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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.

Definition 4.6.1: Average Value of a Function

If $f$ is a continuous function on the bounded interval $\left[a\,b\right]$, its average value on $\left[a\,b\right]$ is
${f}_{\mathrm{avg}}\=\frac{1}{ba}$ ${\int}_{a}^{b}f\left(x\right)\mathit{DifferentialD;}x$



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

Theorem 4.6.1: Mean Value Theorem

1.

$f$ is continuous on the bounded interval $\left[a\,b\right]$

⇒
1.

$f$ attains its average value for at least one $c$ in $\left(a\,b\right)$




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

${f}_{\mathrm{avg}}\=f\left(c\right)\=\frac{1}{ba}{\int}_{a}^{b}f\left(x\right)\mathit{DifferentialD;}xequals;\frac{F\left(b\right)F\left(a\right)}{ba}equals;F\prime \left(c\right)$
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For the function $F\left(x\right)\,$the fraction $\frac{F\left(b\right)F\left(a\right)}{ba}$ 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.

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

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

${\int}_{a}^{b}f\left(x\right)\mathit{DifferentialD;}xequals;\left(ba\right){f}_{\mathrm{avg}}equals;\left(ba\right)f\left(c\right)$
for some $c$ in $\left[a\,b\right]$. This is the essence of the Integral Mean Value theorem, which gives the value of a definite integral in terms of the integrand evaluated at a single point.
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Maple has both a
tutor and a FunctionAverage command, which are used in Examples 4.6.(12).

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