' ' ' ' i! I! I! 11 ' SALAS AND HILLE'S CALCULUS I ;' 1 1 ONE VARIABLE SEVENTH EDITION REVISED BY GARRET J. ETGEI y.-'' ' / ' ' ' / / // X / / / /-.-.,</' SA ::: T ::; ;T: : r f -. 1 -z-'^*""'j- --' ; ; " :; ;,,,,, i,., ; ', i ; JOHN WILEY & SONS, INC. ; ;' ; New York Chichester Brisbane Toronto Singapore t: ~ )
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 XV
XVi PREFACE CHAPTER 3 DIFFERENTIATION 129 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 y, CHAPTER 5 INTEGRATION 277 _r fl. /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) = f* a f(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 w-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 XVII 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 yja 2 ± x 2 and yjx 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) * Denotes optional section.
xviii 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 (<b,b) 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 (co/ 00 ); Other Indeterminate Forms 673 10.7 Improper Integrals 680 Chapter Highlights 690 Projects and Explorations Using Technology 691 CHAPTER 11 INFINITE SERIES 693.1 Sigma Notation 693.2 Infinite Series 696.3 The Integral Test; Comparison Theorems 706.4 The Root Test; the Ratio Test 715.5 Absolute and Conditional Convergence; Alternating Series 720 1.6 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 XIX Chapter Highlights 768 Projects and Explorations Using Technology 769 APPENDIX A SOME SUPPORT TOPICS A-l A.I Rotation of Axes; Equations of Second Degree 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