Antiderivatives, Differential Equations, and Slope Fields.ppt

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AP Calculus AB

Antiderivatives,

Differential Equations,and Slope Fields

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Solution

Review

Consider the equation2xy

2xdy

dx

Find

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Antiderivatives

What is an inverse operation?

Examples include:

Addition and subtraction

Multiplication and division

Exponents and logarithms

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Antiderivatives

Differentiation also has an inverse

antidefferentiation

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Antiderivatives

Consider the function whose derivative is givenby .

What is ?

F 45xxf

xF

xF xf

Solution

We say that is an antiderivative of .

5

F x x

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Antiderivatives

Notice that we say is an antiderivative andnot the antiderivative. Why?

Since is an antiderivative of , we cansay that .

If and , find

and .

xF

xF xf xfxF '

3

5 xxG 25 xxH

xg xh

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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.

dxdy x2

xdxdy 2

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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

2. Invert both sidesin other words, find

the antiderivative

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Differential Equations

Since we are only finding indefinitesolutions, we must indicate the ambiguityof the constant.

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

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Solution

Differential Equations

Solvedx

dyx2

Cxy

2

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Slope Fields

Consider the following:

HippoCampus

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Slope Fields

Aslope fieldshows the general flow of adifferential equations solution.

Often, slope fields are used in lieu ofactually solving differential equations.

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Slope Fields

To construct a slope field, start with adifferential equation. For simplicitys sake welluse Slope Fields

Rather than solving the differential equation,well construct a slope field

Pick points in the coordinate plane

Plug in the x and y values The result is the slope of the tangent line at that

point

xdxdy 2

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Slope Fields

Notice that since there is no y in our equation,horizontal rows all contain parallel segments.The same would be true for vertical columns if

there were no x.

Construct a slope field for .yxdx

dy