/ / / ' ' ' / / ' '' ' - -'/-' yy xy xy' y- y/ /: - y/ yy y /'}' / >' // yy,y-' 'y '/' /y <-y' y^yy' // // // y >, I '////I! li II ii: ' MiIIIIIIH IIIIII!l ii r-i: V /- A' /; // ;.1 " SALAS AND HILLE'S CALCULUS ONE AND SEVERAL VARIABLES SEVENTH EDITION REVISED BY \ GARRET J. ETGEN JOHN WILEY & SONS, INC. New York! Chichester Brisbane Toronto Singapore
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 69 2.1 The Idea of Limit 69 2.2 Definition of Limit 81 2.3 Some Limit Theorems 92 2.4 Continuity 101 2.5 The Pinching Theorem; Trigonometric Limits 111 2.6 Two Basic Theorems 117 Chapter Highlights 124 Projects and Explorations Using Technology 125 xiii
XIV PREFACE CHAPTER 3 DIFFERENTIATION 129 h I 3.1 The Derivative 129 3.2 Some Differentiation Formulas 132 3.3 The djdx Notation; Derivatives of Higher Order 152 3.4 The Derivative as a Rate of Change 158 3.5 The Chain Rule 170 3.6 Differentiating the Trigonometric Functions 179 3.7 Implicit Differentiation; Rational Powers 185 3.8 Rates of Change Per Unit Time 192 3.9 Differentials; Newton-Raphson Approximations 199 Chapter Highlights 207 Projects and Explorations Using Technology 209 CHAPTER 4 THE MEAN-VALUE THEOREM AND APPLICATIONS 211 4.1 The Mean-Value Theorem 211 4.2 Increasing and Decreasing Functions 218 4.3 Local Extreme Values 226 4.4 Endpoint and Absolute Extreme Values 234 4.5 Some Max-Min Problems 242 4.6 Concavity and Points of Inflection 252 4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps 257 4.8 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 277 5.2 The Definite Integral of a Continuous Function 280 5.3 The Function F(x) = J x af(t) dt 289 5.4 The Fundamental Theorem of Integral Calculus 297 5.5 Some Area Problems 305 5.6 Indefinite Integrals 311 5.7 The K-Substitution; Change of Variables 319 5.8 Some Further Properties of the Definite Integral 329 5.9 Mean-Value Theorem for Integrals; Average Values 334 5.10 The Integral as the Limit of Riemann Sums 339 Chapter Highlights 343 Projects and Explorations Using Technology 344
CONTENTS XV CHAPTER 6 SOME APPLICATIONS OF THE INTEGRAL 347 6.1 More on Area 347 6.2 Volume by Parallel Cross Sections; Discs and Washers 353 6.3 Volume by the Shell Method 364 6.4 The Centroid of a Region; Pappus's Theorem on Volumes 371 6.5- The Notion of Work 379 6.6 Fluid Pressure and Fluid Forces 386 Chapter Highlights 391 Projects and Explorations Using Technology 392 (2,4) CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS 395 7.1 One-to-One Functions; Inverses 395 7.2 The Logarithm Function, Part I 406 7.3 The Logarithm Function, Part II 415 7.4 The Exponential Function 424 7.5 Arbitrary Powers; Other Bases; Estimating e 433 7.6 Exponential Growth and Decay 442 7.7 More on the Integration of the Trigonometric Functions 457 7.8 The Inverse Trigonometric Functions 461 7.9 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 489 8.1 Review 489 8.2 Integration by Parts 493 8.3 Powers and Products of Sine and Cosine 502 8.4 Other Trigonometric Powers 507 8.5 Integrals Involving %ja 2 ± x 2 and V* 2 ~ a 2 ', Trigonometric Substitutions 512 8.6 Partial Fractions 520 8.7 Some Rationalizing Substitutions 531 8.8 Numerical Integration 535 Chapter Highlights 545 Projects and Explorations Using Technology 547 (r, 0) 1 Denotes optional section.
XVi CONTENTS CHAPTER 9 CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS 549 9.1 Conic Sections 549 9.2 Polar Coordinates 568 9.3 Graphing in Polar Coordinates 575 *9.4 The Conic Sections in Polar Coordinates 582 9.5 The Intersection of Polar Curves 586 9.6 Area in Polar Coordinates 589 9.7 Curves Given Parametrically 595 9.8 Tangents to Curves Given Parametrically 605 9.9 Arc Length and Speed 612 9.10 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 637 10.1 The Least Upper Bound Axiom 637 10.2 Sequences of Real Numbers 642 10.3 Limit of a Sequence 650 10.4 Some Important Limits 663 10.5 The Indeterminate Form (0/0) 667 10.6 The Indeterminate Form C 00 / 00 ); Other Indeterminate Forms 673 10.7 Improper Integrals 680 Chapter Highlights 690 Projects and Explorations Using Technology 691 CHAPTER 11 INFINITE SERIES 693 11.1 Sigma Notation 693 11.2 Infinite Series 696 11.3 The Integral Test; Comparison Theorems 706 11.4 The Root Test; the Ratio Test 715 11.5 Absolute and Conditional Convergence; Alternating Series 720 11.6 Taylor Polynomials in x; Taylor Series in x 727 11.7 Taylor Polynomials and Taylor Series in x a 741 11.8 Power Series 745 11.9 Differentiation and Integration of Power Series 752 11.10 The Binomial Series 766
CONTENTS XVH Chapter Highlights 768 Projects and Explorations Using Technology 769 CHAPTER 12 VECTORS 773 12.1 Cartesian Space Coordinates 773 12.2 Displacements; Forces and Velocities; Vectors 778 12.3 The Dot Product 793 12.4 The Cross Product 805 12.5 Lines 814 12.6 Planes 824 12.7 Some Geometry by Vector Methods 835 Chapter Highlights 836 Projects and Explorations Using Technology 837 CHAPTER 13 VECTOR CALCULUS 839 13.1 Vector Functions 839 13.2 Differentiation Formulas 848 13.3 Curves 853 13.4 Arc Length 863 13.5 Curvilinear Motion; Vector Calculus in Mechanics 870 *13.6 Planetary Motion 882 13.7 Curvature 888 Chapter Highlights 896 Projects and Explorations Using Technology 897 CHAPTER 14 FUNCTIONS OF SEVERAL VARIABLES 899 14.1 Elementary Examples 899 14.2 A Brief Catalog of the Quadric Surfaces; Projections 903 14.3 Graphs; Level Curves and Level Surfaces 910 14.4 Partial Derivatives 919 14.5 Open Sets and Closed Sets 927 14.6 Limits and Continuity; Equality of Mixed Partials 931 Chapter Highlights 941 Projects and Explorations Using Technology 942 CHAPTER 15 GRADIENTS; EXTREME VALUES; DIFFERENTIALS 945 15.1 Differentiability and Gradient 945 15.2 Gradients and Directional Derivatives 953 15.3 The Mean-Value Theorem; Two Intermediate-Value Theorems 963
xviii CONTENTS 15.4 Chain Rules 968 15.5 The Gradient as a Normal; Tangent Lines and Tangent Planes 979 15.6 Maximum and Minimum Values 991 15.7 Second-Partials Test 999 15.8 Maxima and Minima with Side Conditions 1007 15.9 Differentials 1015 15.10 Reconstructing a Function From Its Gradient 1021 Chapter Highlights 1028 Projects and Explorations Using Technology 1030 CHAPTER 16 DOUBLE AND TRIPLE INTEGRALS 1033 16.1 Multiple-Sigma Notation 1033 16.2 The Double Integral Over a Rectangle 1036 16.3 The Double Integral Over a Region 1044 16.4 The Evaluation of Double Integrals by Repeated Integrals 1048 16.5 The Double Integral as a Limit of Riemann Sums; Polar Coordinates 1061 16.6 Some Applications of Double Integration 1068 16.7 Triple Integrals 1075 16.8 Reduction to Repeated Integrals 1082 16.9 Cylindrical Coordinates 1092 16.10 The Triple Integral as a Limit of Riemann Sums; Spherical Coordinates 1098 16.11 Jacobians; Changing Variables in Multiple Integration 1106 Chapter Highlights 1112 Projects and Explorations Using Technology 1113 z=l~(x 2 17.1 Line Integrals 1115 CHAPTER 17 LINE INTEGRALS AND SURFACE INTEGRALS 1115 17.2 The Fundamental Theorem for Line Integrals 1124 "17.3 Work-Energy Formula; Conservation of Mechanical Energy 1129 17.4 Another Notation for Line Integrals; Line Integrals with Respect to Arc Length 1132 17.5 Green's Theorem 1137 17.6 Parametrized Surfaces; Surface Area 1147 17.7 Surface Integrals 1159 17.8 The Vector Differential Operator V 1170 17.9 The Divergence Theorem 1176 17.10 Stokes's Theorem 1183 Chapter Highlights 1190 Projects and Explorations Using Technology 1192
CONTENTS XIX CHAPTER 18 ELEMENTARY DIFFERENTIAL EQUATIONS 1195 18.1 Introduction 1195 18.2 First Order Linear Differential Equations 1200 18.3 Separable Equations; Homogeneous Equations 1208 18.4 Exact Equations; Integrating Factors 1219 18.5 The Equation/' + ay' + by = 0 1224 18.6 The Equation/' + ay' + by = 4>(x) 1234 18.7 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