Table of Contents:

Chapter 0: Preliminaries

• 0.1 Polynomials and Rational Functions"

• 0.2 Graphing Calculators and Computer Algebra Systems"

• 0.3 Inverse Functions"

• 0.4 Trigonometric and Inverse Trigonometric Functions"

• 0.5 Exponential and Logarithmic Functions"

• 0.6 Transformations of Functions"

Chapter 1: Limits and Continuity

• 1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve

• 1.2 The Concept of Limit"

• 1.3 Computation of Limits"

• 1.4 Continuity and Its Consequences"

• 1.5 Limits Involving Infinity; Asymptotes"

• 1.6 Formal Definition of the Limit"

• 1.7 Limits and Loss-of-Significance Errors"

Chapter 2: Differentiation

• 2.1 Tangent Lines and Velocity"

• 2.2 The Derivative"

• 2.3 Computation of Derivatives: The Power Rule"

• 2.4 The Product and Quotient Rules"

• 2.5 The Chain Rule"

• 2.6 Derivatives of Trigonometric Functions"

• 2.7 Derivatives of Exponential and Logarithmic Functions"

• 2.8 Implicit Differentiation and Inverse Trigonometric Functions"

• 2.9 The Hyperbolic Functions"

• 2.10 The Mean Value Theorem"

Chapter 3: Applications of the Derivative

• 3.1 Linear Approximations and Newton's Method""

• 3.2 Indeterminate Forms and L'Hôpital's Rule "

• 3.3 Maximum and Minimum Values"

• 3.4 Increasing and Decreasing Functions"

• 3.5 Concavity and the Second Derivative Test"

• 3.6 Overview of Curve Sketching"

• 3.7 Optimization"

• 3.8 Related Rates"

• 3.9 Rates of Change in Economics and the Sciences"

Chapter 4: Integration

• 4.1 Antiderivatives"

• 4.2 Sums and Sigma Notation"

• 4.3 Area"

• 4.4 The Definite Integral"

• 4.5 The Fundamental Theorem of Calculus"

• 4.6 Integration by Substitution"

• 4.7 Numerical Integration"

• 4.8 The Natural Logarithm as an Integral"

Chapter 5: Applications of the Definite Integral

• 5.1 Area Between Curves"

• 5.2 Volume: Slicing, Disks and Washers"

• 5.3 Volumes by Cylindrical Shells"

• 5.4 Arc Length and Surface Area"

• 5.5 Projectile Motion"

• 5.6 Applications of Integration to Physics and Engineering"

• 5.7 Probability"

Chapter 6: Integration Techniques

• 6.1 Review of Formulas and Techniques"

• 6.2 Integration by Parts"

• 6.3 Trigonometric Techniques of Integration"

• 6.4 Integration of Rational Functions Using Partial Fractions"

• 6.5 Integration Tables and Computer Algebra Systems"

• 6.6 Improper Integrals"

Chapter 7: First-Order Differential Equations

• 7.1 Modeling with Differential Equations"

• 7.2 Separable Differential Equations"

• 7.3 Direction Fields and Euler's Method"

• 7.4 Systems of First-Order Differential Equations"

Chapter 8: Infinite Series

• 8.1 Sequences of Real Numbers"

• 8.2 Infinite Series"

• 8.3 The Integral and Comparison Tests"

• 8.4 Alternating Series"

• 8.5 Absolute Convergence and the Ratio Test"

• 8.6 Power Series"

• 8.7 Taylor Series"

• 8.8 Applications of Taylor Series"

• 8.9 Fourier Series"

Chapter 9: Parametric Equations and Polar Coordinates

• 9.1 Plane Curves and Parametric Equations"

• 9.2 Calculus and Parametric Equations"

• 9.3 Arc Length and Surface Area in Parametric Equations"

• 9.4 Polar Coordinates"

• 9.5 Calculus and Polar Coordinates"

• 9.6 Conic Sections"

• 9.7 Conic Sections in Polar Coordinates"

Chapter 10: Vectors and the Geometry of Space

• 10.1 Vectors in the Plane"

• 10.2 Vectors in Space"

• 10.3 The Dot Product"

• 10.4 The Cross Product"

• 10.5 Lines and Planes in Space"

• 10.6 Surfaces in Space"

Chapter 11: Vector-Valued Functions

• 11.1 Vector-Valued Functions"

• 11.2 The Calculus of Vector-Valued Functions"

• 11.3 Motion in Space"

• 11.4 Curvature"

• 11.5 Tangent and Normal Vectors"

• 11.6 Parametric Surfaces"

Chapter 12: Functions of Several Variables and Partial Differentiation

• 12.1 Functions of Several Variables"

• 12.2 Limits and Continuity"

• 12.3 Partial Derivatives"

• 12.4 Tangent Planes and Linear Approximations"

• 12.5 The Chain Rule"

• 12.6 The Gradient and Directional Derivatives"

• 12.7 Extrema of Functions of Several Variables"

• 12.8 Constrained Optimization and and Lagrange Multipliers"

Chapter 13: Multiple Integrals

• 13.1 Double Integrals"

• 13.2 Area, Volume and Center of Mass"

• 13.3 Double Integrals in Polar Coordinates"

• 13.4 Surface Area"

• 13.5 Triple Integrals"

• 13.6 Cylindrical Coordinates"

• 13.7 Spherical Coordinates"

• 13.8 Change of Variables in Multiple Integrals"

Chapter 14: Vector Calculus

• 14.1 Vector Fields"

• 14.2 Line Integrals"

• 14.3 Independence of Path and Conservative Vector Fields"

• 14.4 Green's Theorem"

• 14.5 Curl and Divergence"

• 14.6 Surface Integrals"

• 14.7 The Divergence Theorem"

• 14.8 Stokes' Theorem"

• 14.9 Applications of Vector Calculus"

Chapter 15: Second Order Differential Equations

• 15.1, Second-Order Equations With Constant Coefficients"

• 15.2 Nonhomogeneous Equations: Undetermined Coefficients"

• 15.3 Applications of Second-Order Equations"

• 15.4 Power Series Solutions of Differential Equations"

Appendix A: Proofs of Selected Theorems

Appendix B: Answers to Odd-Numbered Exercises

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