Modeling with Differential Equations

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Integration by Parts

Integration by Parts
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Integration by Parts with a definite integral
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Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
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Trig Substitutions

How Trig Substitution Works
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Completing the Square

Partial Fractions

Introduction to Partial Fractions
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Irreducible Quadratic Factors
Improper Rational Functions and Long Division
Summary

Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type 1 - Improper Integrals with Infinite Intervals of Integration
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Comparison Tests for Convergence

Modeling with Differential Equations

Introduction
Separable Equations
A Second Order Problem

Euler's Method and Direction Fields

Euler's Method (follow your nose)
Direction Fields
Euler's method revisited

Separable Equations

The Simplest Differential Equations
Separable differential equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
The Zombie Apocalypse (Logistic Growth)

Linear Equations

Linear ODEs: Working an Example
The Solution in General
Saving for Retirement

Parametrized Curves

Three kinds of functions, three kinds of curves
The Cycloid
Visualizing Parametrized Curves
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Lissajous Figures

Calculus with Parametrized Curves

Video: Slope and Area
Video: Arclength and Surface Area
Summary and Simplifications
Higher Derivatives

Polar Coordinates

Definitions of Polar Coordinates
Graphing polar functions
Video: Computing Slopes of Tangent Lines

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Area Inside a Polar Curve
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Arc Length of Polar Curves

Conic sections

Slicing a Cone
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Shifting the Center by Completing the Square

Conic Sections in Polar Coordinates

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Astronomy and Equations in Polar Coordinates

Infinite Sequences

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Infinite Series

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Integral Test

Preview of Coming Attractions
The Integral Test
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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 and Root Tests

The Ratio Test
The Root Test
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Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Radius and Interval of Convergence
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$x=a$

Representing Functions as Power Series

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Taylor and Maclaurin Series

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Taylor Polynomials
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Functions of 2 and 3 variables

Functions of several variables
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Partial Derivatives

One variable at a time (yet again)
Definitions and Examples
An Example from DNA
Geometry of partial derivatives
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Differentials and Taylor Expansions

Differentiability and the Chain Rule

Differentiability
The First Case of the Chain Rule
Chain Rule, General Case
Video: Worked problems

Multiple Integrals

General Setup and Review of 1D Integrals
What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Swapping the Order of Integration
Area and Volume Revisited

Double integrals in polar coordinates

dA = r dr (d theta)
Examples

Multiple integrals in physics

Double integrals in physics
Triple integrals in physics

Integrals in Probability and Statistics

Single integrals in probability
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Change of Variables

Review: Change of variables in 1 dimension
Mappings in 2 dimensions
Jacobians
Examples
Bonus: Cylindrical and spherical coordinates

Differential Equations

A differential equation is an equation that relates the rate $\displaystyle\frac{dy}{dt}$ at which a quantity $y$ is changing (or sometimes a higher derivative) to some function $f(t,y)$ of that quantity and time.

Examples:

$\displaystyle\frac{dy}{dt} = 3y; \qquad \frac{dy}{dt}= 5t^2;\qquad \frac{dy}{dt} = 5t^2 + 3y$ are examples of explicit first-order equations, i.e., equations of the form

$\frac{dy}{dt} = f(t,y)$ while

$\frac{d^2y}{dt^2} = -4 x;\qquad \frac{d^2y}{dt^2}=y\sin(t)+\frac{dy}{dt}$ are examples of explicit second-order equations, i.e., equations of the form

$\frac{d^2y}{dt^2} = f\left(t,y, y'\right).$

Modeling with differential equations boils down to four steps.

Understand the science behind what we're trying to model. For example, if we are studying populations of animals, we need to know something about population biology, and what might cause the number of animals to increase or decrease.

Express the rules for how the system changes in mathematical form. The result is a differential equation.

Use calculus to solve the differential equation.

Interpret the solution(s) in context to predict future behaviour.

The following video describes this general philosophy.