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Copyright, iv
Photo Credits, x
About the Authors, xi
Preface, xiii
Chapter 1. Functions, Graphs, and Models, 1
1.1. Functions and Mathematical Modeling, 2
1.2. Graphs of Equations and Functions, 12
1.3. Polynomials and Algebraic Functions, 24
1.4. Transcendental Functions, 34
1.5. Preview: What Is Calculus?, 45
Review: Understanding: Concepts and Definitions, 50
Objectives: Methods and Techniques, 50
Miscellaneous Problems, 51
Chapter 2. Prelude to Calculus, 53
2.1. Tangent Lines and Slope Predictors, 54
2.2. The Limit Concept, 64
2.3. More About Limits, 76
2.4. The Concept of Continuity, 90
Review: Understanding: Concepts and Definitions, 102
Objectives: Methods and Techniques, 102
Miscellaneous Problems, 103
Chapter 3. The Derivative, 105
3.1. The Derivative and Rates of Change, 106
3.2. Basic Differentiation Rules, 119
3.3. The Chain Rule, 130
3.4. Derivatives of Algebraic Functions, 138
3.5. Maxima and Minima of Functions on Closed Intervals, 146
3.6. Applied Optimization Problems, 156
3.7. Derivatives of Trigonometric Functions, 169
3.8. Exponential and Logarithmic Functions, 180
3.9. Implicit Differentiation and Related Rates, 194
3.10. Successive Approximations and Newton’s Method, 204
Review: Understanding: Concepts, Definitions, Formulas, 218
Objectives: Methods and Techniques, 219
Miscellaneous Problems, 219
Chapter 4. Additional Applications of the Derivative, 225
4.1. Introduction, 226
4.2. Increments, Differentials, and Linear Approximation, 226
4.3. Increasing and Decreasing Functions and the Mean Value Theorem, 235
4.4. The First Derivative Test and Applications, 246
4.5. Simple Curve Sketching, 256
4.6. Higher Derivatives and Concavity, 266
4.7. Curve Sketching and Asymptotes, 280
4.8. Indeterminate Forms and L’Hôpital’s Rule, 293
4.9. More Indeterminate Forms, 301
Review: Understanding: Concepts, Definitions, Results, 308
Objectives: Methods and Techniques, 309
Miscellaneous Problems, 309
Chapter 5. The Integral, 313
5.1. Introduction, 314
5.2. Antiderivatives and Initial Value Problems, 314
5.3. Elementary Area Computations, 329
5.4. Riemann Sums and the Integral, 341
5.5. Evaluation of Integrals, 352
5.6. The Fundamental Theorem of Calculus, 362
5.7. Integration by Substitution, 373
5.8. Areas of Plane Regions, 382
5.9. Numerical Integration, 393
Review: Understanding: Concepts, Definitions, Results, 409
Objectives: Methods and Techniques, 410
Miscellaneous Problems, 410
Chapter 6. Applications of the Integral, 413
6.1. Riemann Sum Approximations, 414
6.2. Volumes by the Method of Cross Sections, 425
6.3. Volumes by the Method of Cylindrical Shells, 437
6.4. Arc Length and Surface Area of Revolution, 446
6.5. Force and Work, 457
6.6. Centroids of Plane Regions and Curves, 468
6.7. The Natural Logarithm as an Integral, 476
6.8. Inverse Trigonometric Functions, 488
6.9. Hyperbolic Functions, 499
Review: Understanding: Concepts, Definitions, Formulas, 508
Objectives: Methods and Techniques, 510
Miscellaneous Problems, 510
Chapter 7. Techniques of Integration, 515
7.1. Introduction, 516
7.2. Integral Tables and Simple Substitutions, 516
7.3. Integration by Parts, 521
7.4. Trigonometric Integrals, 528
7.5. Rational Functions and Partial Fractions, 535
7.6. Trigonometric Substitution, 543
7.7. Integrals Involving Quadratic Polynomials, 549
7.8. Improper Integrals, 554
Summary: Integration Strategies, 568
Review: Understanding: Concepts and Techniques, 570
Objectives: Methods and Techniques, 570
Miscellaneous Problems, 571
Chapter 8. Differential Equations, 575
8.1. Simple Equations and Models, 576
8.2. Slope Fields and Euler’s Method, 588
8.3. Separable Equations and Applications, 599
8.4. Linear Equations and Applications, 607
8.5. Population Models, 619
8.6. Linear Second-Order Equations, 631
8.7. Mechanical Vibrations, 641
Review: Understanding: Concepts, Definitions, and Methods, 653
Objectives: Methods and Techniques, 654
Miscellaneous Problems, 654
Chapter 9. Polar Coordinates and Parametric Curves, 659
9.1. Analytic Geometry and the Conic Sections, 660
9.2. Polar Coordinates, 665
9.3. Area Computations in Polar Coordinates, 674
9.4. Parametric Curves, 680
9.5. Integral Computations with Parametric Curves, 690
9.6. Conic Sections and Applications, 698
Review: Understanding: Concepts, Definitions, and Formulas, 717
Objectives: Methods and Techniques, 718
Miscellaneous Problems, 718
Chapter 10. Infinite Series, 721
10.1. Introduction, 722
10.2. Infinite Sequences, 722
10.3. Infinite Series and Convergence, 732
10.4. Taylor Series and Taylor Polynomials, 743
10.5. The Integral Test, 757
10.6. Comparison Tests for Positive-Term Series, 765
10.7. Alternating Series and Absolute Convergence, 771
10.8. Power Series, 780
10.9. Power Series Computations, 794
10.10. Series Solutions of Differential Equations, 803
Review: Understanding: Concepts, Definitions, and Results, 812
Objectives: Methods and Techniques, 813
Miscellaneous Problems, 813
Chapter 11. Vectors, Curves, and Surfaces in Space, 817
11.1. Vectors in the Plane, 818
11.2. Three-Dimensional Vectors, 824
11.3. The Cross Product of Vectors, 835
11.4. Lines and Planes in Space, 843
11.5. Curves and Motion in Space, 851
11.6. Curvature and Acceleration, 865
11.7. Cylinders and Quadric Surfaces, 879
11.8. Cylindrical and Spherical Coordinates, 887
Review: Understanding: Concepts, Definitions, and Results, 895
Objectives: Methods and Techniques, 896
Miscellaneous Problems, 896
Chapter 12. Partial Differentiation, 899
12.1. Introduction, 900
12.2. Functions of Several Variables, 900
12.3. Limits and Continuity, 910
12.4. Partial Derivatives, 919
12.5. Multivariable Optimization Problems, 931
12.6. Increments and Linear Approximation, 942
12.7. The Multivariable Chain Rule, 951
12.8. Directional Derivatives and the Gradient Vector, 962
12.9. Lagrange Multipliers and Constrained Optimization, 973
12.10. Critical Points of Functions of Two Variables, 984
Review: Understanding: Concepts, Definitions, and Results, 993
Objectives: Methods and Techniques, 994
Miscellaneous Problems, 994
Chapter 13. Multiple Integrals, 997
13.1. Double Integrals, 998
13.2. Double Integrals over More General Regions, 1006
13.3. Area and Volume by Double Integration, 1013
13.4. Double Integrals in Polar Coordinates, 1020
13.5. Applications of Double Integrals, 1028
13.6. Triple Integrals, 1039
13.7. Integration in Cylindrical and Spherical Coordinates, 1049
13.8. Surface Area, 1057
13.9. Change of Variables in Multiple Integrals, 1064
Review: Understanding: Concepts, Definitions, and Results, 1073
Objectives: Methods and Techniques, 1074
Miscellaneous Problems, 1075
Chapter 14. Vector Calculus, 1079
14.1. Vector Fields, 1080
14.2. Line Integrals, 1086
14.3. The Fundamental Theorem and Independence of Path, 1097
14.4. Green’s Theorem, 1105
14.5. Surface Integrals, 1116
14.6. The Divergence Theorem, 1127
14.7. Stokes’ Theorem, 1136
Review: Understanding: Concepts, Definitions, and Results, 1143
Objectives: Methods and Techniques, 1145
Miscellaneous Problems, 1145
Appendices, A-1
A. Real Numbers and Inequalities, A-1
B. The Coordinate Plane and Straight Lines, A-6
C. Review of Trigonometry, A-13
D. Proofs of the Limit Laws, A-19
E. The Completeness of the Real Number System, A-23
F. Existence of the Integral, A-27
G. Approximations and Riemann Sums, A-33
H. L’Hôpital’s Rule and Cauchy’s Mean Value Theorem, A-35
I. Proof of Taylor’s Formula, A-38
J. Conic Sections as Sections of a Cone, A-39
K. Proof of the Linear Approximation Theorem, A-40
L. Units of Measurement and Conversion Factors, A-41
M. Formulas from Algebra, Geometry, and Trigonometry, A-41
N. The Greek Alphabet, A-43
T/F Study Guides—Hints & Answers, A-45
Answers to Odd-Numbered Problems, A-65
References for Further Study, A-155
Indices, I-1

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