CALCULUS SALAS AND HILLE'S REVISED BY GARRET J. ETGEI ONE VARIABLE SEVENTH EDITION ' ' ' ' i! I! I! 11 ' ;' 1 ::: T.

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1 ' ' ' ' 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 z-'^*""'j- --' ; ; " :; ;,,,,, i,., ; ', i ; JOHN WILEY & SONS, INC. ; ;' ; New York Chichester Brisbane Toronto Singapore t: ~ )

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 XV

3 XVi PREFACE CHAPTER 3 DIFFERENTIATION 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 y, CHAPTER 5 INTEGRATION 277 _r fl. /odd 5.1 An Area Problem; A Speed-Distance Problem The Definite Integral of a Continuous Function The Function F(x) = f* a f(t) dt The Fundamental Theorem of Integral Calculus Some Area Problems Indefinite Integrals The w-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 XVII 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 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 One-to-One Functions; Inverses The Logarithm Function, Part I 406 *- 7.3 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 yja 2 ± x 2 and yjx 2 a 2 ; Trigonometric Substitutions Partial Fractions Some Rationalizing Substitutions Numerical Integration 535 Chapter Highlights 545 Projects and Explorations Using Technology 547 (r, 0) * Denotes optional section.

5 xviii 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 (<b,b) 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 (co/ 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 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