Euler's Method and Direction Fields

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Substitution
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Comparison Tests for Convergence

Modeling with Differential Equations

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Euler's Method and Direction Fields

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

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Linear Equations

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Parametrized Curves

Three kinds of functions, three kinds of curves
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Calculus with Parametrized Curves

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Video: Arclength and Surface Area
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Polar Coordinates

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Conic Sections in Polar Coordinates

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Convergence of Series with Negative Terms

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Partial Derivatives

One variable at a time (yet again)
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An Example from DNA
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Differentiability and the Chain Rule

Differentiability
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Chain Rule, General Case
Video: Worked problems

Multiple Integrals

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

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Integrals in Probability and Statistics

Single integrals in probability
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Change of Variables

Review: Change of variables in 1 dimension
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Jacobians
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Bonus: Cylindrical and spherical coordinates

Euler's Method Revisited

Euler's method and direction fields go hand in hand. In terms of direction fields, the algorithm for Euler's method goes as follows:

Pick a step size $h$.

Look at the direction field at $\left(x_0, y_0\right)$. Draw the line in that direction through $\left(x_0, y_0\right)$.

Follow the line out to a point $\left(x_1, y_1\right)$, where $x_1 = x_0 + h$.

Look at the direction field at $\left(x_1, y_1\right)$. Draw the line in that direction through $\left(x_1, y_1\right)$.

Follow that line out to a point $\left(x_2, y_2\right)$, where $x_2 = x_1 + h = x_0 + 2h$.

Rinse, lather, repeat.