Notes

Calculus Notes

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A Preview of Calculus Outline

1

Functions and Models

1.1	Four Ways to Represent a Function
1.2	Mathematical Models
1.3	New Functions from Old Functions
1.4	Graphing Calculators and Computers
1.5	Exponential Functions
1.6	Inverse Functions and Logarithms
1.7	Parametric Curves

2

Limits and Derivatives

2.1	The Tangent and Velocity Problems
2.2	The Limit of a Function
2.3	Calculating Limits Using the Limit Laws
2.4	Continuity
2.5	Limits Involving Infinity
2.6	Tangents, Velocities, and Other Rates of Change
2.7	Derivatives
2.8	The Derivative as a Function
2.9	Linear Approximations
2.10	What does f ' Say About f ?

3

Differentiation Rules

3.1	Derivatives of Polynomials and Exponential Functions
3.2	The Product and Quotient Rules
3.3	Rates of Change in the Natural and Social Sciences
3.4	Derivatives of Trigonometric Functions
3.5	The Chain Rule
3.6	Implicit Differentiation
3.7	Derivatives of Logarithmic Functions
3.8	Linear Approximations and Differentials

4

Applications of Differentiation

4.1	Related Rates
4.2	Maximum and Minimum Values
4.3	Derivatives and the Shapes of Curves
4.4	Graphing with Calculus and Calculators
4.5	Indeterminate Forms and L'Hopital's Rule
4.6	Optimization Problems
4.7	Applications to Economics
4.8	Newton's Method
4.9	Antiderivatives

5

Integrals

5.1	Areas and Distances
5.2	The Definite Integral
5.3	Evaluating Definite Integrals
5.4	The Fundamental Theorem of Calculus
5.5	The Substitution Rule
5.6	Integration by Parts
5.7	Additional Techniques of Integration
5.8	Integration Using Tables and Computer Algebra Systems
5.9	Approximate I
5.10	Improper Integrals

6

Applications of Integration

6.1	More About Areas
6.2	Volumes
6.3	Arc Length
6.4	Average Value of a Function
6.5	Applications to Physics and Engineering
6.6	Applications to Economics and Biology
6.7	Probability

7

Differential Equations

7.1	Modeling with Differential Equations
7.2	Direction Fields
7.3	Separable Equations
7.4	Exponential Growth and Decay
7.5	The Logistic Equation
7.6	Predator-Prey Systems

8

Infinite Sequences and Series

8.1	Sequences
8.2	Series
8.3	The Integral and Comparison Tests: Estimating Sums
8.4	Other Convergence Tests
8.5	Power Series
8.6	Representations of Functions as Power Series
8.7	Taylor and MacLaurin Series
8.8	The Binomial Series
8.9	Applications of Taylor Polynomials
8.10	Using Series to Solve Differential Equations