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This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest.

Copyright 2010 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.

R E F E R E N C E PA G E 1

Cut here and keep for reference

ALGEBRA

GEOMETRY

Arithmetic Operations

Geometric Formulas a c ad bc 苷 b d bd a d ad b a 苷 苷 c b c bc d

a共b c兲 苷 ab ac a c ac 苷 b b b

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

Circle

Sector of Circle

A 苷 12 bh

A 苷 r 2

A 苷 12 r 2

C 苷 2 r

s 苷 r 共 in radians兲

苷 ab sin 1 2

a

Exponents and Radicals x 苷 x mn xn 1 xn 苷 n x

x m x n 苷 x mn 共x 兲 苷 x m n

mn

冉冊 x y

共xy兲n 苷 x n y n

n

苷

xn yn

n n x m兾n 苷 s x m 苷 (s x )m

n x 1兾n 苷 s x

冑

n n n xy 苷 s xs y s

n

r

h

¨

m

r

s

¨

b

r

Sphere V 苷 43 r 3

Cylinder V 苷 r 2h

Cone V 苷 13 r 2h

A 苷 4 r 2

A 苷 rsr 2 h 2

n x x s 苷 n y sy

r r

h

h

Factoring Special Polynomials

r

x 2 y 2 苷 共x y兲共x y兲 x 3 y 3 苷 共x y兲共x 2 xy y 2兲 x 3 y 3 苷 共x y兲共x 2 xy y 2兲

Distance and Midpoint Formulas

Binomial Theorem 共x y兲2 苷 x 2 2xy y 2

共x y兲2 苷 x 2 2xy y 2

Distance between P1共x1, y1兲 and P2共x 2, y2兲: d 苷 s共x 2 x1兲2 共 y2 y1兲2

共x y兲3 苷 x 3 3x 2 y 3xy 2 y 3 共x y兲3 苷 x 3 3x 2 y 3xy 2 y 3 共x y兲n 苷 x n nx n1y

冉冊

n共n 1兲 n2 2 x y 2

冉冊

n nk k x y nxy n1 y n k

n共n 1兲 共n k 1兲 n where 苷 k 1 ⴢ 2 ⴢ 3 ⴢ

ⴢ k

Midpoint of P1 P2 :

冉

x1 x 2 y1 y2 , 2 2

Lines Slope of line through P1共x1, y1兲 and P2共x 2, y2兲:

Quadratic Formula

m苷

If ax 2 bx c 苷 0, then x 苷

冊

b sb 2 4ac . 2a

y2 y1 x 2 x1

Point-slope equation of line through P1共x1, y1兲 with slope m:

Inequalities and Absolute Value

y y1 苷 m共x x1兲

If a b and b c, then a c.

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

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

y 苷 mx b

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

ⱍxⱍ 苷 a ⱍxⱍ a ⱍxⱍ a

means

x 苷 a or

x 苷 a

means a x a means

x a or

x a

Circles Equation of the circle with center 共h, k兲 and radius r: 共x h兲2 共 y k兲2 苷 r 2

Copyright 2010 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.

R E F E R E N C E PA G E 2

TRIGONOMETRY Angle Measurement

Fundamental Identities

radians 苷 180⬚

csc 苷

1 sin

sec 苷

1 cos

tan 苷

sin cos

cot 苷

cos sin

共 in radians兲

cot 苷

1 tan

sin 2 ⫹ cos 2 苷 1

Right Angle Trigonometry

1 ⫹ tan 2 苷 sec 2

1 ⫹ cot 2 苷 csc 2

sin共⫺兲 苷 ⫺sin

cos共⫺兲 苷 cos

tan共⫺兲 苷 ⫺tan

sin

1⬚ 苷

rad 180

1 rad 苷

180⬚

¨ r

s 苷 r

sin 苷 cos 苷 tan 苷

opp hyp

csc 苷

adj hyp

sec 苷

opp adj

cot 苷

s

r

hyp opp

hyp

hyp adj

opp

¨ adj

冉 冊

adj opp

cos

Trigonometric Functions sin 苷

y r

csc 苷

r y

cos 苷

x r

sec 苷

r x

tan 苷

y x

cot 苷

x y

B a

r

C c

¨

The Law of Cosines

x

b

a 2 苷 b 2 ⫹ c 2 ⫺ 2bc cos A b 2 苷 a 2 ⫹ c 2 ⫺ 2ac cos B y

A

c 2 苷 a 2 ⫹ b 2 ⫺ 2ab cos C

y=tan x

y=cos x

1

1 π

⫺ 苷 cot 2

sin A sin B sin C 苷 苷 a b c

(x, y)

y y=sin x

tan

⫺ 苷 cos 2

The Law of Sines

y

Graphs of Trigonometric Functions y

⫺ 苷 sin 2

冉 冊 冉 冊

2π

Addition and Subtraction Formulas

2π x

_1

π

2π x

π

x

sin共x ⫹ y兲 苷 sin x cos y ⫹ cos x sin y sin共x ⫺ y兲 苷 sin x cos y ⫺ cos x sin y

_1

cos共x ⫹ y兲 苷 cos x cos y ⫺ sin x sin y y

y

y=csc x

y

y=sec x

cos共x ⫺ y兲 苷 cos x cos y ⫹ sin x sin y

y=cot x

1

1 π

2π x

π

2π x

π

2π x

tan共x ⫹ y兲 苷

tan x ⫹ tan y 1 ⫺ tan x tan y

tan共x ⫺ y兲 苷

tan x ⫺ tan y 1 ⫹ tan x tan y

_1

_1

Double-Angle Formulas sin 2x 苷 2 sin x cos x

Trigonometric Functions of Important Angles

cos 2x 苷 cos 2x ⫺ sin 2x 苷 2 cos 2x ⫺ 1 苷 1 ⫺ 2 sin 2x

radians

sin

cos

tan

0⬚ 30⬚ 45⬚ 60⬚ 90⬚

0 兾6 兾4 兾3 兾2

0 1兾2 s2兾2 s3兾2 1

1 s3兾2 s2兾2 1兾2 0

0 s3兾3 1 s3 —

tan 2x 苷

2 tan x 1 ⫺ tan2x

Half-Angle Formulas sin 2x 苷

1 ⫺ cos 2x 2

cos 2x 苷

1 ⫹ cos 2x 2

Copyright 2010 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.

CA L C U L U S SEVENTH EDITION

JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO

Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States

Copyright 2010 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.

Calculus, Seventh Edition James Stewart Executive Editor: Liz Covello Assistant Editor: Liza Neustaetter Editorial Assistant: Jennifer Staller Media Editor : Maureen Ross Marketing Manager: Jennifer Jones Marketing Coordinator: Michael Ledesma Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Cheryll Linthicum Art Director: Vernon T. Boes Print Buyer: Becky Cross Rights Acquisitions Specialist: Don Schlotman Production Service: TECH· arts Text Designer: TECH· arts Photo Researcher: Terri Wright, www.terriwright.com Copy Editor: Kathi Townes Cover Designer: Irene Morris Cover Illustration: Irene Morris Compositor: Stephanie Kuhns, TECH· arts

© 2012, 2008 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to [email protected]

Library of Congress Control Number: 2010936608 Student Edition: ISBN-13: 978-0-538-49781-7 ISBN-10: 0-538-49781-5 Loose-leaf Edition: ISBN-13: 978-0-8400-5818-8 ISBN-10: 0-8400-5818-7 Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with oﬃce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local oﬃce at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Brooks/Cole, visit www.cengage.com/brookscole.

Printed in the United States of America 1 2 3 4 5 6 7 1 4 1 3 1 2 11 1 0

Trademarks ExamView ® and ExamViewPro ® are registered trademarks of FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation and used herein under license. Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc. Used herein under license. Derive is a registered trademark of Soft Warehouse, Inc. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. Tools for Enriching is a trademark used herein under license.

Copyright 2010 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.

K10T10

Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com.

Contents Preface

xi

To the Student

xxiii

Diagnostic Tests

xxiv

A Preview of Calculus

1

1

Functions and Limits 9 1.1

Four Ways to Represent a Function

1.2

Mathematical Models: A Catalog of Essential Functions

1.3

New Functions from O...

Copyright 2010 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.

R E F E R E N C E PA G E 1

Cut here and keep for reference

ALGEBRA

GEOMETRY

Arithmetic Operations

Geometric Formulas a c ad bc 苷 b d bd a d ad b a 苷 苷 c b c bc d

a共b c兲 苷 ab ac a c ac 苷 b b b

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

Circle

Sector of Circle

A 苷 12 bh

A 苷 r 2

A 苷 12 r 2

C 苷 2 r

s 苷 r 共 in radians兲

苷 ab sin 1 2

a

Exponents and Radicals x 苷 x mn xn 1 xn 苷 n x

x m x n 苷 x mn 共x 兲 苷 x m n

mn

冉冊 x y

共xy兲n 苷 x n y n

n

苷

xn yn

n n x m兾n 苷 s x m 苷 (s x )m

n x 1兾n 苷 s x

冑

n n n xy 苷 s xs y s

n

r

h

¨

m

r

s

¨

b

r

Sphere V 苷 43 r 3

Cylinder V 苷 r 2h

Cone V 苷 13 r 2h

A 苷 4 r 2

A 苷 rsr 2 h 2

n x x s 苷 n y sy

r r

h

h

Factoring Special Polynomials

r

x 2 y 2 苷 共x y兲共x y兲 x 3 y 3 苷 共x y兲共x 2 xy y 2兲 x 3 y 3 苷 共x y兲共x 2 xy y 2兲

Distance and Midpoint Formulas

Binomial Theorem 共x y兲2 苷 x 2 2xy y 2

共x y兲2 苷 x 2 2xy y 2

Distance between P1共x1, y1兲 and P2共x 2, y2兲: d 苷 s共x 2 x1兲2 共 y2 y1兲2

共x y兲3 苷 x 3 3x 2 y 3xy 2 y 3 共x y兲3 苷 x 3 3x 2 y 3xy 2 y 3 共x y兲n 苷 x n nx n1y

冉冊

n共n 1兲 n2 2 x y 2

冉冊

n nk k x y nxy n1 y n k

n共n 1兲 共n k 1兲 n where 苷 k 1 ⴢ 2 ⴢ 3 ⴢ

ⴢ k

Midpoint of P1 P2 :

冉

x1 x 2 y1 y2 , 2 2

Lines Slope of line through P1共x1, y1兲 and P2共x 2, y2兲:

Quadratic Formula

m苷

If ax 2 bx c 苷 0, then x 苷

冊

b sb 2 4ac . 2a

y2 y1 x 2 x1

Point-slope equation of line through P1共x1, y1兲 with slope m:

Inequalities and Absolute Value

y y1 苷 m共x x1兲

If a b and b c, then a c.

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

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

y 苷 mx b

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

ⱍxⱍ 苷 a ⱍxⱍ a ⱍxⱍ a

means

x 苷 a or

x 苷 a

means a x a means

x a or

x a

Circles Equation of the circle with center 共h, k兲 and radius r: 共x h兲2 共 y k兲2 苷 r 2

Copyright 2010 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.

R E F E R E N C E PA G E 2

TRIGONOMETRY Angle Measurement

Fundamental Identities

radians 苷 180⬚

csc 苷

1 sin

sec 苷

1 cos

tan 苷

sin cos

cot 苷

cos sin

共 in radians兲

cot 苷

1 tan

sin 2 ⫹ cos 2 苷 1

Right Angle Trigonometry

1 ⫹ tan 2 苷 sec 2

1 ⫹ cot 2 苷 csc 2

sin共⫺兲 苷 ⫺sin

cos共⫺兲 苷 cos

tan共⫺兲 苷 ⫺tan

sin

1⬚ 苷

rad 180

1 rad 苷

180⬚

¨ r

s 苷 r

sin 苷 cos 苷 tan 苷

opp hyp

csc 苷

adj hyp

sec 苷

opp adj

cot 苷

s

r

hyp opp

hyp

hyp adj

opp

¨ adj

冉 冊

adj opp

cos

Trigonometric Functions sin 苷

y r

csc 苷

r y

cos 苷

x r

sec 苷

r x

tan 苷

y x

cot 苷

x y

B a

r

C c

¨

The Law of Cosines

x

b

a 2 苷 b 2 ⫹ c 2 ⫺ 2bc cos A b 2 苷 a 2 ⫹ c 2 ⫺ 2ac cos B y

A

c 2 苷 a 2 ⫹ b 2 ⫺ 2ab cos C

y=tan x

y=cos x

1

1 π

⫺ 苷 cot 2

sin A sin B sin C 苷 苷 a b c

(x, y)

y y=sin x

tan

⫺ 苷 cos 2

The Law of Sines

y

Graphs of Trigonometric Functions y

⫺ 苷 sin 2

冉 冊 冉 冊

2π

Addition and Subtraction Formulas

2π x

_1

π

2π x

π

x

sin共x ⫹ y兲 苷 sin x cos y ⫹ cos x sin y sin共x ⫺ y兲 苷 sin x cos y ⫺ cos x sin y

_1

cos共x ⫹ y兲 苷 cos x cos y ⫺ sin x sin y y

y

y=csc x

y

y=sec x

cos共x ⫺ y兲 苷 cos x cos y ⫹ sin x sin y

y=cot x

1

1 π

2π x

π

2π x

π

2π x

tan共x ⫹ y兲 苷

tan x ⫹ tan y 1 ⫺ tan x tan y

tan共x ⫺ y兲 苷

tan x ⫺ tan y 1 ⫹ tan x tan y

_1

_1

Double-Angle Formulas sin 2x 苷 2 sin x cos x

Trigonometric Functions of Important Angles

cos 2x 苷 cos 2x ⫺ sin 2x 苷 2 cos 2x ⫺ 1 苷 1 ⫺ 2 sin 2x

radians

sin

cos

tan

0⬚ 30⬚ 45⬚ 60⬚ 90⬚

0 兾6 兾4 兾3 兾2

0 1兾2 s2兾2 s3兾2 1

1 s3兾2 s2兾2 1兾2 0

0 s3兾3 1 s3 —

tan 2x 苷

2 tan x 1 ⫺ tan2x

Half-Angle Formulas sin 2x 苷

1 ⫺ cos 2x 2

cos 2x 苷

1 ⫹ cos 2x 2

Copyright 2010 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.

CA L C U L U S SEVENTH EDITION

JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO

Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States

Copyright 2010 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.

Calculus, Seventh Edition James Stewart Executive Editor: Liz Covello Assistant Editor: Liza Neustaetter Editorial Assistant: Jennifer Staller Media Editor : Maureen Ross Marketing Manager: Jennifer Jones Marketing Coordinator: Michael Ledesma Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Cheryll Linthicum Art Director: Vernon T. Boes Print Buyer: Becky Cross Rights Acquisitions Specialist: Don Schlotman Production Service: TECH· arts Text Designer: TECH· arts Photo Researcher: Terri Wright, www.terriwright.com Copy Editor: Kathi Townes Cover Designer: Irene Morris Cover Illustration: Irene Morris Compositor: Stephanie Kuhns, TECH· arts

© 2012, 2008 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to [email protected]

Library of Congress Control Number: 2010936608 Student Edition: ISBN-13: 978-0-538-49781-7 ISBN-10: 0-538-49781-5 Loose-leaf Edition: ISBN-13: 978-0-8400-5818-8 ISBN-10: 0-8400-5818-7 Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with oﬃce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local oﬃce at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Brooks/Cole, visit www.cengage.com/brookscole.

Printed in the United States of America 1 2 3 4 5 6 7 1 4 1 3 1 2 11 1 0

Trademarks ExamView ® and ExamViewPro ® are registered trademarks of FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation and used herein under license. Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc. Used herein under license. Derive is a registered trademark of Soft Warehouse, Inc. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. Tools for Enriching is a trademark used herein under license.

Copyright 2010 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.

K10T10

Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com.

Contents Preface

xi

To the Student

xxiii

Diagnostic Tests

xxiv

A Preview of Calculus

1

1

Functions and Limits 9 1.1

Four Ways to Represent a Function

1.2

Mathematical Models: A Catalog of Essential Functions

1.3

New Functions from O...

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