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Mean Value Theorem for Derivatives - PowerPoint PPT Presentation


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Photo by Vickie Kelly, 2002. Greg Kelly, Hanford High School, Richland, Washington. Mean Value Theorem for Derivatives. Teddy Roosevelt National Park, North Dakota. If f ( x ) is a continuous function over [ a , b ], and differentiable over (a,b), then at some point between a and b :.

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Presentation Transcript
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Photo by Vickie Kelly, 2002

Greg Kelly, Hanford High School, Richland, Washington

Mean Value Theorem for Derivatives

Teddy Roosevelt National Park, North Dakota

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If f (x) is a continuous function over [a,b], and differentiable over (a,b), then at some point between a and b:

Mean Value Theorem for Derivatives

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If f (x) is a continuous function over [a,b], and differentiable over (a,b), then at some point between a and b:

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

Mean Value Theorem for Derivatives

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Tangent parallel to chord.

Slope of tangent:

Slope of chord:

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A function is increasing over an interval if the derivative is always positive.

A function is decreasing over an interval if the derivative is always negative.

A couple of somewhat obvious definitions:

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Functions with the same derivative differ by a constant.

These two functions have the same slope at any value of x.

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could be

or could vary by some constant .

Example 6:

Find the function whose derivative is and whose graph passes through .

so:

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Example 6:

Find the function whose derivative is and whose graph passes through .

so:

Notice that we had to have initial values to determine the value of C.

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Antiderivative

A function is an antiderivative of a function

if for all x in the domain of f. The process of finding an antiderivative is antidifferentiation.

The process of finding the original function from the derivative is so important that it has a name:

You will hear much more about antiderivatives in the future.

This section is just an introduction.