Euler's Method and Direction Fields

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Substitution
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Modeling with Differential Equations

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Separable Equations
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Euler's Method and Direction Fields

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

Separable Equations

The Simplest Differential Equations
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Exponential Growth and Decay
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Linear Equations

Linear ODEs: Working an Example
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Parametrized Curves

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

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

Polar Coordinates

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Video: Computing Slopes of Tangent Lines

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Conic Sections in Polar Coordinates

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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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Video: Worked problems

Multiple Integrals

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What is a Double Integral?
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Iterated Integrals over Rectangles

How To Compute Iterated Integrals
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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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Bonus: Cylindrical and spherical coordinates

Direction Fields

Direction fields are a good way to visualize the solutions of a differential equation.

Since $\displaystyle\frac{dy}{dx} = f(x,y)$, at each point $\left(x_0,y_0\right)$ in the $xy$-plane we can draw a short line with slope $f\left(x_0,y_0\right)$. This is called a direction field. A solution to the differential equation $\displaystyle\frac{dy}{dx} = f(x,y)$ is then a curve that is everywhere tangent to the direction field.

Examples: The direction fields of $$\displaystyle\frac{dy}{dx} = x+y,\qquad \displaystyle\frac{dy}{dx} = 2x, \quad\text{ and } \quad \displaystyle\frac{dy}{dx}=y,$$ are (respectively)

The following video goes over the details and shows how to deduce information about the solutions of the corresponding differential equations.