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‎other/stem/math/pure/changes/calculus-cheat-sheet/.markdownlint.json

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{
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"MD013": false
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"MD013": false,
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"MD033": false,
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"MD036": false
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}

‎other/stem/math/pure/changes/calculus-cheat-sheet/README.md

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* [INTEGRAL CALCULUS (THE INTEGRAL)](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#integral-calculus-the-integral)
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* [DEFINITION OF AN INTEGRAL](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#definition-of-an-integral)
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* [BASIC INTEGRAL RULES](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#basic-integral-rules)
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* [A REAL WORLD EXAMPLE USING DERIVATIVES AND INTEGRALS](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#a-real-world-example-using-derivatives-and-integrals)
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* [EXAMPLES](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#examples)
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* [FINDING VELOCITY (Using a Derivative)](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#finding-velocity-using-a-derivative)
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* [FINDING DISTANCE FUNCTION (Using an Integral)](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet#finding-distance-function-using-an-integral)
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cheat sheet
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* [my-latex-renders](https://github.com/JeffDeCola/my-latex-renders)
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## OVERVIEW
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| CONTEXT | WHAT f(x) REPRESENTS | WHAT WE WANT TO FIND |
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|------------------------|------------------------------------|--------------------------------------|
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| CALCULUS (Derivatives) | $f(x)$ is the original function | The derivative $f'(x)=\frac{dy}{dx}$ |
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| CALCULUS (Integrals) | $f(x)$ is function to integrate | Function $F(x)=\int f(x)dx$ |
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| DIFFERENTIAL EQUATIONS | $f(x)$ is the derivative of $y(x)$ | Solve for $y(x)$ by integrating |
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## OVERVIEW
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Calculus is a branch of mathematics that studies continuous change.
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It was developed independently by Isaac Newton and
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Gottfried Leibniz in the 17th century, and it has become one of
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\end{aligned}
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$$
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## A REAL WORLD EXAMPLE USING DERIVATIVES AND INTEGRALS
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## EXAMPLES
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Lets take everything we learned and use an example of a car driving.
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### FINDING VELOCITY (Using a Derivative)
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Lets say a car travels 50 miles every hour.
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Instead of $x,y$ we will use $t,s$ for time and distance.
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We can write the distance function as
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$$
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s(t) = 50t
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s = f(t) = 50t
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$$
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where `t` is time in hours and `s` is distance in miles.
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The velocity of the car is the **derivative** of the distance function.
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$$
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v(t) = \frac{ds}{dt} = 50
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\frac{ds}{dt} = f'(t) = 50
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$$
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So the velocity of the car is 50 mph.
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We can write the velocity function as
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$$
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v(t) = 50
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\frac{ds}{dt} = f'(t) = 50
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$$
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The distance the car travels is the **integral** of the velocity function.
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The distance function is the **integral** of the velocity function.
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$$
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s(t) = \int v(t) dt = 50t + C
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$$
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So the distance the car travels is 50t + C.

‎other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet/.markdownlint.json

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{
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"MD013": false
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"MD013": false,
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"MD036": false
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}

‎other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet/README.md

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Table of Contents
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* [OVERVIEW](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#overview)
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* [HOW TO SOLVE A DIFFERENTIAL EQUATION](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#how-to-solve-a-differential-equation)
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* [UNDERSTANDING f(x)](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#understanding-fx)
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* [f(x) in CALCULUS](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#fx-in-calculus)
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* [f(x) in DIFFERENTIAL EQUATIONS](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#fx-in-differential-equations)
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* [SUMMARY](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#summary)
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* [CLASSIFICATION OF DIFFERENTIAL EQUATIONS](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#classification-of-differential-equations)
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* [BY TYPE](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#by-type)
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* [BY ORDER](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet/differential-equations-cheat-sheet#by-order)
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* [calculus](https://github.com/JeffDeCola/my-cheat-sheets/tree/master/other/stem/math/pure/changes/calculus-cheat-sheet)
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cheat sheet
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## OVERVIEW
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| CONTEXT | WHAT f(x) REPRESENTS | WHAT WE WANT TO FIND |
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|------------------------|------------------------------------|--------------------------------------|
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| CALCULUS (Derivatives) | $f(x)$ is the original function | The derivative $f'(x)=\frac{dy}{dx}$ |
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| CALCULUS (Integrals) | $f(x)$ is function to integrate | Function $F(x)=\int f(x)dx$ |
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| DIFFERENTIAL EQUATIONS | $f(x)$ is the derivative of $y(x)$ | Solve for $y(x)$ by integrating |
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## OVERVIEW
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* **Calculus** is a broad field of mathematics that includes
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differentiation (finding derivatives) and
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integration (finding integrals).
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The goal of diffEQ is to find the function y(x) that satisfies the equation.
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```
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## HOW TO SOLVE A DIFFERENTIAL EQUATION
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Given this first-order ordinary differential equation
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let's walk through the steps to solve it.
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$$
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\frac{dy}{dx} = 2x
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$$
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Separate the variables,
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$$
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dy = 2x dx
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$$
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Integrate both sides.
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$$
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\int dy = \int 2x dx
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$$
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$$
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y = x² + C
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$$
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Solve for the constant C using initial conditions.
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$$
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y(0) = 0² + C = 0
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$$
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Hence
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$$
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C = 0
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$$
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Therefore, the solution to the differential equation is
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$$
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y = x²
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$$
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## UNDERSTANDING f(x)
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Understanding the role of $f(x)$ is important because:
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In calculus, we define f(x) as the original function, and we
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find the derivative or integral on that function.
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#### Derivative
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**Derivative**
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Given a function, find the rate of change.
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f'(x) = \frac{dy}{dx} = 2x + 3
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$$
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#### Integral
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**Integral**
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Given the rate of change, find the function.
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y'(x) = \frac{dy}{dx} = f(x)
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$$
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To solve for y(x), we integrate both sides of the equation,
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Separate the variables
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$$
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{dy} = f(x){dx}
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$$
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integrate both sides of the equation,
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$$
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\int \frac{dy}{dx} dx = \int f(x) dx
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\int {dy} = \int f(x) dx
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$$
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$$
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y'(x) = \frac{dy}{dx} = f(x) = 2x + 3
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$$
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Solve for y(x),
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or,
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$$
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\frac{dy}{dx} = 2x + 3
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$$
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Separate the variables,
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$$
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dy = (2x + 3) dx
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$$
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Integrate both sides,
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$$
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\int \frac{dy}{dx} dx = \int (2x + 3) dx
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\int dy = \int (2x + 3) dx
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$$
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$$

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