1. Functions
1.1 Functions and Their Graphs
1.2 Combining Functions; Shifting and Scaling Graphs
1.3 Trigonometric Functions
1.4 Graphing with Calculators and Computers
2. Limits and Continuity
2.1 Rates of Change and Tangents to Curves
2.2 Limit of a Function and Limit Laws
2.3 The Precise Definition of a Limit
2.4 One-Sided Limits and Limits at Infinity
2.5 Infinite Limits and Vertical Asymptotes
2.6 Continuity
2.7 Tangents and Derivatives at a Point
3. Differentiation
3.1 The Derivative as a Function
3.2 Differentiation Rules
3.3 The Derivative as a Rate of Change
3.4 Derivatives of Trigonometric Functions
3.5 The Chain Rule
3.6 Implicit Differentiation
3.7 Related Rates
3.8 Linearization and Differentials
3.9 Parametrizations of Plane Curves
4. Applications of Derivatives
4.1 Extreme Values of Functions
4.2 The Mean Value Theorem
4.3 Monotonic Functions and the First Derivative Test
4.4 Concavity and Curve Sketching
4.5 Applied Optimization
4.6 Newton's Method
4.7 Antiderivatives
5. Integration
5.1 Area and Estimating with Finite Sums
5.2 Sigma Notation and Limits of Finite Sums
5.3 The Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Indefinite Integrals and the Substitution Rule
5.6 Substitution and Area Between Curves
6. Applications of Definite Integrals
6.1 Volumes by Slicing and Rotation About an Axis
6.2 Volumes by Cylindrical Shells
6.3 Lengths of Plane Curves
6.4 Areas of Surfaces of Revolution
6.5 Work
6.6 Moments and Centers of Mass
6.7 Fluid Pressures and Forces
7. Transcendental Functions
7.1 Inverse Functions and Their Derivatives
7.2 Natural Logarithms
7.3 Exponential Functions
7.4 Inverse Trigonometric Functions
7.5 Exponential Change and Separable Differential Equations
7.6 Indeterminate Forms and L'Hopital's Rule
7.7 Hyperbolic Functions
8. Techniques of Integration
8.1 Integration by Parts
8.2 Trigonometric Integrals
8.3 Trigonometric Substitutions
8.4 Integration of Rational Functions by Partial Fractions
8.5 Integral Tables and Computer Algebra Systems
8.6 Numerical Integration
8.7 Improper Integrals
9. Infinite Sequences and Series
9.1 Sequences
9.2 Infinite Series
9.3 The Integral Test
9.4 Comparison Tests
9.5 The Ratio and Root Tests
9.6 Alternating Series, Absolute and Conditional Convergence
9.7 Power Series
9.8 Taylor and Maclaurin Series
9.9 Convergence of Taylor Series
9.10 The Binomial Series
10. Polar Coordinates and Conics
10.1 Polar Coordinates
10.2 Graphing in Polar Coordinates
10.3 Areas and Lengths in Polar Coordinates
10.4 Conic Sections
10.5 Conics in Polar Coordinates
10.6 Conics and Parametric Equations; The Cycloid
11. Vectors and the Geometry of Space
11.1 Three-Dimensional Coordinate Systems
11.2 Vectors
11.3 The Dot Product
11.4 The Cross Product
11.5 Lines and Planes in Space
11.6 Cylinders and Quadric Surfaces
12. Vector-Valued Functions and Motion in Space
12.1 Vector Functions and Their Derivatives
12.2 Integrals of Vector Functions
12.3 Arc Length in Space
12.4 Curvature of a Curve
12.5 Tangential and Normal Components of Acceleration
12.6 Velocity and Acceleration in Polar Coordinates
13. Partial Derivatives
13.1 Functions of Several Variables
13.2 Limits and Continuity in Higher Dimensions
13.3 Partial Derivatives
13.4 The Chain Rule
13.5 Directional Derivatives and Gradient Vectors
13.6 Tangent Planes and Differentials
13.7 Extreme Values and Saddle Points
13.8 Lagrange Multipliers
13.9 Taylor's Formula for Two Variables
14. Multiple Integrals
14.1 Double and Iterated Integrals over Rectangles
14.2 Double Integrals over General Regions
14.3 Area by Double Integration
14.4 Double Integrals in Polar Form
14.5 Triple Integrals in Rectangular Coordinates
14.6 Moments and Centers of Mass
14.7 Triple Integrals in Cylindrical and Spherical Coordinates
14.8 Substitutions in Multiple Integrals
15. Integration in Vector Fields
15.1 Line Integrals
15.2 Vector Fields, Work, Circulation, and Flux
15.3 Path Independence, Potential Functions, and Conservative Fields
15.4 Green's Theorem in the Plane
15.5 Surfaces and Area
15.6 Surface Integrals and Flux
15.7 Stokes' Theorem
15.8 The Divergence Theorem and a Unified Theory
16. First-Order Differential Equations (online)
16.1 Solutions, Slope Fields, and Picard's Theorem
16.2 First-Order Linear Equations
16.3 Applications
16.4 Euler's Method
16.5 Graphical Solutions of Autonomous Equations
16.6 Systems of Equations and Phase Planes
17. Second-Order Differential Equations (online)
17.1 Second-Order Linear Equations
17.2 Nonhomogeneous Linear Equations
17.3 Applications
17.4 Euler Equations
17.5 Power Series Solutions
Appendices
1 Real Numbers and the Real Line
2 Mathematical Induction
3 Lines, Circles, and Parabolas
4 Trigonometry Formulas
5 Proofs of Limit Theorems
6 Commonly Occurring Limits
7 Theory of the Real Numbers
8 The Distributive Law for Vector Cross Products
9 The Mixed Derivative Theorem and the Increment Theorem