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#### The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

#### The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

#### Substitution

Substitution for Indefinite Integrals
Revised Table of Integrals
Substitution for Definite Integrals

#### Area Between Curves

The Slice and Dice Principle
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
Volumes Rotating About the $y$-axis

Behind IBP
Examples
Going in Circles

#### Integrals of Trig Functions

Basic Trig Functions
Product of Sines and Cosines (1)
Product of Sines and Cosines (2)
Product of Secants and Tangents
Other Cases

#### Trig Substitutions

How it works
Examples
Completing the Square

#### Partial Fractions

Introduction
Linear Factors
Improper Rational Functions and Long Division
Summary

#### Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

#### Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

#### Differential Equations

Introduction
Separable Equations
Mixing and Dilution

#### Models of Growth

Exponential Growth and Decay
Logistic Growth

#### Infinite Sequences

Close is Good Enough (revisited)
Examples
Limit Laws for Sequences
Monotonic Convergence

#### Infinite Series

Introduction
Geometric Series
Limit Laws for Series
Telescoping Sums and the FTC

#### Integral Test

The Integral Test
When the Integral Diverges
When the Integral Converges

#### Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

#### Convergence of Series with Negative Terms

Introduction
Alternating Series and the AS Test
Absolute Convergence
Rearrangements

The Ratio Test
The Root Test
Examples

#### Strategies for testing Series

List of Major Convergence Tests
Examples

#### Power Series

Finding the Interval of Convergence
Other Power Series

#### Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

#### Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

#### Applications of Taylor Polynomials

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
Other Uses of Taylor Polynomials

#### Partial Derivatives

Definitions and Rules
The Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

#### Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

#### Iterated Integrals over Rectangles

One Variable at the Time
Fubini's Theorem
Notation and Order

#### Double Integrals over General Regions

Type I and Type II regions
Examples
Order of Integration
Area and Volume Revisited

### The Limit Comparison Test

There are two extensions of the basic comparison test:

 Theorem: Suppose that $\displaystyle{\sum_{n=1}^\infty a_n}$ and $\displaystyle{\sum_{n=1}^\infty b_n}$ are positive series; that $c$ is a positive constant; and that $N$ is some positive integer. If $\displaystyle{\sum_{n=1}^\infty b_n}$ converges and $a_n \le c b_n$ for all $n>N$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ converges. If $\displaystyle{\sum_{n=1}^\infty b_n}$ diverges and $a_n \ge c b_n$ for all $n>N$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ diverges.

 Limit Comparison Test: Let $\displaystyle{\sum_{n=1}^\infty a_n}$ and $\displaystyle{\sum_{n=1}^\infty b_n}$ be positive series. If $$\displaystyle{\lim_{n \to \infty} \frac{a_n}{b_n}}=c,$$ for some positive number $c$, then either both series converge or both diverge.

 Example: To determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{4^n}{2^n+3^n}$ converges or diverges, we'll look for a series that "behaves like" it when $n$ is large. Since $$\frac{4^n}{2^n+3^n}\approx \frac{4^n}{3^n},$$ when $n$ is large, we'll use $\displaystyle\sum_{n=1}^\infty\left(\frac43\right)^n$ for comparison. Since $$\lim_{n\to\infty}\frac{ \frac{4^n}{2^n+3^n}}{\frac{4^n}{3^n}}= \lim_{n\to\infty}\frac{ 3^n}{2^n+3^n}= \lim_{n\to\infty}\frac{ 3^n}{3^n\left(\frac{2^n}{3^n}+1\right)}= 1>0,$$ and the geometric series $\displaystyle\sum_{n=1}^\infty\left(\frac43\right)^n$ diverges, we can conclude that our original series diverges as well.