目次

0 BEFORE CALCULUS 1 0.1 Functions 1 0.2 New Functions from Old 15 0.3 Families of Functions 27 0.4 Inverse Functions; Inverse Trigonometric Functions 38 0.5 Exponential and Logarithmic Functions 52 1 LIMITS AND CONTINUITY 67 1.1 Limits (An Intuitive Approach) 67 1.2 Computing Limits 80 1.3 Limits at Infinity; End Behavior of a Function 89 1.4 Limits (Discussed More Rigorously) 100 1.5 Continuity 110 1.6 Continuity of Trigonometric, Exponential, and Inverse Functions 121 2 THE DERIVATIVE 131 2.1 Tangent Lines and Rates of Change 131 2.2 The Derivative Function 143 2.3 Introduction to Techniques of Differentiation 155 2.4 The Product and Quotient Rules 163 2.5 Derivatives of Trigonometric Functions 169 2.6 The Chain Rule 174 3 TOPICS IN DIFFERENTIATION 185 3.1 Implicit Differentiation 185 3.2 Derivatives of Logarithmic Functions 192 3.3 Derivatives of Exponential and Inverse Trigonometric Functions 197 3.4 Related Rates 204 3.5 Local Linear Approximation; Differentials 212 3.6 L???Hôpital???s Rule; Indeterminate Forms 219 4 THE DERIVATIVE IN GRAPHING AND APPLICATIONS 232 4.1 Analysis of Functions I: Increase, Decrease, and Concavity 232 4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 244 4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 254 4.4 Absolute Maxima and Minima 266 4.5 Applied Maximum and Minimum Problems 274 4.6 Rectilinear Motion 288 4.7 Newton???s Method 296 4.8 Rolle???s Theorem; Mean–Value Theorem 302 5 INTEGRATION 316 5.1 An Overview of the Area Problem 316 5.2 The Indefinite Integral 322 5.3 Integration by Substitution 332 5.4 The Definition of Area as a Limit; Sigma Notation 340 5.5 The Definite Integral 353 5.6 The Fundamental Theorem of Calculus 362 5.7 Rectilinear Motion Revisited Using Integration 376 5.8 Average Value of a Function and its Applications 385 5.9 Evaluating Definite Integrals by Substitution 390 5.10 Logarithmic and Other Functions Defined by Integrals 396 6 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 413 6.1 Area Between Two Curves 413 6.2 Volumes by Slicing; Disks and Washers 421 6.3 Volumes by Cylindrical Shells 432 6.4 Length of a Plane Curve 438 6.5 Area of a Surface of Revolution 444 6.6 Work 449 6.7 Moments, Centers of Gravity, and Centroids 458 6.8 Fluid Pressure and Force 467 6.9 Hyperbolic Functions and Hanging Cables 474 7 PRINCIPLES OF INTEGRAL EVALUATION 488 7.1 An Overview of Integration Methods 488 7.2 Integration by Parts 491 7.3 Integrating Trigonometric Functions 500 7.4 Trigonometric Substitutions 508 7.5 Integrating Rational Functions by Partial Fractions 514 7.6 Using Computer Algebra Systems and Tables of Integrals 523 7.7 Numerical Integration; Simpson???s Rule 533 7.8 Improper Integrals 547 8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 561 8.1 Modeling with Differential Equations 561 8.2 Separation of Variables 568 8.3 Slope Fields; Euler???s Method 579 8.4 First–Order Differential Equations and Applications 586 9 INFINITE SERIES 596 9.1 Sequences 596 9.2 Monotone Sequences 607 9.3 Infinite Series 614 9.4 Convergence Tests 623 9.5 The Comparison, Ratio, and Root Tests 631 9.6 Alternating Series; Absolute and Conditional Convergence 638 9.7 Maclaurin and Taylor Polynomials 648 9.8 Maclaurin and Taylor Series; Power Series 659 9.9 Convergence of Taylor Series 668 9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 678 10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS 692 10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 692 10.2 Polar Coordinates 705 10.3 Tangent Lines, Arc Length, and Area for Polar Curves 719 10.4 Conic Sections 730 10.5 Rotation of Axes; Second–Degree Equations 748 10.6 Conic Sections in Polar Coordinates 754 11 THREE–DIMENSIONAL SPACE; VECTORS 767 11.1 Rectangular Coordinates in 3–Space; Spheres; Cylindrical Surfaces 767 11.2 Vectors 773 11.3 Dot Product; Projections 785 11.4 Cross Product 795 11.5 Parametric Equations of Lines 805 11.6 Planes in 3–Space 813 11.7 Quadric Surfaces 821 11.8 Cylindrical and Spherical Coordinates 832 12 VECTOR–VALUED FUNCTIONS 841 12.1 Introduction to Vector–Valued Functions 841 12.2 Calculus of Vector–Valued Functions 848 12.3 Change of Parameter; Arc Length 858 12.4 Unit Tangent, Normal, and Binormal Vectors 868 12.5 Curvature 873 12.6 Motion Along a Curve 882 12.7 Kepler???s Laws of Planetary Motion 895 13 PARTIAL DERIVATIVES 906 13.1 Functions of Two or More Variables 906 13.2 Limits and Continuity 917 13.3 Partial Derivatives 927 13.4 Differentiability, Differentials, and Local Linearity 940 13.5 The Chain Rule 949 13.6 Directional Derivatives and Gradients 960 13.7 Tangent Planes and Normal Vectors 971 13.8 Maxima and Minima of Functions of Two Variables 977 13.9 Lagrange Multipliers 989 14 MULTIPLE INTEGRALS 1000 14.1 Double Integrals 1000 14.2 Double Integrals over Nonrectangular Regions 1009 14.3 Double Integrals in Polar Coordinates 1018 14.4 Surface Area; Parametric Surfaces 1026 14.5 Triple Integrals 1039 14.6 Triple Integrals in Cylindrical and Spherical Coordinates 1048 14.7 Change of Variables in Multiple Integrals; Jacobians 1058 14.8 Centers of Gravity Using Multiple Integrals 1071 15 TOPICS IN VECTOR CALCULUS 1084 15.1 Vector Fields 1084 15.2 Line Integrals 1094 15.3 Independence of Path; Conservative Vector Fields 1111 15.4 Green???s Theorem 1122 15.5 Surface Integrals 1130 15.6 Applications of Surface Integrals; Flux 1138 15.7 The Divergence Theorem 1148 15.8 Stokes??? Theorem 1158 A APPENDICES A GRAPHING FUNCTIONS USING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS A1 B TRIGONOMETRY REVIEW A13 C SOLVING POLYNOMIAL EQUATIONS A27 D SELECTED PROOFS A34 ANSWERS TO ODD–NUMBERED EXERCISES A45 INDEX I–1 WEB APPENDICES (online only) Available for download at www.wiley.com/go/global/anton and in WileyPLUS. E REAL NUMBERS, INTERVALS, AND INEQUALITIES F ABSOLUTE VALUE G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS H DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS I EARLY PARAMETRIC EQUATIONS OPTION J MATHEMATICAL MODELS K THE DISCRIMINANT L SECOND–ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS WEB PROJECTS: Expanding the Calculus Horizon (online only) Available for download at www.wiley.com/go/global/anton and in WileyPLUS. BLAMMO THE HUMAN CANNONBALL COMET COLLISION HURRICANE MODELING ITERATION AND DYNAMICAL SYSTEMS RAILROAD DESIGN ROBOTICS