Calculus

Cover Page......Page 1
Title Page......Page 2
Copyright Page......Page 3
Dedication......Page 4
CONTENTS......Page 5
About the Authors......Page 7
Preface......Page 8
Media and Supplements......Page 10
Features......Page 12
Acknowledgments......Page 15
To the Student......Page 17
Chapter 0 Functions and Precalculus......Page 18
 0.1 Functions and Graphs......Page 19
 0.2 Operations, Transformations, and Inverses......Page 36
 0.3 Algebraic Functions......Page 52
 0.4 Exponential and Trigonometric Functions......Page 64
 0.5 Logic and Mathematical Thinking......Page 80
 Chapter Review, Self-Test, and Capstones......Page 90
Chapter 1 Limits......Page 94
 1.1 An Intuitive Introduction to Limits......Page 95
 1.2 Formal Definition of Limit......Page 107
 1.3 Delta–Epsilon Proofs......Page 117
 1.4 Continuity and Its Consequences......Page 126
 1.5 Limit Rules and Calculating Basic Limits......Page 140
 1.6 Infinite Limits and Indeterminate Forms......Page 155
 Chapter Review, Self-Test, and Capstones......Page 169
Chapter 2 Derivatives......Page 172
 2.1 An Intuitive Introduction to Derivatives......Page 173
 2.2 Formal Definition of the Derivative......Page 186
 2.3 Rules for Calculating Basic Derivatives......Page 204
 2.4 The Chain Rule and Implicit Differentiation......Page 218
 2.5 Derivatives of Exponential and Logarithmic Functions......Page 229
 2.6 Derivatives of Trigonometric and Hyperbolic Functions......Page 241
 Chapter Review, Self-Test, and Capstones......Page 253
Chapter 3 Applications of the Derivative......Page 256
 3.1 The Mean Value Theorem......Page 257
 3.2 The First Derivative and Curve Sketching......Page 267
 3.3 The Second Derivative and Curve Sketching......Page 281
 3.4 Optimization......Page 295
 3.5 Related Rates......Page 308
 3.6 L’Hôpital’s Rule......Page 319
 Chapter Review, Self-Test, and Capstones......Page 330
Chapter 4 Definite Integrals......Page 332
 4.1 Addition and Accumulation......Page 334
 4.2 Riemann Sums......Page 345
 4.3 Definite Integrals......Page 359
 4.4 Indefinite Integrals......Page 371
 4.5 The Fundamental Theorem of Calculus......Page 381
 4.6 Areas and Average Values......Page 392
 4.7 Functions Defined by Integrals......Page 405
 Chapter Review, Self-Test, and Capstones......Page 419
Chapter 5 Techniques of Integration......Page 424
 5.1 Integration by Substitution......Page 425
 5.2 Integration by Parts......Page 437
 5.3 Partial Fractions and Other Algebraic Techniques......Page 449
 5.4 Trigonometric Integrals......Page 461
 5.5 Trigonometric Substitution......Page 471
 5.6 Improper Integrals......Page 484
 5.7 Numerical Integration......Page 497
 Chapter Review, Self-Test, and Capstones......Page 511
Chapter 6 Applications of Integration......Page 516
 6.1 Volumes by Slicing......Page 517
 6.2 Volumes by Shells......Page 531
 6.3 Arc Length and Surface Area......Page 543
 6.4 Real-World Applications of Integration......Page 559
 6.5 Differential Equations......Page 576
 Chapter Review, Self-Test, and Capstones......Page 590
Chapter 7 Sequences and Series......Page 594
 7.1 Sequences......Page 596
 7.2 Limits of Sequences......Page 611
 7.3 Series......Page 623
 7.4 Introduction to Convergence Tests......Page 634
 7.5 Comparison Tests......Page 643
 7.6 The Ratio and Root Tests......Page 650
 7.7 Alternating Series......Page 658
 Chapter Review, Self-Test, and Capstones......Page 672
Chapter 8 Power Series......Page 676
 8.1 Power Series......Page 677
 8.2 Maclaurin Series and Taylor Series......Page 689
 8.3 Convergence of Power Series......Page 698
 8.4 Differentiating and Integrating Power Series......Page 711
 Chapter Review, Self-Test, and Capstones......Page 720
Chapter 9 Parametric Equations, Polar Coordinates, and Conic Sections......Page 724
 9.1 Parametric Equations......Page 725
 9.2 Polar Coordinates......Page 741
 9.3 Graphing Polar Equations......Page 750
 9.4 Computing Arc Length and Area with Polar Functions......Page 766
 9.5 Conic Sections......Page 775
 Chapter Review, Self-Test, and Capstones......Page 791
Chapter 10 Vectors......Page 794
 10.1 Cartesian Coordinates......Page 795
 10.2 Vectors......Page 809
 10.3 Dot Product......Page 820
 10.4 Cross Product......Page 831
 10.5 Lines in Three-Dimensional Space......Page 844
 10.6 Planes......Page 853
 Chapter Review, Self-Test, and Capstones......Page 864
Chapter 11 Vector Functions......Page 868
 11.1 Vector-valued Functions......Page 869
 11.2 The Calculus of Vector Functions......Page 879
 11.3 Unit Tangent and Unit Normal Vectors......Page 891
 11.4 Arc Length Parametrizations and Curvature......Page 898
 11.5 Motion......Page 908
 Chapter Review, Self-Test, and Capstones......Page 917
Chapter 12 Multivariable Functions......Page 920
 12.1 Functions of Two and Three Variables......Page 921
 12.2 Open Sets, Closed Sets, Limits, and Continuity......Page 936
 12.3 Partial Derivatives......Page 950
 12.4 Directional Derivatives and Differentiability......Page 963
 12.5 The Chain Rule and the Gradient......Page 972
 12.6 Extreme Values......Page 983
 12.7 Lagrange Multipliers......Page 994
 Chapter Review, Self-Test, and Capstones......Page 1004
Chapter 13 Double and Triple Integrals......Page 1008
 13.1 Double Integrals over Rectangular Regions......Page 1009
 13.2 Double Integrals over General Regions......Page 1023
 13.3 Double Integrals using Polar Coordinates......Page 1034
 13.4 Applications of Double Integrals......Page 1046
 13.5 Triple Integrals......Page 1058
 13.6 Integration using Cylindrical and Spherical Coordinates......Page 1075
 13.7 Jacobians and Change of Variables......Page 1086
 Chapter Review, Self-Test, and Capstones......Page 1098
Chapter 14 Vector Analysis......Page 1102
 14.1 Vector Fields......Page 1103
 14.2 Line Integrals......Page 1114
 14.3 Surfaces and Surface Integrals......Page 1126
 14.4 Green’s Theorem......Page 1139
 14.5 Stokes’ Theorem......Page 1151
 14.6 The Divergence Theorem......Page 1160
 Chapter Review, Self-Test, and Capstones......Page 1169
Answers to Odd-Numbered Problems......Page 1174
Index......Page 1264