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Section 3.4 Average value and average rates
Objectives
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Interpret an average as redistributing a quantity evenly across an interval.
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View average value as the mean height of a function over a region.
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Relate average rates of change to accumulated change over an interval.
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The average value of \(f(x)\) on the interval \([a,b]\) is
\begin{equation*}
\frac{\DS\int_{a}^{b} f(x)\dd{x}}{b-a}
\end{equation*}
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If we think of \(f'\) as the instantaneous rate of change of \(f\text{,}\) then the average rate of change of \(f\) is
\begin{equation*}
\frac{\DS\int_{a}^{b} f'(x)\dd{x}}{b-a}=\frac{f(b)-f(a)}{b-a}
\end{equation*}
Aside
