Revisiting Average Rates of Change, Using Integrals

A function's rate of change is itself a function: the original function's derivative. Thus, we can think of the average rate of change as the average value of the derivative, and calculate it in the method described previously. Say we wish to find the average rate of change of $f(x)$ over the range $[a, b]$. It is

$\frac{1}{b-a} \int_{a}^{b} f'(x)dx$

By the Fundamental Theorem of Calculus, this is

$\frac{f(b) - f(a)}{b-a}$,

the same value we learned before.