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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

### List of Major Convergence Tests

 Standard examples: When using comparison tests, these are the things we are most likely to compare to: The geometric series $\sum c r^n$ converges absolutely if $\lvert r\rvert<1$ and diverges if $\lvert r\rvert\ge 1$. The $p$-series $\sum \frac{1}{n^p}$ converges if $p>1$ and diverges if $p \le 1$. Divergence test: If $\displaystyle\lim_{n \to \infty} a_n \ne 0$, then $\sum_n a_n$ diverges. Integral test: If $a_n = f(n)$, where $f(x)$ is a non-negative non-increasing function, then we look at the improper integral $\int_1^\infty f(x)\, dx$. The sum $\sum_{n=1}^\infty f(n)$ converges if and only if the integral $\int_1^\infty f(x) \,dx$ converges. Comparison test: This applies only to positive series. If $0 \le a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $0 \le b_n \le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges. Limit comparison test: If $\sum a_n$ and $\sum b_n$ are positive series, and $\displaystyle\lim_{n \to \infty} \frac{a_n}{b_n} = L$, with $L>0$, then either $\sum a_n$ and $\sum b_n$ both converge or both diverge. Alternating series: If {$b_n$} is a positive sequence with $b_{n+1} \le b_n$ and $\displaystyle\lim_{n \to \infty} b_n = 0$, then $\sum (-1)^{n+1} b_n$ converges. Absolute convergence: If $\sum |a_n|$ converges, then $\sum a_n$ converges. Ratio test: Let $R= \displaystyle{\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}}$. If $R < 1$, then $\sum a_n$ converges absolutely. If $R > 1$, then $\sum a_n$ diverges. If $R=1$ or if $R$ does not exist, then we can't tell. Root test: Let $\rho = \displaystyle{\lim_{n \to \infty} |a_n|^{1/n}}$. If $\rho<1$, then $\sum a_n$ converges absolutely $\big($like the geometric series $\sum \rho^n$$\big) If \rho>1, then \sum a_n diverges \big(like the geometric series \sum \rho^n$$\big)$ If $\rho=1$, or if $\rho$ does not exist, then we can't tell. For the root and ratio tests, limits that go to infinity count as "$>1$", not as "does not exist". If the ratio and root tests both work, then $\rho$ will equal $R$. However, one test may be easier to use than the other.