# Antiderivatives, Differential Equations, and Slope FieldsPage 1

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AP Calculus AB

Antiderivatives,

Differential Equations,

and Slope Fields

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## Review

Solution

Consider the equation

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## Antiderivatives

What is an inverse operation?

Examples include:

Addition and subtraction

Multiplication and division

Exponents and logariths

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Antiderivatives

Differentiation also has an inverse…

antidefferentiation

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Antiderivatives

Consider the function whose derivative is given by .

What is ?

Solution

We say that is an antiderivative of .

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Antiderivatives

Notice that we say is an antiderivative and not the antiderivative. Why?

Since is an antiderivative of , we can

say that .

If and , find

and .

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## Differential Equations

Recall the earlier equation .

This is called a differential equation and could also be written as .

We can think of solving a differential equation as being similar to solving any other equation.

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Differential Equations

Trying to find y as a function of x

Can only find indefinite solutions

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Differential Equations

There are two basic steps to follow:

1. Isolate the differential

Invert both sides…in other words, find

the antiderivative

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Differential Equations

Since we are only finding indefinite solutions, we must indicate the ambiguity of the constant.

Normally, this is done through using a letter to represent any constant. Generally, we use C.

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Solution

Differential Equations

Solve

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## Slope Fields

Consider the following:

HippoCampus

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Slope Fields

A slope field shows the general “flow” of a differential equation’s solution.

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