Calculus

INTRODUCTION: The Roots of Calculus 1 LIMITS AND CONTINUITY 1.1 Limits (An Intuitive Approach) 1.2 Computing Limits 1.3 Limits at Infinity; End Behavior of a Function 1.4 Limits (Discussed More Rigorously) 1.5 Continuity 1.6 Continuity of Trigonometric Functions 1.7 Inverse Trigonometric Functions 1.8 Exponential and Logarithmic Functions 2 THE DERIVATIVE 2.1 Tangent Lines and Rates of Change 2.2 The Derivative Function 2.3 Introduction to Techniques of Differentiation 2.4 The Product and Quotient Rules 2.5 Derivatives of Trigonometric Functions 2.6 The Chain Rule 3 TOPICS IN DIFFERENTIATION 3.1 Implicit Differentiation 3.2 Derivatives of Logarithmic Functions 3.3 Derivatives of Exponential and Inverse Trigonometric Functions 3.4 Related Rates 3.5 Local Linear Approximation; Differentials 3.6 L’Hôpital’s Rule; Indeterminate Forms 4 THE DERIVATIVE IN GRAPHING AND APPLICATIONS 4.1 Analysis of Functions I: Increase, Decrease, and Concavity 4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 4.4 Absolute Maxima and Minima 4.5 Applied Maximum and Minimum Problems 4.6 Rectilinear Motion 4.7 Newton’s Method 4.8 Rolle’s Theorem; Mean-Value Theorem 5 INTEGRATION 5.1 An Overview of the Area Problem 5.2 The Indefinite Integral 5.3 Integration by Substitution 5.4 The Definition of Area as a Limit; Sigma Notation 5.5 The Definite Integral 5.6 The Fundamental Theorem of Calculus 5.7 Rectilinear Motion Revisited Using Integration 5.8 Average Value of a Function and its Applications 5.9 Evaluating Definite Integrals by Substitution 5.10 Logarithmic and Other Functions Defined by Integrals 6 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 6.1 Area Between Two Curves 6.2 Volumes by Slicing; Disks and Washers 6.3 Volumes by Cylindrical Shells 6.4 Length of a Plane Curve 6.5 Area of a Surface of Revolution 6.6 Work 6.7 Moments, Centers of Gravity, and Centroids 6.8 Fluid Pressure and Force 6.9 Hyperbolic Functions and Hanging Cables 7 PRINCIPLES OF INTEGRAL EVALUATION 7.1 An Overview of Integration Methods 7.2 Integration by Parts 7.3 Integrating Trigonometric Functions 7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.7 Numerical Integration; Simpson’s Rule 7.8 Improper Integrals 8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 8.1 Modeling with Differential Equations 8.2 Separation of Variables 8.3 Slope Fields; Euler’s Method 8.4 First-Order Differential Equations and Applications 9 INFINITE SERIES 9.1 Sequences 9.2 Monotone Sequences 9.3 Infinite Series 9.4 Convergence Tests 9.5 The Comparison, Ratio, and Root Tests 9.6 Alternating Series; Absolute and Conditional Convergence 9.7 Maclaurin and Taylor Polynomials 9.8 Maclaurin and Taylor Series; Power Series 9.9 Convergence of Taylor Series 9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS 10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length, and Area for Polar Curves 10.4 Conic Sections 10.5 Rotation of Axes; Second-Degree Equations 10.6 Conic Sections in Polar Coordinates 11 THREE-DIMENSIONAL SPACE; VECTORS 11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 11.2 Vectors 11.3 Dot Product; Projections 11.4 Cross Product 11.5 Parametric Equations of Lines 11.6 Planes in 3-Space 11.7 Quadric Surfaces 11.8 Cylindrical and Spherical Coordinates 12 VECTOR-VALUED FUNCTIONS 12.1 Introduction to Vector-Valued Functions 12.2 Calculus of Vector-Valued Functions 12.3 Change of Parameter; Arc Length 12.4 Unit Tangent, Normal, and Binormal Vectors 12.5 Curvature 12.6 Motion Along a Curve 12.7 Kepler’s Laws of Planetary Motion 13 PARTIAL DERIVATIVES 13.1 Functions of Two or More Variables 13.2 Limits and Continuity 13.3 Partial Derivatives 13.4 Differentiability, Differentials, and Local Linearity 13.5 The Chain Rule 13.6 Directional Derivatives and Gradients 13.7 Tangent Planes and Normal Vectors 13.8 Maxima and Minima of Functions of Two Variables 13.9 Lagrange Multipliers 14 MULTIPLE INTEGRALS 14.1 Double Integrals 14.2 Double Integrals over Nonrectangular Regions 14.3 Double Integrals in Polar Coordinates 14.4 Surface Area; Parametric Surfaces 14.5 Triple Integrals 14.6 Triple Integrals in Cylindrical and Spherical Coordinates 14.7 Change of Variables in Multiple Integrals; Jacobians 14.8 Centers of Gravity Using Multiple Integrals 15 TOPICS IN VECTOR CALCULUS 15.1 Vector Fields 15.2 Line Integrals 15.3 Independence of Path; Conservative Vector Fields 15.4 Green’s Theorem 15.5 Surface Integrals 15.6 Applications of Surface Integrals; Flux 15.7 The Divergence Theorem 15.8 Stokes’ Theorem APPENDICES A TRIGONOMETRY SUMMARY B FUNCTIONS (SUMMARY) C NEW FUNCTIONS FROM OLD (SUMMARY) D FAMILIES OF FUNCTIONS (SUMMARY)