The average value of a function over an interval is made precise by Definition 4.6.1. Theorem 4.6.1 then says that a continuous function attains its average value at least once on a closed, bounded interval.

A formal statement of the Mean Value theorem for integrals is given in Theorem 4.6.1.

For a geometric interpretation of the Mean Value theorem, let $F\left(x\right)$ be an antiderivative for $f\left(x\right)$ in

For the function $F\left(x\right)\,$the fraction $\frac{F\left(b\right)-F\left(a\right)}{b-a}$ is the slope of the (secant) line connecting the endpoints $\left[a\,F\left(a\right)\right]$ and $\left[b\,F\left(b\right)\right]$. The number $F\prime \left(c\right)$ is the slope of the tangent line. Hence, the Mean Value theorem implies that under suitable conditions, there is at least one point on the graph of $F\left(x\right)$ where the tangent line is parallel to the secant through the endpoints.

Incidentally, this geometric interpretation of the Mean Value theorem is consistent with the linear approximation afforded by Theorem 3.4.1.

From the definition of $f_{\mathrm{avg}}$ and from the existence of $x\=c$ in the Mean Value theorem, it follows that

Maple has both a  tutor and a FunctionAverage command, which are used in Examples 4.6.(1-2).

Maple also has both a  tutor and a MeanValueTheorem command, used in Example 4.6.4.