Smith R., Minton R. Calculus
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Calculus
Fourth Edition
ROBERT T. SMITH
Millersville University of Pennsylvania
ROLAND B. MINTON
Roanoke College
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CALCULUS, FOURTH EDITION
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,
New York, NY 10020. Copyright
c
2012 by The McGraw-Hill Companies, Inc. All rights reserved. Previous
editions
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Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or
broadcast for distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the
United States.
This book is printed on acid-free paper.
1234567890QVR/QVR 10987654321
ISBN 978–0–07–338311–8
MHID 0–07–338311–2
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Library of Congress Cataloging-in-Publication Data
Smith, Robert T. (Robert Thomas), 1955-
Calculus / Robert T. Smith, Roland B. Minton.— 4th ed.
p. cm.
Includes index.
ISBN 978–0–07–338311–8—ISBN 0–07–338311–2 (hard copy : alk. paper)
1. Transcendental functions—Textbooks. 2. Calculus—Textbooks.
I. Minton, Roland B., 1956– II. Title.
QA353.S649 2012
515
.22—dc22 2010030314
www.mhhe.com
ii
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DEDICATION
To Pam, Katie and Michael
To Jan, Kelly and Greg
And in memory of our parents:
George and Anne Smith
and
Paul and Mary Frances Minton
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About the Authors
Robert T. Smith is Professor of Mathematics and Dean of the School of Science and
MathematicsatMillersvilleUniversityofPennsylvania,wherehehasbeenafacultymember
since 1987. Prior to that, he was on the faculty at Virginia Tech. He earned his Ph.D. in
mathematics from the University of Delaware in 1982.
Professor Smith’s mathematical interests are in the application of mathematics to prob-
lemsinengineeringandthephysicalsciences.Hehaspublishedanumberofresearcharticles
on the applications of partial differential equations as well as on computational problems in
x-raytomography.HeisamemberoftheAmericanMathematicalSociety,theMathematical
Association of America, and the Society for Industrial and Applied Mathematics.
ProfessorSmith livesin Lancaster,Pennsylvania,with his wife Pam, his daughter Katie
and his son Michael. His ongoing extracurricular goal is to learn to play golf well enough
to not come in last in his annual mathematicians/statisticians tournament.
Roland B. Minton is Professor of Mathematics and Chair of the Departmentof Mathemat-
ics, Computer Science and Physics at Roanoke College, where he has taught since 1986.
Prior to that, he was on the faculty at Virginia Tech. He earned his Ph.D. from Clemson
University in 1982. He is the recipient of Roanoke College awards for teaching excellence
and professional achievement, as well as the 2005 Virginia Outstanding Faculty Award and
the 2008 George Polya Award for mathematics exposition.
Professor Minton’s current research program is in the mathematics of golf, especially
the analysis of ShotLink statistics. He has published articles on various aspects of sports
science,and co-authoredwithTimPenningsanarticle onPennings’dogElvis andhisability
to solve calculus problems. He is co-author of a technical monograph on control theory.
He has supervised numerous independent studies and held workshops for local high school
teachers. He is an active member of the Mathematical Association of America.
Professor Minton lives in Salem, Virginia, with his wife Jan and occasionally with his
daughter Kelly and son Greg when they visit. He enjoys playing golf when time permits
and watching sports events even when time doesn’t permit. Jan also teaches at Roanoke
College and is very active in mathematics education.
In addition to Calculus: Early Transcendental Functions, Professors Smith and Minton
are also coauthors of Calculus: Concepts and Connections
c
2006, and three earlier books
for McGraw-Hill Higher Education. Earlier editions of Calculus have been translated into
Spanish, Chinese and Korean and are in use around the world.
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Brief Table of Contents
CHAPTER 0
..
Preliminaries 1
CHAPTER 1
..
Limits and Continuity 47
CHAPTER 2
..
Differentiation 107
CHAPTER 3
..
Applications of Differentiation 173
CHAPTER 4
..
Integration 251
CHAPTER 5
..
Applications of the Definite Integral 315
CHAPTER 6
..
Exponentials, Logarithms and Other Transcendental
Functions 375
CHAPTER 7
..
Integration Techniques 421
CHAPTER 8
..
First-Order Differential Equations 491
CHAPTER 9
..
Infinite Series 531
CHAPTER 10
..
Parametric Equations and Polar Coordinates 625
CHAPTER 11
..
Vectors and the Geometry of Space 687
CHAPTER 12
..
Vector-Valued Functions 749
CHAPTER 13
..
Functions of Several Variables and Partial Differentiation 809
CHAPTER 14
..
Multiple Integrals 901
CHAPTER 15
..
Vector Calculus 977
CHAPTER 16
..
Second-Order Differential Equations 1073
APPENDIX A
..
Proofs of Selected Theorems A-1
APPENDIX B
..
Answers to Odd-Numbered Exercises A-13
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Table of Contents
Seeing the Beauty and Power of Mathematics xiii
Applications Index xxiv
CHAPTER 0
..
Preliminaries 1
0.1 The Real Numbers and the Cartesian Plane 2
.
The Real Number System and Inequalities
.
The Cartesian Plane
0.2 Lines and Functions 9
.
Equations of Lines
.
Functions
0.3 Graphing Calculators and Computer Algebra Systems 21
0.4 Trigonometric Functions 27
0.5 Transformations of Functions 36
CHAPTER 1
..
Limits and Continuity 47
1.1 A Brief Preview of Calculus: Tangent Lines and the Length
of a Curve 47
1.2 The Concept of Limit 52
1.3 Computation of Limits 59
1.4 Continuity and Its Consequences 68
.
The Method of Bisections
1.5 Limits Involving Infinity; Asymptotes 78
.
Limits at Infinity
1.6 Formal Definition of the Limit 87
.
Exploring the Definition of Limit Graphically
.
Limits Involving Infinity
1.7 Limits and Loss-of-Significance Errors 98
.
Computer Representation of Real Numbers
CHAPTER 2
..
Differentiation 107
2.1 Tangent Lines and Velocity 107
.
The General Case
.
Velocity
2.2 The Derivative 118
.
Alternative Derivative Notations
.
Numerical Differentiation
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Table of Contents vii
2.3 Computation of Derivatives: The Power Rule 127
.
The Power Rule
.
General Derivative Rules
.
Higher Order Derivatives
.
Acceleration
2.4 The Product and Quotient Rules 135
.
Product Rule
.
Quotient Rule
.
Applications
2.5 The Chain Rule 142
2.6 Derivatives of Trigonometric Functions 147
.
Applications
2.7 Implicit Differentiation 155
2.8 The Mean Value Theorem 162
CHAPTER 3
..
Applications of Differentiation 173
3.1 Linear Approximations and Newton’s Method 174
.
Linear Approximations
.
Newton’s Method
3.2 Maximum and Minimum Values 185
3.3 Increasing and Decreasing Functions 195
.
What You See May Not Be What You Get
3.4 Concavity and the Second Derivative Test 203
3.5 Overview of Curve Sketching 212
3.6 Optimization 223
3.7 Related Rates 234
3.8 Rates of Change in Economics and the Sciences 239
CHAPTER 4
..
Integration 251
4.1 Antiderivatives 252
4.2 Sums and Sigma Notation 259
.
Principle of Mathematical Induction
4.3 Area 266
4.4 The Definite Integral 273
.
Average Value of a Function
4.5 The Fundamental Theorem of Calculus 284
4.6 Integration by Substitution 292
.
Substitution in Definite Integrals
4.7 Numerical Integration 298
.
Simpson’s Rule
.
Error Bounds for Numerical Integration
CHAPTER 5
..
Applications of the Definite
Integral 315
5.1 Area Between Curves 315
5.2 Volume: Slicing, Disks and Washers 324
.
Volumes by Slicing
.
The Method of Disks
.
The Method of Washers
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viii Table of Contents
5.3 Volumes by Cylindrical Shells 338
5.4 Arc Length and Surface Area 345
.
Arc Length
.
Surface Area
5.5 Projectile Motion 352
5.6 Applications of Integration to Physics and Engineering 361
CHAPTER 6
..
Exponentials, Logarithms and Other
Transcendental Functions 375
6.1 The Natural Logarithm 375
.
Logarithmic Differentiation
6.2 Inverse Functions 384
6.3 The Exponential Function 391
.
Derivative of the Exponential Function
6.4 The Inverse Trigonometric Functions 399
6.5 The Calculus of the Inverse Trigonometric Functions 405
.
Integrals Involving the Inverse Trigonometric Functions
6.6 The Hyperbolic Functions 411
.
The Inverse Hyperbolic Functions
.
Derivation of the Catenary
CHAPTER 7
..
Integration Techniques 421
7.1 Review of Formulas and Techniques 422
7.2 Integration by Parts 426
7.3 Trigonometric Techniques of Integration 433
.
Integrals Involving Powers of Trigonometric Functions
.
Trigonometric Substitution
7.4 Integration of Rational Functions Using Partial Fractions 442
.
Brief Summary of Integration Techniques
7.5 Integration Tables and Computer Algebra Systems 450
.
Using Tables of Integrals
.
Integration Using a Computer Algebra System
7.6 Indeterminate Forms and L’H
ˆ
opital’s Rule 457
.
Other Indeterminate Forms
7.7 Improper Integrals 467
.
Improper Integrals with a Discontinuous Integrand
.
Improper Integrals with an Infinite Limit of Integration
.
A Comparison Test
7.8 Probability 479
CHAPTER 8
..
First-Order Differential
Equations 491
8.1 Modeling with Differential Equations 491
.
Growth and Decay Problems
.
Compound Interest
8.2 Separable Differential Equations 501
.
Logistic Growth