Mean Value Theorem for Derivatives

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# Mean Value Theorem for Derivatives - PowerPoint PPT Presentation

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

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:

Mean Value Theorem for Derivatives

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

Tangent parallel to chord.

Slope of tangent:

Slope of chord:

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:

Functions with the same derivative differ by a constant.

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

could be

or could vary by some constant .

Example 6:

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

so:

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.

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.