Includes index

chapter one FUNCTIONS 11.1 Functions 11.2 Graphing Functions Using Calculators and Computer Algebra Systems161.3 New Functions from Old 271.4 Families of Functions401.5 Inverse Functions; Inverse Trigonometric Functions 511.6 Exponential and Logarithmic Functions 651.7 Mathematical Models 761.8 Parametric Equations 86chapter two LIMITS AND CONTINUITY 1012.1 Limits (An Intuitive Approach) 1012.2 Computing Limits 1132.3 Limits at Infinity; End Behavior of a Function 1222.4 Limits (Discussed More Rigorously) 1342.5 Continuity 1442.6 Continuity of Trigonometric and Inverse Functions 155chapter three THE DERIVATIVE 1653.1 Tangent Lines, Velocity, and General Rates of Change 1653.2 The Derivative Function 1783.3 Techniques of Differentiation 1903.4 The Product and Quotient Rules 1983.5 Derivatives of Trigonometric Functions 2043.6 The Chain Rule 2093.7 Related Rates 2173.8 Local Linear Approximation; Differentials 224 chapter four EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS 2354.1 Implicit Differentiation 2354.2 Derivatives of Logarithmic Functions 2434.3 Derivatives of Exponential and Inverse Trigonometric Functions 2484.4 L'Hopital's Rule; Indeterminate Forms 256chapter five THE DERIVATIVE IN GRAPHING AND APPLICATIONS 2675.1 Analysis of Functions I:Increase, Decrease, and Concavity 2675.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 2795.3 More on Curve Sketching: Rational Functions; Curves with Cusps and Vertical Tangent Lines; Using Technology 2895.4 Absolute Maxima and Minima 3015.5 Applied Maximum and Minimum Problems 3095.6 Newton's Method 3235.7 Rolle's Theorem; Mean-Value Theorem 3295.8 Rectilinear Motion 336chapter six INTEGRATION 3496.1 An Overview of the Area Problem 3496.2 The Indefinite Integral 3556.3 Integration by Substitution 3656.4 The Definition of Area as a Limit; Sigma Notation3736.5 The Definite Integral 3866.6 The Fundamental Theorem of Calculus 3966.7 Rectilinear Motion Revisited Using Integration 4106.8 Evaluating Definite Integrals by Substitution 4196.9 Logarithmic Functions from the Integral Point of View 425chapter 7 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 4427.1 Area Between Two Curves 4427.2 Volumes by Slicing; Disks and Washers 4507.3 Volumes by Cylindrical Shells 4597.4 Length of a Plane Curve 4657.5 Area of a Surface of Revolution 4717.6 Average Value of a Function and its Applications 4767.7 Work 4817.8 Fluid Pressure and Force 4907.9 Hyperbolic Functions and Hanging Cables 496 chapter eight PRINCIPLES OF INTEGRAL EVALUATION 5108.1 An Overview of Integration Methods 5108.2 Integration by Parts 5138.3 Trigonometric Integrals 5228.4 Trigonometric Substitutions 5308.5 Integrating Rational Functions by Partial Fractions 5378.6 Using Computer Algebra Systems and Tables of Integrals 5458.7 Numerical Integration; Simpson's Rule 5568.8 Improper Integrals 569 chapter 9 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 5829.1 First-Order Differential Equations and Applications 5829.2 Slope Fields; Euler's Method 5969.3 Modeling with First-Order Differential Equations 6039.4 Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring 612 chapter ten INFINITE SERIES 62410.1 Sequences 62410.2 Monotone Sequences 63510.3 Infinite Series 64310.4 Convergence Tests65210.5 The Comparison, Ratio, and Root Tests 65910.6 Alternating Series; Conditional Convergence 66610.7 Maclaurin and Taylor Polynomials 67510.8 Maclaurin and Taylor Series; PowerSeries 68510.9 Convergence of Taylor Series 69410.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 704chapter eleven ANALYTIC GEOMETRY IN CALCULUS 71711.1 Polar Coordinates 71711.2 Tangent Lines and Arc Length for Parametric and Polar Curves 73111.3 Area in Polar Coordinates 74011.4 Conic Sections in Calculus 74611.5 Rotation of Axes; Second-Degree Equations 76511.6 Conic Sections in Polar Coordinates 771Horizon Module: Comet Collision 783chapter twelve THREE-DIMENSIONAL SPACE; VECTORS 78612.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 78612.2 Vectors 79212.3 Dot Product; Projections 80412.4 Cross Product 81312.5 Parametric Equations of Lines 82412.6 Planes in 3-Space 83112.7 Quadric Surfaces 83912.8 Cylindrical and Spherical Coordinates 850chapter thirteen VECTOR-VALUED FUNCTIONS 85913.1 Introduction to Vector-Valued Functions 85913.2 Calculus of Vector-Valued Functions 86513.3 Change of Parameter; Arc Length 87613.4 Unit Tangent, Normal, and Binormal Vectors 88613.5 Curvature 89213.6 Motion Along a Curve 90113.7 Kepler's Laws of Planetary Motion 914chapter fourteen PARTIAL DERIVATIVES 92414.1 Functions of Two or More Variables 92414.2 Limits and Continuity 93614.3 Partial Derivatives 94514.4 Differentiability, Differentials, and Local Linearity 95914.5 The Chain Rule 96814.6 Directional Derivatives and Gradients 97814.7 Tangent Planes and Normal Vectors 98914.8 Maxima and Minima of Functions of Two Variables 99614.9 Lagrange Multipliers 1008 chapter fifteen MULTIPLE INTEGRALS 101815.1 Double Integrals 101815.2 Double Integrals over Nonrectangular Regions 102615.3 Double Integrals in Polar Coordinates 103515.4 Parametric Surfaces; Surface Area 104315.5 Triple Integrals 105615.6 Centroid, Center of Gravity, Theorem of Pappus 106515.7 Triple Integrals in Cylindrical and Spherical Coordinates 107615.8 Change of Variables in Multiple Integrals; Jacobians 1087 chapter sixteen TOPICS IN VECTOR CALCULUS 110216.1 Vector Fields 110216.2 Line Integrals 111216.3 Independence of Path; Conservative Vector Fields 112916.4 Green's Theorem 113916.5 Surface Integrals 114716.6 Applications of Surface Integrals; Flux 115516.7 The Divergence Theorem 116416.8 Stokes' Theorem 1173Horizon Module: Hurricane Modeling 1183appendix a TRIGONOMETRY REVIEW A1appendix b SOLVING POLYNOMIAL EQUATIONS A15appendix c SELECTED PROOFS A22ANSWERS A33PHOTOCREDITS C1INDEX I-1web appendix d REAL NUMBERS,INTERVALS, AND INEQUALITIES W1web appendix e ABSOLUTE VALUE W11web appendix f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS W16web appendix g DISTANCE, CIRCLES, AND QUADRATIC FUNCTIONS W32web appendix h THE DISCRIMINANT W41