Multivariable Calculus

About the Authors xi Preface xiii Functions, Graphs, and Models 1(52) Functions and Mathematical Modeling 2(10) Project: A Square Wading Pool 11(1) Graphs of Equations and Functions 12(12) Project: A Broken Tree 23(1) Polynomials and Algebraic Functions 24(9) Project: A Leaning Ladder 33(1) Transcendental Functions 33(12) Project: A Spherical Asteroid 45(1) Preview: What Is Calculus? 45(8) Review: Definitions and Concepts 48(5) Prelude to Calculus 53(48) Tangent Lines and Slope Predictors 54(9) Project: Numerical Slope Investigations 63(1) The Limit Concept 63(12) Project: Limits, Slopes, and Logarithms 74(1) More About Limits 75(13) Project: Numerical Epsilon-Delta Limit Investigations 87(1) The Concept of Continuity 88(13) Review: Definitions, Concepts, Results 99(2) The Derivative 101(92) The Derivative and Rates of Change 102(13) Basic Differentiation Rules 115(11) The Chain Rule 126(7) Derivatives of Algebraic Functions 133(9) Maxima and Minima of Functions on Closed Intervals 142(10) Project: When is Your Coffee Cup Stablest? 150(2) Applied Optimization Problems 152(13) Derivatives of Trigonometric Functions 165(11) Successive Approximations and Newton's Method 176(17) Project: How Deep Does a Floating Ball Sink? 188(1) Review: Formulas, Concepts, Definitions 189(4) Additional Applications of the Derivative 193(78) Implicit Functions and Related Rates 194(10) Project: Investigating the Folium of Descartes 203(1) Increments, Differentials, and Linear Approximation 204(8) Increasing and Decreasing Functions and the Mean Value Theorem 212(10) The First Derivative Test and Applications 222(10) Project: Making a Candy Box With Lid 232(1) Simple Curve Sketching 232(10) Higher Derivatives and Concavity 242(14) Curve Sketching and Asymptotes 256(15) Project: Locating Special Points on Exotic Graphs 267(1) Review: Definitions, Concepts, Results 267(4) The Integral 271(94) Introduction 272(1) Antiderivatives and Initial Value Problems 272(14) Elementary Area Computations 286(12) Riemann Sums and the Integral 298(10) Project: Calculator/Computer Riemann Sums 307(1) Evaluation of Integrals 308(10) The Fundamental Theorem of Calculus 318(10) Integration by Substitution 328(8) Areas of Plane Regions 336(10) Numerical Integration 346(19) Project: Trapezoidal and Simpson Approximations 359(2) Review: Definitions, Concepts, Results 361(4) Applications of the Integral 365(62) Riemann Sum Approximations 366(10) Volumes by the Method of Cross Sections 376(11) Volumes by the Method of Cylindrical Shells 387(9) Project: Design Your Own Ring! 395(1) Arc Length and Surface Area of Revolution 396(9) Force and Work 405(11) Centroids of Plane Regions and Curves 416(11) Review: Definitions, Concepts, Results 423(4) Calculus of Transcendental Functions 427(62) Exponential and Logarithmic Functions 428(13) Project: Discovering the Number e for Yourself 441(1) Indeterminate Forms and L'Hopital's Rule 441(8) More Indeterminate Forms 449(6) The Natural Logarithm as an Integral 455(12) Project: Natural Functional Equations 466(1) Inverse Trigonometric Functions 467(10) Hyperbolic Functions 477(12) Review: Formulas, Concepts, Definitions 485(4) Techniques of Integration 489(56) Introduction 490(1) Integral Tables and Simple Substitutions 490(4) Integration by Parts 494(7) Trigonometric Integrals 501(7) Rational Functions and Partial Fractions 508(7) Trigonometric Substitution 515(6) Integrals Involving Quadratic Polynomials 521(5) Improper Integrals 526(19) Summary 539(6) Differential Equations 545(78) Simple Equations and Models 546(12) Slope Fields and Euler's Method 558(10) Project: Computer-Assisted Slope Fields and Euler's Method 567(1) Separable Equations and Applications 568(7) Linear Equations and Applications 575(12) Population Models 587(11) Project: Predator-Prey Equations and Your Own Game Preserve 597(1) Linear Second-Order Equations 598(9) Mechanical Vibrations 607(16) Review: Definitions, Concepts, Results 618(5) Polar Coordinates and Parametric Curves 623(58) Analytic Geometry and the Conic Sections 624(5) Polar Coordinates 629(9) Area Computations in Polar Coordinates 638(5) Parametric Curves 643(10) Project:: Trochoid Investigations 652(1) Integral Computations with Parametric Curves 653(8) Project:: Moon Orbits and Race Tracks 660(1) Conic Sections and Applications 661(20) Review: Concepts and Definitions 679(2) Infinite Series 681(90) Introduction 682(1) Infinite Sequences 682(9) Project:: Nested Radicals and Continued Fractions 691(1) Infinite Series and Convergence 691(11) Project:: Numerical Summation and Geometric Series 701(1) Taylor Series and Taylor Polynomials 702(13) Project:: Calculating Logarithms on a Deserted Island 715(1) The Integral Test 715(7) Project:: The Number π, Once and for All 722(1) Comparison Tests for Positive-Term Series 722(6) Alternating Series and Absolute Convergence 728(9) Power Series 737(13) Power Series Computations 750(8) Project:: Calculating Trigonometric Functions on a Deserted Island 758(1) Series Solutions of Differential Equations 758(13) Review: Definitions, Concepts, Results 767(4) Vectors, Curves, and Surfaces in Space 771(78) Vectors in the Plane 772(6) Three-Dimensional Vectors 778(10) The Cross Product of Vectors 788(8) Lines and Planes in Space 796(7) Curves and Motion in Space 803(14) Project:: Does a Pitched Baseball Really Curve? 816(1) Curvature and Acceleration 817(13) Cylinders and Quadric Surfaces 830(8) Cylindrical and Spherical Coordinates 838(11) Review: Definitions, Concepts, Results 845(4) Partial Differentiation 849(90) Introduction 850(1) Functions of Several Variables 850(10) Limits and Continuity 860(8) Partial Derivatives 868(10) Multivariable Optimization Problems 878(11) Increments and Linear Approximation 889(7) The Multivariable Chain Rule 896(11) Directional Derivatives and the Gradient Vector 907(11) Lagrange Multipliers and Constrained Optimization 918(9) Project:: Numerical Solution of Lagrange Multiplier Systems 927(1) Critical Points of Functions of Two Variables 927(12) Project:: Critical Point Investigations 935(1) Review: Definitions, Concepts, Results 936(3) Multiple Integrals 939(74) Double Integrals 940(7) Project:: Midpoint Sums Approximating Double Integrals 947(1) Double Integrals over More General Regions 947(7) Area and Volume by Double Integration 954(7) Double Integrals in Polar Coordinates 961(7) Applications of Double Integrals 968(11) Project:: Optimal Design of Downhill Race-Car Wheels 978(1) Triple Integrals 979(9) Project:: Archimedes' Floating Paraboloid 987(1) Integration in Cylindrical and Spherical Coordinates 988(8) Surface Area 996(5) Change of Variables in Multiple Integrals 1001(12) REVIEW: Definitions, Concepts, Results 1009(4) Vector Calculus 1013(1) Vector Fields 1014(5) Line Integrals 1019(11) The Fundamental Theorem and Independence of Path 1030(7) Green's Theorem 1037(10) Surface Integrals 1047(10) Project:: Surface Integrals and Rocket Nose Cones 1057(1) The Divergence Theorem 1057(8) Stokes' Theorem 1065 Review: Definitions, Concepts, Results 1072 Appendices A: Real Numbers and Inequalities 1(5) B: The Coordinate Plane and Straight Lines 6(7) C: Review of Trigonometry 13(6) D: Proofs of the Limit Laws 19(4) E: The Completeness of the Real Number System 23(5) F: Existence of the Integral 28(5) G: Approximations and Riemann Sums 33(3) H: L'Hopital's Rule and Cauchy's Mean Value Theorem 36(2) I: Proof of Taylor's Formula 38(1) J: Conic Sections as Sections of a Cone 39(1) K: Proof of the Linear Approximation Theorem 40(1) L: Units of Measurement and Conversion Factors 41(1) M: Formulas from Algebra, Geometry, and Trigonometry 42(2) N: The Greek Alphabet 44(1) Answers to Odd-Numbered Problems 45(23) References for Further Study 68 Index 1(1) Table of Integrals 1