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7-8.

For each function below, calculate the average value over the given interval and state the value of such that equals the average value.

  1. Read the Math Note about how to compute the Mean Value of , given .

    To find the the time the function is at its average value, let the average value and solve for .

  1. Average Value . Now find the time, , that its average value.

Average (Mean) Values

To calculate the mean (average) value of a finite set of items, add up the values of items and divide by the number of items.

Integrals help us add over a continuous interval. Therefore, for any continuous function :

mean value of over

Since , we can also calculate the average value of any function using its antiderivative . Its average slope gives the average rate of change of , which is the same as the average value of

  mean rate of change of over 

The Mean Value Theorem states that a differentiable function will reach its average (mean) value at least once on any closed interval.
Check your values of t in parts (a) and (b). Are they within the the given closed intervals?

The Mean Value Theorem

The Mean Value Theorem for Integrals

If is continuous on , then there exists at least one point in such that .

The Mean Value Theorem for Derivatives

If is continuous on and differentiable on , then there exists at least one point in such that .