иии ϩ

ͩͪ

n͑n Ϫ 1͒ nϪ2 2
x y
2

Midpoint of P1 P2 :

ͩͪ

n nϪk k
x y ϩ и и и ϩ nxy nϪ1 ϩ y n
k

ͩ

x1 ϩ x 2 y1 ϩ y2
,
2
2

LINES

n
n͑n Ϫ 1͒ и и и ͑n Ϫ k ϩ 1͒
where
෇
k
1 ؒ 2 ؒ 3 ؒ иии ؒ k

Slope of line through P1͑x1, y1͒ and P2͑x 2, y2͒:
m෇

QUADRATIC FORMULA
If ax 2 ϩ bx ϩ c ෇ 0, then x ෇

ͪ

Ϫb Ϯ sb 2 Ϫ 4ac

.
2a

y2 Ϫ y1
x 2 Ϫ x1

Point-slope equation of line through P1͑x1, y1͒ with slope m:
y Ϫ y1 ෇ m͑x Ϫ x1͒

INEQUALITIES AND ABSOLUTE VALUE
If a Ͻ b and b Ͻ c, then a Ͻ c.

Slope-intercept equation of line with slope m and y-intercept b:

If a Ͻ b, then a ϩ c Ͻ b ϩ c.

y ෇ mx ϩ b

If a Ͻ b and c Ͼ 0, then ca Ͻ cb.
If a Ͻ b and c Ͻ 0, then ca Ͼ cb.
If a Ͼ 0, then

ԽxԽ ෇ a
ԽxԽ Ͻ a
ԽxԽ Ͼ a

means

x ෇ a or

CIRCLES

x ෇ Ϫa

Equation of the circle with center ͑h, k͒ and radius r:

means Ϫa Ͻ x Ͻ a
means

x Ͼ a or

͑x Ϫ h͒2 ϩ ͑ y Ϫ k͒2 ෇ r 2

x Ͻ Ϫa
1


REFERENCE PAGES
T R I G O N O M E T RY
ANGLE MEASUREMENT

FUNDAMENTAL IDENTITIES

␲ radians ෇ 180Њ
1Њ ෇

␲
rad
180

180Њ
␲

1 rad ෇

s

r
r

͑␪ in radians͒

RIGHT ANGLE TRIGONOMETRY

cos ␪ ෇
tan ␪ ෇

hyp
csc ␪ ෇
opp

adj
hyp

sec ␪ ෇

opp
adj

cot ␪ ෇

1
sin ␪

sec ␪ ෇

1
cos ␪

tan ␪ ෇

sin ␪
cos ␪

cot ␪ ෇

cos ␪
sin ␪

cot ␪ ෇

1
tan ␪

sin 2␪ ϩ cos 2␪ ෇ 1

¨

s ෇ r␪

opp
sin ␪ ෇
hyp

csc ␪ ෇

hyp

hyp
adj

opp

¨
adj

1 ϩ tan 2␪ ෇ sec 2␪

1 ϩ cot 2␪ ෇ csc 2␪

sin͑Ϫ␪͒ ෇ Ϫsin ␪

cos͑Ϫ␪͒ ෇ cos ␪

tan͑Ϫ␪͒ ෇ Ϫtan ␪

sin

␲
Ϫ ␪ ෇ cos ␪

2

tan

␲
Ϫ ␪ ෇ cot ␪
2

ͩ ͪ
ͩ ͪ

ͩ ͪ

adj
opp

cos

␲
Ϫ ␪ ෇ sin ␪
2

TRIGONOMETRIC FUNCTIONS
sin ␪ ෇

y
r

csc ␪ ෇

cos ␪ ෇

x
r

sec ␪ ෇

r
x

tan ␪ ෇

y
x

cot ␪ ෇

x
y

y

sin A
sin B
sin C
෇
෇
a
b
c

(x, y)

C
c

¨

THE LAW OF COSINES
x

y
y=sin x

y

b

a 2 ෇ b 2 ϩ c 2 Ϫ 2bc cos A
b 2 ෇ a 2 ϩ c 2 Ϫ 2ac cos B
c 2 ෇ a 2 ϩ b 2 Ϫ 2ab cos C

y=tan x

A

y=cos x

1

1
π

a

r

GRAPHS OF THE TRIGONOMETRIC FUNCTIONS
y

B

THE LAW OF SINES

r
y

2π

ADDITION AND SUBTRACTION FORMULAS

2π
x

_1

π

2π x

sin͑x ϩ y͒ ෇ sin x cos y ϩ cos x sin y

x

π

sin͑x Ϫ y͒ ෇ sin x cos y Ϫ cos x sin y

_1

cos͑x ϩ y͒ ෇ cos x cos y Ϫ sin x sin y
y

y=csc x

y

y=sec x

y

cos͑x Ϫ y͒ ෇ cos x cos y ϩ sin x sin y

y=cot x

1

1
π

2π x

π

π

2π x

2π x

tan͑x ϩ y͒ ෇

tan x ϩ tan y
1 Ϫ tan x tan y

tan͑x Ϫ y͒ ෇

tan x Ϫ tan y
1 ϩ tan x tan y

_1

_1

DOUBLE-ANGLE FORMULAS
sin 2x ෇ 2 sin x cos x

TRIGONOMETRIC FUNCTIONS OF IMPORTANT ANGLES

␪

radians

sin ␪

cos ␪

tan ␪

0Њ
30Њ
45Њ
60Њ
90Њ

0
␲͞6
␲͞4
␲͞3
␲͞2

0
1͞2
s2͞2
s3͞2
1

1
s3͞2
s2͞2

1͞2
0

0
s3͞3
1
s3
—

cos 2x ෇ cos 2x Ϫ sin 2x ෇ 2 cos 2x Ϫ 1 ෇ 1 Ϫ 2 sin 2x
tan 2x ෇

2 tan x
1 Ϫ tan2x

HALF-ANGLE FORMULAS
sin 2x ෇

2

1 Ϫ cos 2x
2

cos 2x ෇

1 ϩ cos 2x
2


ESSENTIAL CALCULUS

EARLY TRANSCENDENTALS
JAMES STEWART
McMaster University and University of Toronto

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Essential Calculus: Early Transcendentals
James Stewart
Publisher: Bob Pirtle
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CONTENTS

1

FUNCTIONS AND LIMITS
1.1
1.2
1.3
1.4
1.5

1.6

2

DERIVATIVES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

3

1

Functions and Their Representations 1
A Catalog of Essential Functions 10
The Limit of a Function 24
Calculating Limits 35
Continuity
45
Limits Involving Infinity
56
Review
69

73

Derivatives and Rates of Change 73
The Derivative as a Function 83
Basic Differentiation Formulas 94
The Product and Quotient Rules 106
The Chain Rule 113
Implicit Differentiation 121
Related Rates 127
Linear Approximations and Differentials
Review
138

133

INVERSE FUNCTIONS: Exponential, Logarithmic, and Inverse Trigonometric Functions 142
3.1
3.2
3.3
3.4
3.5
3.6
3.7

Exponential Functions 142
Inverse Functions and Logarithms 148
Derivatives of Logarithmic and Exponential Functions
Exponential Growth and Decay 167
Inverse Trigonometric Functions 175
Hyperbolic Functions 181
Indeterminate Forms and L’Hospital’s Rule 187
Review

195

160

iii


iv

■

CONTENTS

4

APPLICATIONS OF DIFFERENTIATION
4.1
4.2
4.3
4.4
4.5
4.6
4.7

5

INTEGRALS
5.1
5.2
5.3

5.4
5.5

6

7

251
Areas and Distances 251
The Definite Integral 262
Evaluating Definite Integrals 274
The Fundamental Theorem of Calculus
The Substitution Rule 293
Review
300

SERIES

304

357

Areas between Curves 357
Volumes
362
Volumes by Cylindrical Shells 373
Arc Length 378
Applications to Physics and Engineering
Differential Equations 397
Review

407

410
8.1
8.2

284

Integration by Parts 304
Trigonometric Integrals and Substitutions 310
Partial Fractions 320
Integration with Tables and Computer Algebra Systems
Approximate Integration 333
Improper Integrals 345
Review
354

APPLICATIONS OF INTEGRATION
7.1
7.2
7.3
7.4
7.5
7.6

8

Maximum and Minimum Values 198
The Mean Value Theorem 205
Derivatives and the Shapes of Graphs 211

Curve Sketching
220
Optimization Problems 226
Newton’s Method
236
Antiderivatives 241
Review
247

TECHNIQUES OF INTEGRATION
6.1
6.2
6.3
6.4
6.5
6.6

198

Sequences 410
Series
420

384

328


CONTENTS

8.3
8.4
8.5
8.6
8.7
8.8

9

PARAMETRIC EQUATIONS AND POLAR COORDINATES
9.1
9.2

9.3
9.4
9.5

10

The Integral and Comparison Tests 429
Other Convergence Tests 437
Power Series 447
Representing Functions as Power Series 452
Taylor and Maclaurin Series 458
Applications of Taylor Polynomials 471
Review
479

Parametric Curves 482
Calculus with Parametric Curves 488

Polar Coordinates 496
Areas and Lengths in Polar Coordinates 504
Conic Sections in Polar Coordinates 509
Review
515

VECTORS AND THE GEOMETRY OF SPACE

517

10.1 Three-Dimensional Coordinate Systems
10.2 Vectors

517

522

10.3 The Dot Product
10.4
10.5
10.6
10.7
10.8
10.9

11

530
The Cross Product
537

Equations of Lines and Planes 545
Cylinders and Quadric Surfaces 553
Vector Functions and Space Curves 559
Arc Length and Curvature 570
Motion in Space: Velocity and Acceleration
Review
587

PARTIAL DERIVATIVES

591

11.1 Functions of Several Variables
11.2 Limits and Continuity

11.5
11.6
11.7
11.8

591

601
609
Tangent Planes and Linear Approximations 617
The Chain Rule 625
Directional Derivatives and the Gradient Vector 633
Maximum and Minimum Values 644
Lagrange Multipliers 652
Review

659

11.3 Partial Derivatives
11.4

578

482

■

v


vi

■

CONTENTS

12

MULTIPLE INTEGRALS
12.1
12.2
12.3
12.4
12.5
12.6
12.7

12.8

13

Double Integrals over Rectangles 663
Double Integrals over General Regions 674
Double Integrals in Polar Coordinates 682
Applications of Double Integrals 688
Triple Integrals 693
Triple Integrals in Cylindrical Coordinates 703
Triple Integrals in Spherical Coordinates 707
Change of Variables in Multiple Integrals 713
Review
722

VECTOR CALCULUS
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9

APPENDIXES
A
B
C

D
E

INDEX

A83

663

725

Vector Fields 725
Line Integrals 731
The Fundamental Theorem for Line Integrals
Green’s Theorem
751
Curl and Divergence 757
Parametric Surfaces and Their Areas 765
Surface Integrals 775
Stokes’ Theorem
786
The Divergence Theorem 791
Review
797

A1
Trigonometry A1
Proofs
A10
Sigma Notation A26

The Logarithm Defined as an Integral
Answers to Odd-Numbered Exercises

A31
A39

742


PREFACE

This book is a response to those instructors who feel that calculus textbooks are too
big. In writing the book I asked myself: What is essential for a three-semester calculus course for scientists and engineers?
The book is about two-thirds the size of my other calculus books (Calculus, Fifth
Edition and Calculus, Early Transcendentals, Fifth Edition) and yet it contains almost
all of the same topics. I have achieved relative brevity mainly by condensing the exposition and by putting some of the features on the website www.stewartcalculus.com.
Here, in more detail are some of the ways I have reduced the bulk:
■
I have organized topics in an efficient way and rewritten some sections
with briefer exposition.
■
The design saves space. In particular, chapter opening spreads and photographs have been eliminated.
■
The number of examples is slightly reduced. Additional examples are
provided online.
■
The number of exercises is somewhat reduced, though most instructors
will find that there are plenty. In addition, instructors have access to the
archived problems on the website.
■

Although I think projects can be a very valuable experience for students,
I have removed them from the book and placed them on the website.
■
A discussion of the principles of problem solving and a collection of challenging problems for each chapter have been moved to the web.
Despite the reduced size of the book, there is still a modern flavor: Conceptual
understanding and technology are not neglected, though they are not as prominent as
in my other books.

CONTENT
This book treats the exponential, logarithmic, and inverse trigonometric functions
early, in Chapter 3. Those who wish to cover such functions later, with the logarithm
defined as an integral, should look at my book titled simply Essential Calculus.
CHAPTER 1 FUNCTIONS AND LIMITS After a brief review of the basic functions, limits and continuity are introduced, including limits of trigonometric functions, limits
involving infinity, and precise definitions.
■

DERIVATIVES The material on derivatives is covered in two sections in
order to give students time to get used to the idea of a derivative as a function. The
CHAPTER 2

■

vii


viii

■

PREFACE

formulas for the derivatives of the sine and cosine functions are derived in the section
on basic differentiation formulas. Exercises explore the meanings of derivatives in
various contexts.
CHAPTER 3

■

INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS

Exponential functions are defined first and the number e is defined as a limit. Logarithms are then defined as inverse functions. Applications to exponential growth and
decay follow. Inverse trigonometric functions and hyperbolic functions are also
covered here. L’Hospital’s Rule is included in this chapter because limits of transcendental functions so often require it.
APPLICATIONS OF DIFFERENTIATION The basic facts concerning extreme
values and shapes of curves are deduced from the Mean Value Theorem. The section
on curve sketching includes a brief treatment of graphing with technology. The section on optimization problems contains a brief discussion of applications to business
and economics.

CHAPTER 4

■

CHAPTER 5 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 C.) A quite general definition of the definite
integral (with unequal subintervals) is given initially before regular partitions are
employed. Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables.
■

TECHNIQUES OF INTEGRATION All the standard methods are covered, as
well as computer algebra systems, numerical methods, and improper integrals.

CHAPTER 6

■

APPLICATIONS 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. The chapter concludes with an
introduction to differential equations, including separable equations and direction
fields.

CHAPTER 7

■

SERIES The convergence tests have intuitive justifications as well as formal proofs. The emphasis is on Taylor series and polynomials and their applications
to physics. Error estimates include those based on Taylor’s Formula (with Lagrange’s
form of the remainder term) and those from graphing devices.

CHAPTER 8

■

CHAPTER 9 PARAMETRIC EQUATIONS AND POLAR COORDINATES This chapter introduces parametric and polar curves and applies the methods of calculus to them. A brief
treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in
Chapter 10.
■

VECTORS AND THE GEOMETRY OF SPACE In addition to the material on
vectors, dot and cross products, lines, planes, and surfaces, this chapter covers vectorvalued functions, length and curvature of space curves, and velocity and acceleration
along space curves, culminating in Kepler’s laws.
CHAPTER 10

■

CHAPTER 11 PARTIAL DERIVATIVES In view of the fact that many students have difficulty forming mental pictures of the concepts of this chapter, I’ve placed a special
emphasis on graphics to elucidate such ideas as graphs, contour maps, directional derivatives, gradients, and Lagrange multipliers.
■

CHAPTER 12 MULTIPLE INTEGRALS Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals.
■


PREFACE

■

ix

VECTOR CALCULUS The similarities among the Fundamental Theorem
for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem
are emphasized.
CHAPTER 13

■

WEBSITE
The website www.stewartcalculus.com includes the following.
■

Review of Algebra, Analytic Geometry, and Conic Sections

■

Additional Examples

■

Projects

■

Archived Problems (drill exercises that have appeared in previous editions
of my other books), together with their solutions

■

Challenge Problems

■

Complex Numbers

■

Graphing Calculators and Computers

■

Lies My Calculator and Computer Told Me

■

Additional Topics (complete with exercise sets): Principles of Problem
Solving, Strategy for Integration, Strategy for Testing Series, Fourier Series,
Area of a Surface of Revolution, Linear Differential Equations, SecondOrder Linear Differential Equations, Nonhomogeneous Linear Equations,
Applications of Second-Order Differential Equations, Using Series to Solve
Differential Equations, Complex Numbers, Rotation of Axes

■

Links, for particular topics, to outside web resources

■

History of Mathematics, with links to the better historical websites

ACKNOWLEDGMENTS
I thank the following reviewers for their thoughtful comments.
Ulrich Albrecht, Auburn University
Christopher Butler, Case Western Reserve University
Joe Fisher, University of Cincinnati
John Goulet, Worchester Polytechnic Institute
Irvin Hentzel, Iowa State University
Joel Irish, University of Southern Maine
Mary Nelson, University of Colorado, Boulder
Ed Slaminka, Auburn University
Li (Jason) Zhongshan, Georgia State University
I also thank Marv Riedesel for accuracy in proofreading and Dan Clegg for detailed
discussions on how to achieve brevity. In addition, I thank Kathi Townes, Stephanie

x

■

PREFACE

Kuhns, Jenny Turney, and Brian Betsill of TECHarts for their production services and
the following Brooks/Cole staff: Cheryll Linthicum, editorial production project manager; Vernon Boes, art director; Karin Sandberg and Darlene Amidon-Brent, marketing team; Earl Perry, technology project manager; Stacy Green, assistant editor;
Magnolia Molcan, editorial assistant; Bob Kauser, permissions editor; Rebecca Cross,
print/media buyer; and William Stanton, cover designer. They have all done an outstanding job.
The idea for this book came from my editor, Bob Pirtle, who had been hearing
of the desire for a much shorter calculus text from numerous instructors. I thank him
for encouraging me to pursue this idea and for his advice and assistance whenever I
needed it.
JA M E S S T E WA RT


PREFACE

ANCILLARIES FOR INSTRUCTORS

ANCILLARIES FOR STUDENTS

COMPLETE SOLUTIONS MANUAL
ISBN 0495014303

STUDENT SOLUTIONS MANUAL
ISBN 049501429X

The Complete Solutions Manual provides worked-out

solutions to all of the problems in the text.

The Student Solutions Manual provides completely
worked-out solutions to all odd-numbered exercises
within the text, giving students a way to check their
answers and ensure that they took the correct steps to
arrive at an answer.

SOLUTIONS BUILDER CD
ISBN 0495106925

This CD is an electronic version of the complete solutions manual. It provides instructors with an efficient
method for creating solution sets to homework or
exams. Instructors can easily view, select, and save
solution sets that can then be printed or posted.
TOOLS FOR ENRICHING CALCULUS
ISBN 0495107638

TEC contains Visuals and Modules for use as classroom demonstrations. Exercises for each Module allow
instructors to make assignments based on the classroom
demonstration. TEC also includes Homework Hints for
representative exercises. Students can benefit from this
additional help when instructors assign these exercises.

■

INTERACTIVE VIDEO SKILLBUILDER CD
ISBN 0495113719

The Interactive Video Skillbuilder CD-ROM contains

more than eight hours of instruction. To help students
evaluate their progress, each section contains a tenquestion web quiz (the results of which can be e-mailed
to the instructor) and each chapter contains a chapter
test, with the answer to each problem on each test.
TOOLS FOR ENRICHING CALCULUS
ISBN 0495107638

TEC provides a laboratory environment in which students can enrich their understanding by revisiting and
exploring selected topics. TEC also includes Homework
Hints for representative exercises.

Ancillaries for students are available for purchase at
www.cengage.com

xi


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 E. 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 Ϫ 1 and you obtain 1͞(1 ϩ 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
Derive, Maple, Mathematica, or the TI-89/92) 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.
xii

The CD-ROM Tools for Enriching™ Calculus is referred
to by means of the symbol
. It directs you to Visuals and
Modules in which you can explore aspects of calculus for
which the computer is particularly useful. TEC also provides
Homework Hints for representative exercises that are indicated by printing the exercise number in blue: 43. . These
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. (See the front endsheet for information
on how to purchase this and other useful tools.)
The Interactive Video Skillbuilder CD-ROM contains
videos of instructors explaining two or three of the examples
in every section of the text. (The symbol V has been placed
beside these examples in the text.) Also on the CD is a video in
which I offer advice on how to succeed in your calculus course.
I also want to draw your attention to the website
www.stewartcalculus.com. There you will find an Algebra
Review (in case your precalculus skills are weak) as well as
Additional Examples, Challenging Problems, Projects, Lies
My Calculator and Computer Told Me (explaining why calculators sometimes give the wrong answer), History of Mathematics, Additional Topics, chapter quizzes, and links to
outside resources.
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.
JA M E S S T E WA RT


1

FUNCTIONS
AND LIMITS
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. So in this first chapter we begin our study of calculus by investigating how the values of functions change and approach limits.

1.1

Year

Population
(millions)

1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000

1650
1750
1860
2070

2300
2560
3040
3710
4450
5280
6080

FUNCTIONS AND THEIR REPRESENTATONS
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 P͑t͒ at time t, for certain years. For instance,
P͑1950͒ Ϸ 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 a first-class letter depends on the weight w of the letter.
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.
1


2

■

CHAPTER 1

FUNCTIONS AND LIMITS

A function f is a rule that assigns to each element x in a set A exactly one
element, called f ͑x͒, in a set B.

x
(input)

f

ƒ
(output)

FIGURE 2

Machine diagram for a function ƒ

x

ƒ
a

f(a)

f

A

We usually consider functions for which the sets A and B are sets of real numbers.
The set A is called the domain of the function. The number f ͑x͒ 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 ͑x͒ 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.
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 ͑x͒ 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.
Another way to picture a function is by an arrow diagram as in Figure 3. Each
arrow connects an element of A to an element of B. The arrow indicates that f ͑x͒ is
associated with x, f ͑a͒ 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 A, then its graph is the set of ordered pairs
͕͑x, f ͑x͒͒

B

Խ x ʦ A͖

(Notice that these are input-output pairs.) In other words, the graph of f consists of all
points ͑x, y͒ in the coordinate plane such that y ෇ f ͑x͒ 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 ͑x, y͒ on the graph is y ෇ f ͑x͒, we
can read the value of f ͑x͒ 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 ƒ

y

y

{ x, ƒ}

y ϭ ƒ(x)

range

ƒ
f (2)
f(1)
0

1

2

x

x

x

0

domain
FIGURE 4

FIGURE 5

y

EXAMPLE 1 The graph of a function f is shown in Figure 6.

(a) Find the values of f ͑1͒ and f ͑5͒.
(b) What are the domain and range of f ?
1
0

FIGURE 6

SOLUTION
1

x

(a) We see from Figure 6 that the point ͑1, 3͒ lies on the graph of f , so the value of
f at 1 is f ͑1͒ ෇ 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 unit below the x-axis, so we estimate that
f ͑5͒ Ϸ Ϫ0.7.

SECTION 1.1

■ The notation for intervals is given on
Reference Page 3. The Reference Pages
are located at the front and back of the
book.

FUNCTIONS AND THEIR REPRESENTATIONS

■

3

(b) We see that f ͑x͒ is defined when 0 ഛ x ഛ 7, so the domain of f is the closed
interval ͓0, 7͔. Notice that f takes on all values from Ϫ2 to 4, so the range of f is
͕y

Խ Ϫ2 ഛ y ഛ 4͖ ෇ ͓Ϫ2, 4͔

■

REPRESENTATIONS OF FUNCTIONS

There are four possible ways to represent a function:
■
■

verbally (by a description in words)
numerically (by a table of values)

■
■

visually (by a graph)
algebraically (by an explicit formula)

If a single function can be represented in all four ways, it is often useful to go from
one representation to another to gain additional insight into the function. 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.
A. The most useful representation of the area of a circle as a function of its radius

is probably the algebraic formula A͑r͒ ෇ ␲ 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 ͕r r Ͼ 0͖ ෇ ͑0, ϱ͒, and the range is also ͑0, ϱ͒.

Խ

Year

Population
(millions)

1900
1910
1920
1930
1940

1950
1960
1970
1980
1990
2000

1650
1750
1860
2070
2300
2560
3040
3710
4450
5280
6080

B. We are given a description of the function in words: P͑t͒ is the human population

of the world at time t. 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 7. 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 P͑t͒ at any time
t. But it is possible to find an expression for a function that approximates P͑t͒. In
fact, we could use a graphing calculator with exponential regression capabilities to
obtain the approximation
P͑t͒ Ϸ f ͑t͒ ෇ ͑0.008079266͒ и ͑1.013731͒t
and Figure 8 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

6x10'

6x10'

1900

1920

1940

1960

1980

2000 t

FIGURE 7 Scatter plot of data points for population growth

1900

1920

1940

1960

1980

2000 t

FIGURE 8 Graph of a mathematical model for population growth


4

■

CHAPTER 1

FUNCTIONS AND LIMITS

■ A function defined by a table of values
is called a tabular function.

w (ounces)

C͑w͒ (dollars)

0Ͻwഛ1
1Ͻwഛ2
2Ͻwഛ3

3Ͻwഛ4
4Ͻwഛ5

0.39
0.63
0.87
1.11
1.35

и
и
и

и
и
и

12 Ͻ w ഛ 13

3.27

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: C͑w͒ is the cost of mailing a first-class
letter with weight w. The rule that the US Postal Service used as of 2006 is as follows: The cost is 39 cents for up to one ounce, plus 24 cents for each successive
ounce up to 13 ounces. 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 6).

D. The graph shown in Figure 1 is the most natural representation of the vertical acceleration function a͑t͒. 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 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.
EXAMPLE 2 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.

T

0

FIGURE 9

t

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 hotwater 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 9.
■

EXAMPLE 3 Find the domain of each function.

(a) f ͑x͒ ෇ sx ϩ 2

(b) t͑x͒ ෇

1
x Ϫx
2

SOLUTION
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.
■

(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 ϩ 2 ജ 0. This is equivalent to
x ജ Ϫ2, so the domain is the interval ͓Ϫ2, ϱ͒.
(b) Since
1
1
t͑x͒ ෇ 2
෇
x Ϫx
x͑x Ϫ 1͒
and division by 0 is not allowed, we see that t͑x͒ is not defined when x ෇ 0 or
x ෇ 1. Thus the domain of t is ͕x x 0, x 1͖, which could also be written in
interval notation as ͑Ϫϱ, 0͒ ʜ ͑0, 1͒ ʜ ͑1, ϱ͒.
■

Խ

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.
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.


SECTION 1.1

■

FUNCTIONS AND THEIR REPRESENTATIONS

5

The reason for the truth of the Vertical Line Test can be seen in Figure 10. If each
vertical line x ෇ a intersects a curve only once, at ͑a, b͒, then exactly one functional
value is defined by f ͑a͒ ෇ b. But if a line x ෇ a intersects the curve twice, at ͑a, b͒
and ͑a, c͒, then the curve can’t represent a function because a function can’t assign
two different values to a.
y

y

x=a

(a, c)

x=a

(a, b)

(a, b)
x

a

0

a

0

x

FIGURE 10

PIECEWISE DEFINED FUNCTIONS

The functions in the following three examples are defined by different formulas in different parts of their domains.
V EXAMPLE 4

A function f is defined by
f ͑x͒ ෇

ͭ

1 Ϫ x if x ഛ 1
x2
if x Ͼ 1

Evaluate f ͑0͒, f ͑1͒, and f ͑2͒ 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 ഛ 1, then
the value of f ͑x͒ is 1 Ϫ x. On the other hand, if x Ͼ 1, then the value of f ͑x͒ is x 2.

Since 0 ഛ 1, we have f ͑0͒ ෇ 1 Ϫ 0 ෇ 1.
Since 1 ഛ 1, we have f ͑1͒ ෇ 1 Ϫ 1 ෇ 0.

y

Since 2 Ͼ 1, we have f ͑2͒ ෇ 2 2 ෇ 4.

1

1

x

FIGURE 11

How do we draw the graph of f ? We observe that if x ഛ 1, then f ͑x͒ ෇ 1 Ϫ x,
so the part of the graph of f that lies to the left of the vertical line x ෇ 1 must coincide with the line y ෇ 1 Ϫ x, which has slope Ϫ1 and y-intercept 1. If x Ͼ 1, then
f ͑x͒ ෇ x 2, so the part of the graph of f that lies to the right of the line x ෇ 1 must
coincide with the graph of y ෇ x 2, which is a parabola. This enables us to sketch the
graph in Figure l1. The solid dot indicates that the point ͑1, 0͒ is included on the
graph; the open dot indicates that the point ͑1, 1͒ 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

Խ Խ

www.stewartcalculus.com
For a more extensive review of
absolute values, click on Review of
Algebra.
■

For example,

Խ3Խ ෇ 3

Խ Ϫ3 Խ ෇ 3

ԽaԽ ജ 0
Խ0Խ ෇ 0

for every number a

Խ s2 Ϫ 1 Խ ෇ s2 Ϫ 1

Խ3 Ϫ ␲Խ ෇ ␲ Ϫ 3


6

■

CHAPTER 1

FUNCTIONS AND LIMITS

In general, we have

ԽaԽ ෇ a
Խ a Խ ෇ Ϫa

if a ജ 0
if a Ͻ 0

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

Խ Խ

EXAMPLE 5 Sketch the graph of the absolute value function f ͑x͒ ෇ x .

y

SOLUTION From the preceding discussion we know that

y=| x |

ԽxԽ ෇
0

x

ͭ

x

Ϫx

if x ജ 0
if x Ͻ 0

Using the same method as in Example 4, 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 ෇ Ϫx to the
left of the y-axis (see Figure 12).
■

FIGURE 12

EXAMPLE 6 In Example C at the beginning of this section we considered the cost

C͑w͒ of mailing a first-class letter with weight w. In effect, this is a piecewise
defined function because, from the table of values, we have

C
1

0

1

FIGURE 13

2

3

4

5

0.39
0.63
C͑w͒ ෇
0.87
1.11

w

if
if
if
if

0Ͻwഛ
1Ͻwഛ
2Ͻwഛ
3Ͻwഛ

1
2
3
4

The graph is shown in Figure 13. You can see why functions similar to this one are
called step functions—they jump from one value to the next.
■

SYMMETRY

If a function f satisfies f ͑Ϫx͒ ෇ f ͑x͒ for every number x in its domain, then f is
called an even function. For instance, the function f ͑x͒ ෇ x 2 is even because
f ͑Ϫx͒ ෇ ͑Ϫx͒2 ෇ x 2 ෇ f ͑x͒
The geometric significance of an even function is that its graph is symmetric with
respect to the y-axis (see Figure 14). This means that if we have plotted the graph of
y

y

f(_x)

ƒ
_x

0

x

FIGURE 14 An even function

_x
x

ƒ

0
x

FIGURE 15 An odd function

x


SECTION 1.1

f

_1

■

7

f for x ജ 0, we obtain the entire graph simply by reflecting this portion about the
y-axis.
If f satisfies f ͑Ϫx͒ ෇ Ϫf ͑x͒ for every number x in its domain, then f is called an
odd function. For example, the function f ͑x͒ ෇ x 3 is odd because

y
1

FUNCTIONS AND THEIR REPRESENTATIONS

x

1

f ͑Ϫx͒ ෇ ͑Ϫx͒3 ෇ Ϫx 3 ෇ Ϫf ͑x͒

_1

The graph of an odd function is symmetric about the origin (see Figure 15 on page 6).
If we already have the graph of f for x ജ 0, we can obtain the entire graph by rotating this portion through 180Њ about the origin.

(a)
y
1

V EXAMPLE 7 Determine whether each of the following functions is even, odd, or
neither even nor odd.
(a) f ͑x͒ ෇ x 5 ϩ x
(b) t͑x͒ ෇ 1 Ϫ x 4
(c) h͑x͒ ෇ 2x Ϫ x 2

g
1

SOLUTION

x

f ͑Ϫx͒ ෇ ͑Ϫx͒5 ϩ ͑Ϫx͒ ෇ ͑Ϫ1͒5x 5 ϩ ͑Ϫx͒

(a)

෇ Ϫx 5 Ϫ x ෇ Ϫ͑x 5 ϩ x͒
෇ Ϫf ͑x͒

(b)

Therefore, f is an odd function.

y

t͑Ϫx͒ ෇ 1 Ϫ ͑Ϫx͒4 ෇ 1 Ϫ x 4 ෇ t͑x͒

(b)

h

1

So t is even.
x

1

h͑Ϫx͒ ෇ 2͑Ϫx͒ Ϫ ͑Ϫx͒2 ෇ Ϫ2x Ϫ x 2

(c)
Since h͑Ϫx͒
odd.

h͑x͒ and h͑Ϫx͒

Ϫh͑x͒, we conclude that h is neither even nor

■

(c)

The graphs of the functions in Example 7 are shown in Figure 16. Notice that the
graph of h is symmetric neither about the y-axis nor about the origin.

FIGURE 16

INCREASING AND DECREASING FUNCTIONS
y

B

The graph shown in Figure 17 rises from A to B, falls from B to C, and rises again
from C to D. The function f is said to be increasing on the interval ͓a, b͔, decreasing
on ͓b, c͔, and increasing again on ͓c, d͔. Notice that if x 1 and x 2 are any two numbers
between a and b with x 1 Ͻ x 2, then f ͑x 1 ͒ Ͻ f ͑x 2 ͒. We use this as the defining property of an increasing function.

D

y=ƒ
C
f(x™)
A

f(x¡)

A function f is called increasing on an interval I if
0 a x¡

FIGURE 17

x™

b

c

d

x

f ͑x 1 ͒ Ͻ f ͑x 2 ͒

whenever x 1 Ͻ x 2 in I

It is called decreasing on I if
f ͑x 1 ͒ Ͼ f ͑x 2 ͒

whenever x 1 Ͻ x 2 in I


8

■

CHAPTER 1

FUNCTIONS AND LIMITS

In the definition of an increasing function it is important to realize that the inequality f ͑x 1 ͒ Ͻ f ͑x 2 ͒ must be satisfied for every pair of numbers x 1 and x 2 in I with
x 1 Ͻ x 2.
You can see from Figure 18 that the function f ͑x͒ ෇ x 2 is decreasing on the interval ͑Ϫϱ, 0͔ and increasing on the interval ͓0, ϱ͒.
y

y=≈

1.1

EXERCISES

1. The graph of a function f is given.

(a)
(b)
(c)
(d)
(e)
(f )

x

0

FIGURE 18

■ Determine whether the curve is the graph of a function
of x. If it is, state the domain and range of the function.

3–6

State the value of f ͑Ϫ1͒.
Estimate the value of f ͑2͒.
For what values of x is f ͑x͒ ෇ 2?
Estimate the values of x such that f ͑x͒ ෇ 0.
State the domain and range of f .
On what interval is f increasing?

y

3.

y

4.

1

1
0

0

x

1

1

x

1

x

y

1
0

y

5.

y

6.

x

1

1

1
0

1

0

x

2. The graphs of f and t are given.

(a)
(b)
(c)
(d)
(e)
(f )

State the values of f ͑Ϫ4͒ and t͑3͒.
For what values of x is f ͑x͒ ෇ t͑x͒?
Estimate the solution of the equation f ͑x͒ ෇ Ϫ1.
On what interval is f decreasing?
State the domain and range of f.
State the domain and range of t.

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7. The graph shown gives the weight of a certain person as a

function of age. Describe in words how this person’s weight
varies over time. What do you think happened when this
person was 30 years old?

y
200

g
f

Weight
(pounds)

2

0

2

x

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150
100
50
0

10

20 30 40

50

60 70

Age
(years)


SECTION 1.1

8. The graph shown gives a salesman’s distance from his home

as a function of time on a certain day. Describe in words
what the graph indicates about his travels on this day.

8 AM

■ Evaluate the difference quotient for the given function.
Simplify your answer.

f ͑3 ϩ h͒ Ϫ f ͑3͒
h

f ͑a ϩ h͒ Ϫ f ͑a͒
h

20. f ͑x͒ ෇ x 3,

10

NOON

2

4

6 PM

Time
(hours)

water, and then let the glass sit on a table. Describe how
the temperature of the water changes as time passes. Then
sketch a rough graph of the temperature of the water as a
function of the elapsed time.
10. Sketch a rough graph of the number of hours of daylight as

a function of the time of year.
11. Sketch a rough graph of the outdoor temperature as a func-

tion of time during a typical spring day.
12. Sketch a rough graph of the market value of a new car as a

function of time for a period of 20 years. Assume the car is
well maintained.
13. Sketch the graph of the amount of a particular brand of cof-

fee sold by a store as a function of the price of the coffee.

1
,
x

22. f ͑x͒ ෇

xϩ3
,
xϩ1

15. A homeowner mows the lawn every Wednesday afternoon.

Sketch a rough graph of the height of the grass as a function
of time over the course of a four-week period.
16. A jet takes off from an airport and lands an hour later at

another airport, 400 miles away. If t represents the time in
minutes since the plane has left the terminal, let x͑t͒ be

the horizontal distance traveled and y͑t͒ be the altitude of
the plane.
(a) Sketch a possible graph of x͑t͒.
(b) Sketch a possible graph of y͑t͒.
(c) Sketch a possible graph of the ground speed.
(d) Sketch a possible graph of the vertical velocity.
17. If f ͑x͒ ෇ 3x 2 Ϫ x ϩ 2, find f ͑2͒, f ͑Ϫ2͒, f ͑a͒, f ͑Ϫa͒,

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23–27

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V͑r͒ ෇ 43 ␲ r 3 . Find a function that represents the amount of
air required to inflate the balloon from a radius of r inches
to a radius of r ϩ 1 inches.

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Find the domain of the function.
x
3x Ϫ 1

24. f ͑x͒ ෇

5x ϩ 4
x 2 ϩ 3x ϩ 2

3
t
25. f ͑t͒ ෇ st ϩ s

26. t͑u͒ ෇ su ϩ s4 Ϫ u
27. h͑x͒ ෇
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1
4
x 2 Ϫ 5x
s
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28. Find the domain and range and sketch the graph of the

function h͑x͒ ෇ s4 Ϫ x 2 .
29– 40

■

Find the domain and sketch the graph of the function.

29. f ͑x͒ ෇ 5

30. F͑x͒ ෇ 2 ͑x ϩ 3͒

31. f ͑t͒ ෇ t 2 Ϫ 6t

32. H͑t͒ ෇

33. t͑x͒ ෇ sx Ϫ 5

34. F͑x͒ ෇ 2x ϩ 1

35. G͑x͒ ෇
37. f ͑x͒ ෇
38. f ͑x͒ ෇
39. f ͑x͒ ෇

f ͑a ϩ 1͒, 2 f ͑a͒, f ͑2a͒, f ͑a 2 ͒, [ f ͑a͒] 2, and f ͑a ϩ h͒.

18. A spherical balloon with radius r inches has volume

f ͑x͒ Ϫ f ͑1͒
xϪ1

23. f ͑x͒ ෇

14. You place a frozen pie in an oven and bake it for an

hour. Then you take it out and let it cool before eating it.
Describe how the temperature of the pie changes as time
passes. Then sketch a rough graph of the temperature of the
pie as a function of time.

f ͑x͒ Ϫ f ͑a͒

xϪa

21. f ͑x͒ ෇

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9. You put some ice cubes in a glass, fill the glass with cold

9

19–22

19. f ͑x͒ ෇ 4 ϩ 3x Ϫ x 2 ,

Distance
from home
(miles)

■

FUNCTIONS AND THEIR REPRESENTATIONS

40. f ͑x͒ ෇

■

■

1

Խ Խ

3x ϩ x
x

ͭ
ͭ
ͭ

xϩ2
1Ϫx
3 Ϫ 12 x
2x Ϫ 5

ͭ

36.

4 Ϫ t2
2Ϫt

Խ
x
t͑x͒ ෇ Խ Խ

Խ

x2

if x Ͻ 0

if x ജ 0
if x ഛ 2
if x Ͼ 2

x ϩ 2 if x ഛ Ϫ1
x2
if x Ͼ Ϫ1

Ϫ1
if x ഛ Ϫ1
3x ϩ 2 if x Ͻ 1
7 Ϫ 2x if x ജ 1

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Խ Խ

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