For each function below, calculate the average value over the given interval and state the value of
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 .
. 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
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
The Mean Value Theorem for Derivatives