Calculus

Preface ix Preliminaries 1(54) Real Numbers, Estimation, and Logic 1(7) Inequalities and Absolute Values 8(8) The Rectangular Coordinate System 16(8) Graphs of Equations 24(5) Functions and Their Graphs 29(6) Operations on Functions 35(6) Trigonometric Functions 41(10) Chapter Review 51(4) Review and Preview Problems 54(1) Limits 55(38) Introduction to Limits 55(6) Rigorous Study of Limits 61(7) Limit Theorems 68(5) Limits Involving Trigonometric Functions 73(4) Limits at Infinity; Infinite Limits 77(5) Continuity of Functions 82(8) Chapter Review 90(3) Review and Preview Problems 92(1) The Derivative 93(58) Two Problems with One Theme 93(7) The Derivative 100(7) Rules for Finding Derivatives 107(7) Derivatives of Trigonometric Functions 114(4) The Chain Rule 118(7) Higher-Order Derivatives 125(5) Implicit Differentiation 130(5) Related Rates 135(7) Differentials and Approximations 142(5) Chapter Review 147(4) Review and Preview Problems 150(1) Applications of the Derivative 151(64) Maxima and Minima 151(4) Monotonicity and Concavity 155(7) Local Extrema and Extrema on Open Intervals 162(5) Practical Problems 167(11) Graphing Functions Using Calculus 178(7) The Mean Value Theorem for Derivatives 185(5) Solving Equations Numerically 190(7) Antiderivatives 197(6) Introduction to Differential Equations 203(6) Chapter Review 209(6) Review and Preview Problems 214(1) The Definite Integral 215(60) Introduction to Area 215(9) The Definite Integral 224(8) The First Fundamental Theorem of Calculus 232(11) The Second Fundamental Theorem of Calculus and the Method of Substitution 243(10) The Mean Value Theorem for Integrals and the Use of Symmetry 253(7) Numerical Integration 260(10) Chapter Review 270(5) Review and Preview Problems 274(1) Applications of the Integral 275(50) The Area of a Plane Region 275(6) Volumes of Solids: Slabs, Disks, Washers 281(7) Volumes of Solids of Revolution: Shells 288(6) Length of a Plane Curve 294(7) Work and Fluid Force 301(7) Moments and Center of Mass 308(8) Probability and Random Variables 316(6) Chapter Review 322(3) Review and Preview Problems 324(1) Transcendental Functions 325(58) The Natural Logarithm Function 325(6) Inverse Functions and Their Derivatives 331(6) The Natural Exponential Function 337(5) General Exponential and Logarithmic Functions 342(5) Exponential Growth and Decay 347(8) First-Order Linear Differential Equations 355(4) Approximations for Differential Equations 359(6) The Inverse Trigonometric Functions and Their Derivatives 365(9) The Hyperbolic Functions and Their Inverses 374(6) Chapter Review 380(3) Review and Preview Problems 382(1) Techniques of Integration 383(40) Basic Integration Rules 383(4) Integration by Parts 387(6) Some Trigonometric Integrals 393(6) Rationalizing Substitutions 399(5) Integration of Rational Functions Using Partial Fractions 404(7) Strategies for Integration 411(8) Chapter Review 419(4) Review and Preview Problems 422(1) Indeterminate Forms and Improper Integrals 423(26) Indeterminate Forms of Type 0/0 423(5) Other Indeterminate Forms 428(5) Improper Integrals: Infinite Limits of Integration 433(9) Improper Integrals: Infinite Integrands 442(4) Chapter Review 446(3) Review and Preview Problems 448(1) Infinite Series 449(60) Infinite Sequences 449(6) Infinite Series 455(8) Positive Series: The Integral Test 463(5) Positive Series: Other Tests 468(6) Alternating Series, Absolute Convergence, and Conditional Convergence 474(5) Power Series 479(5) Operations on Power Series 484(5) Taylor and Maclaurin Series 489(8) The Taylor Approximation to a Function 497(7) Chapter Review 504(5) Review and Preview Problems 508(1) Conics and Polar Coordinates 509(46) The Parabola 509(4) Ellipses and Hyperbolas 513(10) Translation and Rotation of Axes 523(7) Parametric Representation of Curves in the Plane 530(7) The Polar Coordinate System 537(5) Graphs of Polar Equations 542(5) Calculus in Polar Coordinates 547(5) Chapter Review 552(3) Review and Preview Problems 554(1) Geometry in Space and Vectors 555(62) Cartesian Coordinates in Three-Space 555(5) Vectors 560(6) The Dot Product 566(8) The Cross Product 574(5) Vector-Valued Functions and Curvilinear Motion 579(10) Lines and Tangent Lines in Three-Space 589(4) Curvature and Components of Acceleration 593(10) Surfaces in Three-Space 603(6) Cylindrical and Spherical Coordinates 609(4) Chapter Review 613(4) Review and Preview Problems 616(1) Derivatives for Functions of Two or More Variables 617(58) Functions of Two or More Variables 617(7) Partial Derivatives 624(5) Limits and Continuity 629(6) Differentiability 635(6) Directional Derivatives and Gradients 641(6) The Chain Rule 647(5) Tangent Planes and Approximations 652(5) Maxima and Minima 657(9) The Method of Lagrange Multipliers 666(6) Chapter Review 672(3) Review and Preview Problems 674(1) Multiple Integrals 675(56) Double Integrals over Rectangles 675(5) Iterated Integrals 680(4) Double Integrals over Nonrectangular Regions 684(7) Double Integrals in Polar Coordinates 691(5) Applications of Double Integrals 696(4) Surface Area 700(6) Triple Integrals in Cartesian Coordinates 706(7) Triple Integrals in Cylindrical and Spherical Coordinates 713(5) Change of Variables in Multiple Integrals 718(10) Chapter Review 728(3) Review and Preview Problems 730(1) Vector Calculus 731 Vector Fields 731(4) Line Integrals 735(7) Independence of Path 742(7) Green's Theorem in the Plane 749(6) Surface Integrals 755(9) Gauss's Divergence Theorem 764(6) Stokes's Theorem 770(3) Chapter Review 773 Appendix 1(6) A.1 Mathematical Induction 1(2) A.2 Proofs of Several Theorems 3(4) Answers to Odd-Numbered Problems 7 Index 1(1) Photo Credits 1