i Calculus T H I R D E D I T I O N DENNIS D. BERKEY Boston University PAUL BLANCHARD Boston University SAUNDERS COLLEGE PUBLISHING Harcourt Brace Jovanovich College Publishers Fort Worth Philadelphia San Diego New York Orlando Austin San Antonio Toronto Montreal London Sydney Tokyo
* Contents Unit One Preliminary Notions xxxii Chapter 1 Review of Precalculus Concepts 2 1.1 The Real Number System 2 1.2 The Coordinate Plane, Distance, and Circles 1.3 Linear Equations 16 1.4 Functions 24 1.5 Trigonometrie Functions 39 Summary Outline 49 Review Exercises 51 Readiness Test 54 11 Chapter 2 Limits of Functions 55 2.1 Tangents, Areas, and Limits 55 2.2 Limits of Functions 62 2.3 The Formal Definition of Limit 71 2.4 Properties of Limits 77 2.5 One-Sided Limits 88 2.6 Continuity 95 Summary Outline 108 Review Exercises 110 Unit Two Differentiation 114 Chapter 3 The Derivative 116 3.1 The Derivative as a Function 116 3.2 Rules for Calculating Derivatives 125 3.3 The Derivative as a Rate of Change 132 3.4 Derivatives of the Trigonometrie Functions 139 3.5 The Chain Rule 144 3.6 Related Rates 152 3.7 Linear Approximation 160 3.8 Implicit Differentiation and Rational Power Functions 3.9 Higher Order Derivatives 177 3.10 Newton's Method 182 Summary Outline 187 Review Exercises 188 168 XXV
xxvi Contents Chapter 4 Applications of the Derivative 191 4.1 Extreme Values 191 4.2 Applied Maximum-Minimum Problems 203 4.3 The Mean Value Theorem 215 4.4 Increasing and Decreasing Functions 221 4.5 Significance of the Second Derivative: Concavity 237 4.6 Large-Scale Behavior: Asymptotes 247 4.7 Curve Sketching 258 4.8 Linear Approximation Revisited 265 Summary Outline 270 Review Exercises 271 Chapter 5 Antidifferentiation 274 5.1 Antiderivatives 274 5.2 Finding Antiderivatives by Substitution 287 5.3 Differential Equations 295 Summary Outline 304 Review Exercises 305 Unit Three Chapter 6 Integration 306 The Definite Integral 308 6.1 Area and Summation 308 6.2 Riemann Sums: The Definite Integral 322 6.3 The Fundamental Theorem of Calculus 339 6.4 Substitution in Definite Integrals 353 6.5 Finding Areas by Integration 357 6.6 Numerical Approximation of the Definite Integral 367 Summary Outline 378 Review Exercises 380 Chapter 7 Applications of the Definite Integral 383 7.1 Calculating Volumes by Slicing 384 7.2 Calculating Volumes by the Method of Cylindrical Shells 7.3 Are Length and Surface Area 402 7.4 Distance and Velocity 411 7.5 Hydrostatic Pressure 417 7.6 Work 422 7.7 Moments and Centers of Mass 428 7.8 The Theorem of Pappus 438 Summary Outline 442 Review Exercises 444 396
Unit Four Chapter 8 The Transcendental Functions 446 Logarithmic and Exponential Functions 448 8.1 Review of Logarithms and Inverse Functions 448 8.2 The Natural Logarithm Function 457 8.3 The Natural Exponential Function 467 8.4 Exponentials and Logs to Other Bases 476 8.5 Exponential Growth and Decay 483 Summary Outline 490 Review Exercises 491 Chapter 9 Trigonometrie and Inverse Trigonometrie Functions 494 9.1 Integrals of the Trigonometrie Functions 494 9.2 Integrals Involving Products of Trigonometrie Functions 9.3 The Inverse Trigonometrie Functions 505 9.4 Derivatives of the Inverse Trigonometrie Functions 511 9.5 The Hyperbolic Functions 517 Summary Outline 524 Review Exercises 526 498 Chapter 10 Techniques of Integration 528 10.1 Integration by Parts 529 10.2 Trigonometrie Substitutions 536 10.3 Integrals Involving Quadratic Expressions 10.4 The Method of Partial Fractions 547 10.5 Miscellaneous Substitutions 557 10.6 The Use of Integral Tables 561 Summary Outline 563 Review Exercises 563 543 Chapter 11 L'Höpital's Rule and Improper Integrals 566 11.1 Indeterminate Forms: l'höpital's Rule 11.2 Other Indeterminate Forms 574 11.3 Improper Integrals 579 Summary Outline 589 Review Exercises 589 566
xxviii Contents Unit Five Chapter 12 The Theory of Infinite Series 592 The Theory of Infinite Series 594 12.1 Infinite Sequences 594 12.2 More on Infinite Sequences 603 12.3 Infinite Series 609 12.4 The Integral Test 621 12.5 The Comparison Test 627 12.6 The Ratio and Root Tests 633 12.7 Absolute and Conditional Convergence 639 Summary Outline 648 Review Exercises 650 Chapter 13 Taylor Polynomials and Power Series 652 13.1 The Approximation Problem and Taylor Polynomials 13.2 Taylor's Theorem 660 13.3 Applications of Taylor's Theorem 664 13.4 Power Series 669 13.5 Differentiation and Integration of Power Series 676 13.6 Taylor and Maclaurin Series 682 Summary Outline 692 Review Exercises 694 652 Unit Six Chapter 14 Geometry in the Plane and in Space 696 The Conic Sections 698 14.1 Parabolas 699 14.2 The Ellipse 704 14.3 TheHyperbola 711 14.4 Rotation of Axes 717 Summary Outline 720 Review Exercises 721 Chapter 15 Polar Coordinates and Parametric Equations 722 15.1 The Polar Coordinate System 722 15.2 Graphing Techniques for Polar Equations 731 15.3 Calculating Area in Polar Coordinates 737 15.4 Parametric Equations in the Plane 745 15.5 Are Length and Surface Area Revisited 758 Summary Outline 764 Review Exercises 765
V Contents xxix Chapter 16 Vectors, Lines, and Planes 767 16.1 Vectors in the Plane and in Space 767 16.2 Lines in Space 782 16.3 The Dot Product 789 16.4 The Cross Product 801 Summary Outline 811 Review Exercises 812 Chapter 17 Curves and Surfaces 814 17.1 Vector-Valued Functions; Curves in Space 814 17.2 Derivatives and Integrals of Vector-Valued Functions 17.3 Curves: Tangents and Are Length 831 17.4 Velocity and Acceleration 840 17.5 Curvature 848 17.6 Surfaces 856 17.7 Cylindrical and Spherical Coordinates 869 Summary Outline 874 Review Exercises 876 823 Unit Seven Chapter 18 Calculus in Higher Dimensions and Differential Equations 880 Differentiation for Functions of Several Variables 882 18.1 Functions of Several Variables 882 18.2 Partial Differentiation 892 18.3 Tangent Planes 904 18.4 Relative and Absolute Extrema 908 18.5 Approximation and Differentiability 918 18.6 Chain Rules 926 18.7 Directional Derivatives: The Gradient 933 18.8 Constrained Extrema: The Method of Lagrange Multipliers 946 18.9 Reconstructing a Function from its Gradient 953 Summary Outline 959 Review Exercises 961 Chapter 19 Double and Triple Integrals 964 19.1 The Double Integral over a Rectangle 964 19.2 Double Integrals over More General Regions 973 19.3 Double Integrals in Polar Coordinates 985 19.4 Calculating Areas and Centers of Mass 993 19.5 Surface Area 999 19.6 Triple Integrals 1004 19.7 Triple Integrals in Cylindrical and Spherical Coordinates 19.8 Change of Variables and Jacobians 1023 Summary Outline 1031 Review Exercises 1033 1015
xxx Contents Chapter 20 Chapter 21 Vector Analysis 1036 20.1 Vector Fields 1036 20.2 Work and Line Integrals 1041 20.3 Line Integrals: Independence of Path 1053 20.4 Green's Theorem 1061 20.5 Surface Integrals 1070 20.6 Stokes' Theorem 1084 20.7 The Divergence Theorem 1096 Summary Outline 1105 Review Exercises 1107 Differential Equations 1109 21.1 First Order Linear Differential Equations 1109 21.2 Exact Equations 1117 21.3 Second Order Linear Equations 1121 21.4 Nonhomogeneous Linear Equations 1128 21.5 Power Series Solutions of Differential Equations 1137 21.6 Approximating Solutions of Differential Equations 1142 Summary Outline 1147 Review Exercises 1148 Appendix I Calculus and the Computer A.l Appendix II Additional Proofs A.30 Appendix III Complex Numbers A.40 Appendix IV Newton' s Method in the Complex Plane A.45 Appendix V Calculus and the Graphing Calculator A.55 Appendix VI Tables of Transcendental Values A.93 Appendix VII Table of Integrals A.95 Appendix VIII Geometry Formulas A.100 Answers to Selected Exercises A.102 Index 1.1