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THOMAS'
CALCULUS
Based on the original work
by
George
B.
Thomas, Jr.
Massachusetts Institute of Technology
as revised
by
Maurice
D.
Weir
Naval Postgraduate School
Joel Hass
University of California, Davis
Addison-Wesley
Twelfth
Edition
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River
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tratioDl: Karen Heyt, lllustraThch
Cover
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Cover
image: Forest Edge, Hokuto, Hokkaido, Japsn 2004 © Michael Kenna
About
the cover: The cover image
of
a tree line on a snow-swept landscape,
by
the
photographer Michael Kenna,
was
taken in Hokkaido, Japan.
The
artist
was
not thinking
of
calculus when
he
composed the image, but rather,
of
a
visual haiku consisting
of
a few elements that would
spaIl<
the viewer's imagination. Similarly, the
minima1
design
of
this text allows the central ideas
of
calculus developed in this book to unfold to igoite the learner's imagination.
For permission
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made
to
the copyright holders
on
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C-I,
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Ubrary of
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Cataloging-in-Publication
Data
Weir,
Maurice
D.
Thomas' Calculus I Maurice
D.
Weir, Joel Hass,
George
B.
Thomas.-12th
ed.
p.em
ISBN 978-0-321-58799-2
1.
CalculUl!-Textbooks. I. Hass, Joel. II. Thoroas, George
B.
(George Brinton), 1914--2006. III. Thomas,
George B. (George Brinton), 1914--2006. Calculus.
Iv. Title
V.
Title: Calculus.
QA303.2.W452009b
515-dc22 2009023069
Copyright
()
2010, 2005, 2001 Pearson Education, Inc. All rights reserved.
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ISBN-10: 0-321-58799-5
ISBN-13: 978-0-321-58799-2

CONTENTS
Preface
ix
1
I
Functions
1
1.1
Functions and Their Graphs I
1.2
CombiJring Functions; Shifting and Scaling Graphs
14
1.3 Trigonometric Functions
22
1.4
Graphing with Calcolators and Computers 30
QuEsTIONS
TO
GUIDE YOUR REVIEW
34
PRACTICE EXERCISES
35
AoDmONAL
AND ADvANCED EXERCISES 37
2
Limits
and
Continuity
39
2.1
Rates
of
Change and Tangents to Curves 39
2.2 Limit
of
a Function and Limit Laws 46
2.3
The Precise Definition
of
a Limit 57
2.4 One-Sided Limits 66
2.5
Continuity
73
2.6 Limits Involving Infinity; Asymptotes
of
Graphs
84
QuEsTIONS
TO
GUIDE YOUR REVIEW
96
PRACTICE EXERCISES 97
AoDmONAL
AND ADvANCED EXERCISES 98
3
Differentiation
102
3.1
Tangents and the Derivative at a Point
102
3.2 The Derivative as a Function
106
3.3
Differentiation Rules
115
3.4 The Derivative as a Rate
of
Change 124
3.5
Derivatives
of
Trigonometric Functions
135
3.6 The Chain Rule
142
iii

iv
Contents
3.7 Implicit Differentiation
149
3.8 Related Rates
155
3.9 Linearization and Differentials
164
QuESTIONS
TO
GUIDE
YOUR
REVIEW
175
PRACTICE
EXERCISES
176
ADDITIONAL
AND
ADvANCED
EXERCISES
180
4
Applications
of
Derivatives
184
4.1
Extreme Values
of
Functions
184
4.2 The Mean Value Theorem
192
4.3
Monotonic Functions and the First Derivative Test
198
4.4 Concavity and
Curve
Sketching 203
4.5 Applied Optimization 214
4.6 Newton's Method 225
4.7 Antiderivatives 230
QuESTIONS
TO
GUIDE
YOUR
REVIEW
239
PRACTICE
EXERCISES
240
ADDITIONAL
AND
ADvANCED
EXERCISES
243
5
I
Integration
246
5.1
Area and Estimating with Finite Sums 246
5.2
Sigma Notation and Limits
of
Finite Sums 256
5.3
The Definite Integral 262
5.4 The Fundamental Theorem
of
Calculus 274
5.5
Indefinite Integrals and the Substitution Method 284
5.6
Substitution and Area Between Curves
291
QuESTIONS
TO
GUIDE
YOUR
REVIEW
300
PRACTICE
EXERCISES
301
ADDITIONAL
AND
ADvANCED
EXERCISES
304
6
Applications
of Definite
Integrals
308
6.1
Volumes Using Cross-Sections 308
6.2
Volumes Using Cylindrical Shells 319
6.3
Arc Length 326
6.4 Areas
of
Surfaces
of
Revolution 332
6.5
Work and Fluid Forces
337
6.6
Moments and Centers
of
Mass 346
QuESTIONS
TO
GUIDE
YOUR
REVIEW
357
PRACTICE
EXERCISES
357
ADDITIONAL
AND
ADvANCED
EXERCISES
359

7
8
9
10
Contents
V
Transcendental
Functions
361
7.1
Inverse Functions and Their Derivatives
361
7.2 Natural Logarithms 369
7.3
Exponential Functions 377
7.4 Exponential Change and Separable Differential Equations 387
7.5
Indeterminate Forms and VHopitai's Rule 396
7.6 Inverse Trigonometric Functions 404
7.7 Hyperbolic Functions 416
7.8 Relative Rates
of
Growth 424
QuEsTIONS
TO
GUIDE YOUR REVIEW 429
PRACTICE EXERCISES 430
ADDmONAL
AND ADvANCED EXERCISES
433
Techniques
of
Integration
8.1
Integration by Parts 436
8.2
Trigonometric Integrals 444
8.3
Trigonometric Substitotions 449
8.4
Integration
of
Rational Functions by Partial Fractions
453
8.5
Integral Tables and Computer Algebra Systems
463
8.6
Numerical Integration 468
8.7
Improper Integrals 478
QuEsTIONS
TO
GUIDE YOUR REVIEW 489
PRACTICE EXERCISES 489
ADDmONAL
AND ADvANCED EXERCISES
491
First-Order
Differential
Equations
9.1
Solutions, Slope Fields, and Euler's Method 496
9.2 First-Order Linear Equations 504
9.3
Applications 510
9.4 Graphical Solutions
of
Autonomous Equations 516
9.5
Systems
of
Equations and Phase Planes 523
QuEsTIONS
TO
GUIDE YOUR REVIEW 529
PRACTICE EXERCISES 529
ADDmONAL
AND ADVANCED EXERCISES 530
Infinite
Sequences
and
Series
10.1
Sequences 532
10.2 Infinite Series 544
10.3
The Integral
Test
553
lOA
Comparison
Tests
558
10.5
The Ratio and Root
Thsts
563
435
496
532
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