DENNIS D. BERKEY. Boston University PAUL BLANCHARD. Boston University

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1 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

2 * Contents Unit One Preliminary Notions xxxii Chapter 1 Review of Precalculus Concepts The Real Number System The Coordinate Plane, Distance, and Circles 1.3 Linear Equations Functions Trigonometrie Functions 39 Summary Outline 49 Review Exercises 51 Readiness Test Chapter 2 Limits of Functions Tangents, Areas, and Limits Limits of Functions The Formal Definition of Limit Properties of Limits One-Sided Limits Continuity 95 Summary Outline 108 Review Exercises 110 Unit Two Differentiation 114 Chapter 3 The Derivative The Derivative as a Function Rules for Calculating Derivatives The Derivative as a Rate of Change Derivatives of the Trigonometrie Functions The Chain Rule Related Rates Linear Approximation Implicit Differentiation and Rational Power Functions 3.9 Higher Order Derivatives Newton's Method 182 Summary Outline 187 Review Exercises XXV

3 xxvi Contents Chapter 4 Applications of the Derivative Extreme Values Applied Maximum-Minimum Problems The Mean Value Theorem Increasing and Decreasing Functions Significance of the Second Derivative: Concavity Large-Scale Behavior: Asymptotes Curve Sketching Linear Approximation Revisited 265 Summary Outline 270 Review Exercises 271 Chapter 5 Antidifferentiation Antiderivatives Finding Antiderivatives by Substitution Differential Equations 295 Summary Outline 304 Review Exercises 305 Unit Three Chapter 6 Integration 306 The Definite Integral Area and Summation Riemann Sums: The Definite Integral The Fundamental Theorem of Calculus Substitution in Definite Integrals Finding Areas by Integration Numerical Approximation of the Definite Integral 367 Summary Outline 378 Review Exercises 380 Chapter 7 Applications of the Definite Integral Calculating Volumes by Slicing Calculating Volumes by the Method of Cylindrical Shells 7.3 Are Length and Surface Area Distance and Velocity Hydrostatic Pressure Work Moments and Centers of Mass The Theorem of Pappus 438 Summary Outline 442 Review Exercises

4 Unit Four Chapter 8 The Transcendental Functions 446 Logarithmic and Exponential Functions Review of Logarithms and Inverse Functions The Natural Logarithm Function The Natural Exponential Function Exponentials and Logs to Other Bases Exponential Growth and Decay 483 Summary Outline 490 Review Exercises 491 Chapter 9 Trigonometrie and Inverse Trigonometrie Functions Integrals of the Trigonometrie Functions Integrals Involving Products of Trigonometrie Functions 9.3 The Inverse Trigonometrie Functions Derivatives of the Inverse Trigonometrie Functions The Hyperbolic Functions 517 Summary Outline 524 Review Exercises Chapter 10 Techniques of Integration Integration by Parts Trigonometrie Substitutions Integrals Involving Quadratic Expressions 10.4 The Method of Partial Fractions Miscellaneous Substitutions The Use of Integral Tables 561 Summary Outline 563 Review Exercises Chapter 11 L'Höpital's Rule and Improper Integrals Indeterminate Forms: l'höpital's Rule 11.2 Other Indeterminate Forms Improper Integrals 579 Summary Outline 589 Review Exercises

5 xxviii Contents Unit Five Chapter 12 The Theory of Infinite Series 592 The Theory of Infinite Series Infinite Sequences More on Infinite Sequences Infinite Series The Integral Test The Comparison Test The Ratio and Root Tests Absolute and Conditional Convergence 639 Summary Outline 648 Review Exercises 650 Chapter 13 Taylor Polynomials and Power Series The Approximation Problem and Taylor Polynomials 13.2 Taylor's Theorem Applications of Taylor's Theorem Power Series Differentiation and Integration of Power Series Taylor and Maclaurin Series 682 Summary Outline 692 Review Exercises Unit Six Chapter 14 Geometry in the Plane and in Space 696 The Conic Sections Parabolas The Ellipse TheHyperbola Rotation of Axes 717 Summary Outline 720 Review Exercises 721 Chapter 15 Polar Coordinates and Parametric Equations The Polar Coordinate System Graphing Techniques for Polar Equations Calculating Area in Polar Coordinates Parametric Equations in the Plane Are Length and Surface Area Revisited 758 Summary Outline 764 Review Exercises 765

6 V Contents xxix Chapter 16 Vectors, Lines, and Planes Vectors in the Plane and in Space Lines in Space The Dot Product The Cross Product 801 Summary Outline 811 Review Exercises 812 Chapter 17 Curves and Surfaces Vector-Valued Functions; Curves in Space Derivatives and Integrals of Vector-Valued Functions 17.3 Curves: Tangents and Are Length Velocity and Acceleration Curvature Surfaces Cylindrical and Spherical Coordinates 869 Summary Outline 874 Review Exercises Unit Seven Chapter 18 Calculus in Higher Dimensions and Differential Equations 880 Differentiation for Functions of Several Variables Functions of Several Variables Partial Differentiation Tangent Planes Relative and Absolute Extrema Approximation and Differentiability Chain Rules Directional Derivatives: The Gradient Constrained Extrema: The Method of Lagrange Multipliers Reconstructing a Function from its Gradient 953 Summary Outline 959 Review Exercises 961 Chapter 19 Double and Triple Integrals The Double Integral over a Rectangle Double Integrals over More General Regions Double Integrals in Polar Coordinates Calculating Areas and Centers of Mass Surface Area Triple Integrals Triple Integrals in Cylindrical and Spherical Coordinates 19.8 Change of Variables and Jacobians 1023 Summary Outline 1031 Review Exercises

7 xxx Contents Chapter 20 Chapter 21 Vector Analysis Vector Fields Work and Line Integrals Line Integrals: Independence of Path Green's Theorem Surface Integrals Stokes' Theorem The Divergence Theorem 1096 Summary Outline 1105 Review Exercises 1107 Differential Equations First Order Linear Differential Equations Exact Equations Second Order Linear Equations Nonhomogeneous Linear Equations Power Series Solutions of Differential Equations 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

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