Calculus

0. Before Calculus
0.1 Functions
0.2New Functions from Old
0.3Families of Functions
0.4Inverse Functions 1. Limits and Continuity
1.1Limits (An Intuitive Approach)
1.2Computing Limits
1.3Limits at Infinity; End Behavior of a Function
1.4Limits (Discussed More Rigorously)
1.5Continuity
1.6Continuity of Trigonometric Functions

2. The Derivative
2.1Tangent Lines and Rates of Change
2.2The Derivative Function
2.3Introduction to Techniques of Differentiation
2.4The Product and Quotient Rules
2.5Derivatives of Trigonometric Functions
2.6The Chain Rule
2.7Implicit Differentiation
2.8Related Rates
2.9Local Linear Approximation; Differentials

3. The Derivative in Graphing and Applications
3.1Analysis of Functions I: Increase, Decrease, and Concavity
3.2Analysis of Functions II: Relative Extrema; Graphing Polynomials
3.3Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents
3.4Absolute Maxima and Minima
3.5Applied Maximum and Minimum Problems
3.6Rectilinear Motion
3.7Newton's Method
3.8Rolle's Theorem; Mean-Value Theorem

4. Integration
4.1An Overview of the Area Problem
4.2The Indefinite Integral
4.3Integration by Substitution
4.4 The Definition of Area as a Limit; Sigma Notation
4.5The Definite Integral
4.6The Fundamental Theorem of Calculus
4.7Rectilinear Motion Revisited: Using Integration
4.8Average Value of a Function and Its Applications
4.9Evaluating Definite Integrals by Substitution

5. Applications of the Definite Integral in Geometry, Science and Engineering
5.1Area Between Two Curves
5.2Volumes by Slicing; Disks and Washers
5.3Volumes by Cylindrical Shells
5.4Length of a Plane Curve
5.5Area of a Surface Revolution
5.6Work
5.7Moments, Centers of Gravity, and Centroids
5.8Fluid Pressure and Force

6. Exponential, Logarithmic, and Inverse Trigonometric Functions
6.1Exponential and Logarithmic Functions
6.2Derivatives and Integrals Involving Logarithmic Functions
6.3Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions
6.4Graphs and Applications Involving Logarithmic and Exponential Functions
6.5L'H?opital's Rule; Indeterminate Forms
6.6Logarithmic and Other Functions Defined by Integrals
6.7Derivatives and Integrals Involving Inverse Trigonometric Functions
6.8Hyperbolic Functions and Hanging Cubes

Ch 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

Ch 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

Ch 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


Ch 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

Ch 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

Ch 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

Ch 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

Ch 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 Variable in Multiple Integrals; Jacobians
14.8 Centers of Gravity Using Multiple Integrals

Ch 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

Appendix [order of sections TBD]
A Graphing Functions Using Calculators and Computer Algebra Systems
B Trigonometry Review
C Solving Polynomial Equations
D Mathematical Models
E Selected Proofs

Web Appendices
F Real Numbers, Intervals, and Inequalities
G Absolute Value
H Coordinate Planes, Lines, and Linear Functions
I Distance, Circles, and Quadratic Functions
J Second-Order Linear Homogeneous Differential Equations; The Vibrating String
K The Discriminant

ANSWERS

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INDEX