CALCULUS GARRET J. ETGEN SALAS AND HILLE'S. ' MiIIIIIIH. I '////I! li II ii: ONE AND SEVERAL VARIABLES SEVENTH EDITION REVISED BY \

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2 CHAPTER 1 INTRODUCTION.1 What Is Calculus? 1.2 Notations and Formulas from Elementary Mathematics 4.3 Inequalities 13.4 Coordinate Plane; Analytic Geometry 21.5 Functions 34.6 The Elementary Functions 44.7 Combinations of Functions 55.8 A Note on Mathematical Proof; Mathematical Induction 61 Chapter Highlights 65 Projects and Explorations Using Technology 66 CHAPTER 2 LIMITS AND CONTINUITY The Idea of Limit Definition of Limit Some Limit Theorems Continuity The Pinching Theorem; Trigonometric Limits Two Basic Theorems 117 Chapter Highlights 124 Projects and Explorations Using Technology 125 xiii

3 XIV PREFACE CHAPTER 3 DIFFERENTIATION 129 h I 3.1 The Derivative Some Differentiation Formulas The djdx Notation; Derivatives of Higher Order The Derivative as a Rate of Change The Chain Rule Differentiating the Trigonometric Functions Implicit Differentiation; Rational Powers Rates of Change Per Unit Time Differentials; Newton-Raphson Approximations 199 Chapter Highlights 207 Projects and Explorations Using Technology 209 CHAPTER 4 THE MEAN-VALUE THEOREM AND APPLICATIONS The Mean-Value Theorem Increasing and Decreasing Functions Local Extreme Values Endpoint and Absolute Extreme Values Some Max-Min Problems Concavity and Points of Inflection Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps Some Curve Sketching 264 Chapter Highlights 273 Projects and Explorations Using Technology 274 CHAPTER 5 INTEGRATION 277 /odd 5.1 An Area Problem; A Speed-Distance Problem The Definite Integral of a Continuous Function The Function F(x) = J x af(t) dt The Fundamental Theorem of Integral Calculus Some Area Problems Indefinite Integrals The K-Substitution; Change of Variables Some Further Properties of the Definite Integral Mean-Value Theorem for Integrals; Average Values The Integral as the Limit of Riemann Sums 339 Chapter Highlights 343 Projects and Explorations Using Technology 344

4 CONTENTS XV CHAPTER 6 SOME APPLICATIONS OF THE INTEGRAL More on Area Volume by Parallel Cross Sections; Discs and Washers Volume by the Shell Method The Centroid of a Region; Pappus's Theorem on Volumes The Notion of Work Fluid Pressure and Fluid Forces 386 Chapter Highlights 391 Projects and Explorations Using Technology 392 (2,4) CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS One-to-One Functions; Inverses The Logarithm Function, Part I The Logarithm Function, Part II The Exponential Function Arbitrary Powers; Other Bases; Estimating e Exponential Growth and Decay More on the Integration of the Trigonometric Functions The Inverse Trigonometric Functions The Hyperbolic Sine and Cosine 474 *7.10 The Other Hyperbolic Functions 479 Chapter Highlights 483 Projects and Explorations Using Technology 485 CHAPTER 8 TECHNIQUES OF INTEGRATION Review Integration by Parts Powers and Products of Sine and Cosine Other Trigonometric Powers Integrals Involving %ja 2 ± x 2 and V* 2 ~ a 2 ', Trigonometric Substitutions Partial Fractions Some Rationalizing Substitutions Numerical Integration 535 Chapter Highlights 545 Projects and Explorations Using Technology 547 (r, 0) 1 Denotes optional section.

5 XVi CONTENTS CHAPTER 9 CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS Conic Sections Polar Coordinates Graphing in Polar Coordinates 575 *9.4 The Conic Sections in Polar Coordinates The Intersection of Polar Curves Area in Polar Coordinates Curves Given Parametrically Tangents to Curves Given Parametrically Arc Length and Speed The Area of a Surface of Revolution; The Centroid of a Curve; Pappus's Theorem on Surface Area 621 *9.11 The Cycloid 629 Chapter Highlights 632 Projects and Explorations Using Technology 634 CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS The Least Upper Bound Axiom Sequences of Real Numbers Limit of a Sequence Some Important Limits The Indeterminate Form (0/0) The Indeterminate Form C 00 / 00 ); Other Indeterminate Forms Improper Integrals 680 Chapter Highlights 690 Projects and Explorations Using Technology 691 CHAPTER 11 INFINITE SERIES Sigma Notation Infinite Series The Integral Test; Comparison Theorems The Root Test; the Ratio Test Absolute and Conditional Convergence; Alternating Series Taylor Polynomials in x; Taylor Series in x Taylor Polynomials and Taylor Series in x a Power Series Differentiation and Integration of Power Series The Binomial Series 766

6 CONTENTS XVH Chapter Highlights 768 Projects and Explorations Using Technology 769 CHAPTER 12 VECTORS Cartesian Space Coordinates Displacements; Forces and Velocities; Vectors The Dot Product The Cross Product Lines Planes Some Geometry by Vector Methods 835 Chapter Highlights 836 Projects and Explorations Using Technology 837 CHAPTER 13 VECTOR CALCULUS Vector Functions Differentiation Formulas Curves Arc Length Curvilinear Motion; Vector Calculus in Mechanics 870 *13.6 Planetary Motion Curvature 888 Chapter Highlights 896 Projects and Explorations Using Technology 897 CHAPTER 14 FUNCTIONS OF SEVERAL VARIABLES Elementary Examples A Brief Catalog of the Quadric Surfaces; Projections Graphs; Level Curves and Level Surfaces Partial Derivatives Open Sets and Closed Sets Limits and Continuity; Equality of Mixed Partials 931 Chapter Highlights 941 Projects and Explorations Using Technology 942 CHAPTER 15 GRADIENTS; EXTREME VALUES; DIFFERENTIALS Differentiability and Gradient Gradients and Directional Derivatives The Mean-Value Theorem; Two Intermediate-Value Theorems 963

7 xviii CONTENTS 15.4 Chain Rules The Gradient as a Normal; Tangent Lines and Tangent Planes Maximum and Minimum Values Second-Partials Test Maxima and Minima with Side Conditions Differentials Reconstructing a Function From Its Gradient 1021 Chapter Highlights 1028 Projects and Explorations Using Technology 1030 CHAPTER 16 DOUBLE AND TRIPLE INTEGRALS Multiple-Sigma Notation The Double Integral Over a Rectangle The Double Integral Over a Region The Evaluation of Double Integrals by Repeated Integrals The Double Integral as a Limit of Riemann Sums; Polar Coordinates Some Applications of Double Integration Triple Integrals Reduction to Repeated Integrals Cylindrical Coordinates The Triple Integral as a Limit of Riemann Sums; Spherical Coordinates Jacobians; Changing Variables in Multiple Integration 1106 Chapter Highlights 1112 Projects and Explorations Using Technology 1113 z=l~(x Line Integrals 1115 CHAPTER 17 LINE INTEGRALS AND SURFACE INTEGRALS The Fundamental Theorem for Line Integrals 1124 "17.3 Work-Energy Formula; Conservation of Mechanical Energy Another Notation for Line Integrals; Line Integrals with Respect to Arc Length Green's Theorem Parametrized Surfaces; Surface Area Surface Integrals The Vector Differential Operator V The Divergence Theorem Stokes's Theorem 1183 Chapter Highlights 1190 Projects and Explorations Using Technology 1192

8 CONTENTS XIX CHAPTER 18 ELEMENTARY DIFFERENTIAL EQUATIONS Introduction First Order Linear Differential Equations Separable Equations; Homogeneous Equations Exact Equations; Integrating Factors The Equation/' + ay' + by = The Equation/' + ay' + by = 4>(x) Mechanical Vibrations 1245 Chapter Highlights 1255 Projects and Explorations Using Technology 1256 APPENDIX A SOME ADDITIONAL TOPICS A-l c<0 A. 1 Rotation of Axes; Equations of Second Degree A-l A.2 Determinants A-6 APPENDIX B SOME ADDITIONAL PROOFS A-ll B.I The Intermediate-Value Theorem A-ll B.2 The Maximum-Minimum Theorem A-12 B.3 Inverses A-13 B.4 The Integrability of Continuous Functions A-14 ANSWERS TO ODD-NUMBERED EXERCISES A-19 INDEX 1-1 TABLE OF INTEGRALS Inside Covers