Calculus of a Single Variable

910 Pages·2008·22.277 MB·English

by  Ron Larson, Bruce H. Edwards

Calculus
of a Single Variable
Ninth Edition
Ron Larson
The Pennsylvania State University
The Behrend College
Bruce H. Edwards
University of Florida
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Calculus of a Single Variable, Ninth Edition © 2010, 2006 Brooks/Cole, Cengage Learning
Larson/Edwards
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C
ontents
A Word from the Authors viii
Textbook Features xii
CHAPTER P Preparation for Calculus 1
P.1 Graphs and Models 2
P.2 Linear Models and Rates of Change 10
P.3 Functions and Their Graphs 19
P.4 Fitting Models to Data 31
Review Exercises 37
P.S. Problem Solving 39
CHAPTER 1 Limits and Their Properties 41
1.1 A Preview of Calculus 42
1.2 Finding Limits Graphically and Numerically 48
1.3 Evaluating Limits Analytically 59
1.4 Continuity and One-Sided Limits 70
1.5 Infinite Limits 83
SECTION PROJECT: Graphs and Limits of
Trigonometric Functions 90
Review Exercises 91
P.S. Problem Solving 93
CHAPTER 2 Differentiation 95
2.1 The Derivative and the Tangent Line Problem 96
2.2 Basic Differentiation Rules and Rates of Change 107
2.3 Product and Quotient Rules and Higher-Order Derivatives 119
2.4 The Chain Rule 130
2.5 Implicit Differentiation 141
SECTION PROJECT: Optical Illusions 148
2.6 Related Rates 149
Review Exercises 158
P.S. Problem Solving 161
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iv Contents
CHAPTER 3 Applications of Differentiation 163
3.1 Extrema on an Interval 164
3.2 Rolle’s Theorem and the Mean Value Theorem 172
3.3 Increasing and Decreasing Functions and
the First Derivative Test 179
SECTION PROJECT: Rainbows 189
3.4 Concavity and the Second Derivative Test 190
3.5 Limits at Infinity 198
3.6 A Summary of Curve Sketching 209
3.7 Optimization Problems 218
SECTION PROJECT: Connecticut River 228
3.8 Newton’s Method 229
3.9 Differentials 235
Review Exercises 242
P.S. Problem Solving 245
CHAPTER 4 Integration 247
4.1 Antiderivatives and Indefinite Integration 248
4.2 Area 259
4.3 Riemann Sums and Definite Integrals 271
4.4 The Fundamental Theorem of Calculus 282
SECTION PROJECT: Demonstrating the
Fundamental Theorem 296
4.5 Integration by Substitution 297
4.6 Numerical Integration 311
Review Exercises 318
P.S. Problem Solving 321
CHAPTER 5 Logarithmic, Exponential, and Other
Transcendental Functions 323
5.1 The Natural Logarithmic Function: Differentiation 324
5.2 The Natural Logarithmic Function: Integration 334
5.3 Inverse Functions 343
5.4 Exponential Functions: Differentiation and Integration 352
5.5 Bases Other Than e and Applications 362
SECTION PROJECT: Using Graphing Utilities to
Estimate Slope 372
5.6 Inverse Trigonometric Functions: Differentiation 373
5.7 Inverse Trigonometric Functions: Integration 382
Contents v
5.8 Hyperbolic Functions 390
SECTION PROJECT: St. Louis Arch 400
Review Exercises 401
P.S. Problem Solving 403
CHAPTER 6 Differential Equations 405
6.1 Slope Fields and Euler’s Method 406
6.2 Differential Equations: Growth and Decay 415
6.3 Separation of Variables and the Logistic Equation 423
6.4 First-Order Linear Differential Equations 434
SECTION PROJECT: Weight Loss 442
Review Exercises 443
P.S. Problem Solving 445
CHAPTER 7 Applications of Integration 447
7.1 Area of a Region Between Two Curves 448
7.2 Volume: The Disk Method 458
7.3 Volume: The Shell Method 469
SECTION PROJECT: Saturn 477
7.4 Arc Length and Surfaces of Revolution 478
7.5 Work 489
SECTION PROJECT: Tidal Energy 497
7.6 Moments, Centers of Mass, and Centroids 498
7.7 Fluid Pressure and Fluid Force 509
Review Exercises 515
P.S. Problem Solving 517
CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and
Improper Integrals 519
8.1 Basic Integration Rules 520
8.2 Integration by Parts 527
8.3 Trigonometric Integrals 536
SECTION PROJECT: Power Lines 544
8.4 Trigonometric Substitution 545
8.5 Partial Fractions 554
8.6 Integration by Tables and Other Integration Techniques 563
8.7 Indeterminate Forms and L’Hôpital’s Rule 569
8.8 Improper Integrals 580
Review Exercises 591
P.S. Problem Solving 593
vi Contents
CHAPTER 9 Infinite Series 595
9.1 Sequences 596
9.2 Series and Convergence 608
SECTION PROJECT: Cantor’s Disappearing Table 618
9.3 The Integral Test and p-Series 619
SECTION PROJECT: The Harmonic Series 625
9.4 Comparisons of Series 626
SECTION PROJECT: Solera Method 632
9.5 Alternating Series 633
9.6 The Ratio and Root Tests 641
9.7 Taylor Polynomials and Approximations 650
9.8 Power Series 661
9.9 Representation of Functions by Power Series 671
9.10 Taylor and Maclaurin Series 678
Review Exercises 690
P.S. Problem Solving 693
CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates 695
10.1 Conics and Calculus 696
10.2 Plane Curves and Parametric Equations 711
SECTION PROJECT: Cycloids 720
10.3 Parametric Equations and Calculus 721
10.4 Polar Coordinates and Polar Graphs 731
SECTION PROJECT: Anamorphic Art 740
10.5 Area and Arc Length in Polar Coordinates 741
10.6 Polar Equations of Conics and Kepler’s Laws 750
Review Exercises 758
P.S. Problem Solving 761
Contents vii
Appendix A Proofs of Selected Theorems A2
Appendix B Integration Tables A20
Answers to Odd-Numbered Exercises A25
Index A115
ADDITIONAL APPENDICES
Appendix C Precalculus Review (Online)
C.1 Real Numbers and the Real Number Line
C.2 The Cartesian Plane
C.3 Review of Trigonometric Functions
Appendix D Rotation and the General Second-Degree Equation (Online)
Appendix E Complex Numbers (Online)
Appendix F Business and Economic Applications (Online)
A
Word from the Authors
Welcome to the Ninth Edition of Calculus of a Single Variable! We are proud to
offer you a new and revised version of our textbook. Much has changed since we
wrote the first edition over 35 years ago. With each edition we have
listened to you, our users, and have incorporated many of your
suggestions for improvement.
6th
7th
9th 8th
Throughout the years, our objective has always been to write in a precise,
readable manner with the fundamental concepts and rules of calculus clearly defined
and demonstrated. When writing for students, we strive to offer features and
materials that enable mastery by all types of learners. For the instructors, we aim to
provide a comprehensive teaching instrument that employs proven pedagogical
techniques,freeing instructors to make the most efficient use of classroom time.
This revision brings us to a new level of change and improvement. For the past
several years, we’ve maintained an independent website—CalcChat.com—that
provides free solutions to all odd-numbered exercises in the text. Thousands of
students using our textbooks have visited the site for practice and help with their
homework. With the Ninth Edition, we were able to use information from
CalcChat.com,including which solutions students accessed most often,to help guide
the revision of the exercises. This edition of Calculuswill be the first calculus textbook
to use actual data from students.
We have also added a new feature called Capstoneexercises to this edition. These
conceptual problems synthesize key topics and provide students with a better under-
standing of each section’s concepts. Capstone exercises are excellent for
classroom discussion or test prep,and instructors may find value in integrating these
problems into their review of the section. These and other new features join our
time-tested pedagogy,with the goal of enabling students and instructors to make the
best use of this text.
We hope you will enjoy the Ninth Edition of Calculus of a Single Variable. As
always,we welcome comments and suggestions for continued improvements.
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