Note: Each chapter concludes with Review Exercises.
1. The Cartesian Plane and Functions
1.1 Real Numbers and the Real Line
1.2 The Cartesian Plane
1.3 Graphs of Equations
1.4 Lines in the Plane
1.5 Functions
2. Limits and Their Properties
2.1 An Introduction to Limits
2.2 Techniques for Evaluating Limits
2.3 Continuity
2.4 Infinite Limits
2.5 e-d Definition of Limits
3. Differentiation
3.1 The Derivative and the Tangent Line Problem
3.2 Velocity, Acceleration, and Other Rates of Change
3.3 Differentiation Rules for Powers, Constant Multiples, and Sums
3.4 Differentiation Rules for Products and Quotients
3.5 The Chain Rule
3.6 Implicit Differentiation
3.7 Related Rates
4. Applications of Differentiation
4.1 Extrema on an Interval
4.2 Rolle's Theorem and the Mean Value Theorem
4.3 Increasing and Decreasing Functions and the First Derivative Test
4.4 Concavity and the Second Derivative Test
4.5 Limits at Infinity
4.6 A Summary of Curve Sketching
4.7 Optimization Problems
4.8 Newton's Method
4.9 Differentials
4.10 Business and Economics Applications
5. Integration
5.1 Antiderivatives and Indefinite Integration
5.2 Area
5.3 Riemann Sums and the Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Integration by Substitution
5.6 Numerical Integration
6. Applications of Integration
6.1 Area of a Region Between Two Curves
6.2 Volume: The Disc Method
6.3 Volume: The Shell Method
6.4 Arc Length and Surfaces of Revolution
6.5 Work
6.6 Fluid Pressure and Fluid Force
6.7 Moments, Centers of Mass, and Centroids
7. Exponential and Logarithmic Functions
7.1 Exponential Functions
7.2 Differentiation and Integration of Exponential Functions
7.3 Inverse Functions
7.4 Logarithmic Functions
7.5 Logarithmic Functions and Differentiation
7.6 Logarithmic Functions and Integration
7.7 Growth and Decay
7.8 Indeterminate Forms and L
Hôpital's Rule
8. Trigonometric Functions and Inverse Trigonometric Functions
8.1 Review of Trigonometric Functions
8.2 Graphs and Limits of Trigonometric Functions
8.3 Derivatives of Trigonometric Functions
8.4 Integrals of Trigonometric Functions
8.5 Inverse Trigonometric Functions and Differentiation
8.6 Inverse Trigonometric Functions: Integration and Completing the Square
8.7 Hyperbolic Functions
9. Integration Techniques and Improper Integrals
9.1 Basic Integration Formulas
9.2 Integration by Parts
9.3 Trigonometric Integrals
9.4 Trigonometric Substitution
9.5 Partial Fractions
9.6 Integration by Tables and Other Integration Techniques
9.7 Improper Integrals
10. Infinite Series
10.1 Sequences
10.2 Series and Convergence
10.3 The Integral Test and p-Series
10.4 Comparisons of Series
10.5 Alternating Series
10.6 The Ratio and Root Tests
10.7 Taylor Polynomials and Approximations
10.8 Power Series
10.9 Representation of Functions by Power Series
10.10 Taylor and Maclaurin Series
11. Conic Sections
11.1 Parabolas
11.2 Ellipses
11.3 Hyperbolas
11.4 Rotation and the General Second-Degree Equation
12. Plane Curves, Parametric Equations, and Polar Coordinates
12.1 Plane Curves and Parametric Equations
12.2 Parametric Equations and Calculus
12.3 Polar Coordinates and Polar Graphs
12.4 Tangent Lines and Curve Sketching in Polar Coordinates
12.5 Area and Arc Length in Polar Coordinates
12.6 Polar Equations for Conics and Kepler's Laws
13. Vectors and Curves in the Plane
13.1 Vectors in the Plane
13.2 The Dot Product of Two Vectors
13.3 Vector-Valued Functions
13.4 Velocity and Acceleration
13.5 Tangent Vectors and Normal Vectors
13.6 Arc Length and Curvature
14. Solid Analytic Geometry and Vectors in Space
14.1 Space Coordinates and Vectors in Space
14.2 The Cross Product of Two Vectors in Space
14.3 Lines and Planes in Space
14.4 Surfaces in Space
14.5 Curves and Vector-Valued Functions in Space
14.6 Tangent Vectors, Normal Vectors, and Curvature in Space
15. Functions of Several Variables
15.1 Introduction to Functions of Several Variables
15.2 Limits and Continuity
15.3 Partial Derivatives
15.4 Differentials
15.5 Chain Rules for Functions of Several Variables
15.6 Directional Derivatives and Gradients
15.7 Tangent Planes and Normal Lines
15.8 Extrema of Functions of Two Variables
15.9 Applications of Extrema of Functions of Two Variables
15.10 Lagrange Multipliers
16. Multiple Integration
16.1 Iterated Integrals and Area in the Plane
16.2 Double Integrals and Volume
16.3 Change of Variables: Polar Coordinates
16.4 Center of Mass and Moments of Inertia
16.5 Surface Area
16.6 Triple Integrals and Applications
16.7 Cylindrical and Spherical Coordinates
16.8 Triple Integrals in Cylindrical and Spherical Coordinates
16.9 Change of Variables: Jacobians
17. Vector Analysis
17.1 Vector Fields
17.2 Line Integrals
17.3 Conservative Vector Fields and Independence of Path
17.4 Green's Theorem
17.5 Parametric Surfaces
17.6 Surface Integrals
17.7 Divergence Theorem
17.8 Stokes's Theorem
18. Differential Equations
18.1 Definitions and Basic Concepts
18.2 Separation of Variables in First-Order Equations
18.3 Exact First-Order Equations
18.4 First-Order Linear Differential Equations
18.5 Second-Order Homogeneous Linear Equations
18.6 Second-Order Nonhomogeneous Linear Equations
18.7 Series Solutions of Differential Equations
- Appendixes: A. Proofs of Selected Theorems; B. Basic Differentiation Rules for Elementary Functions; C. Integration Tables
- Answers to Odd-Numbered Exercises
- Index
