AP Calculus AB and BC

Average Value vs Average Rate of Change

Average value integrates a function over an interval and divides by the interval length, answering what constant height would give the same area. Average rate of change divides the change in output by the change in input, answering how fast the function moved on average.

Average value

Use when: You are given a quantity and asked for its typical size over an interval.

Average rate of change

Use when: You are asked how much a quantity changed per unit of input across an interval.

Side by side

	Average value	Average rate of change
Formula	$\frac{1}{b-a}\int_a^b f(x)\,dx$	$\frac{f(b) - f(a)}{b - a}$
Uses	An integral	Two function values
Geometry	Height of an equal-area rectangle	Secant slope
Related theorem	Mean Value Theorem for integrals	Mean Value Theorem

The trap is a rate function. If $r(t)$ is already a rate, then the average value of $r$ is $\frac{1}{b-a}\int_a^b r(t)\,dt$ , and that is also the average rate of change of the accumulated quantity. Asking for the average rate of change of $r$ itself is a different question entirely.

Both divide by $b - a$ , which is what makes them averages rather than totals. Dropping that division is the single most common slip.

Frequently asked questions

Is the average value ever attained by the function?

For a continuous function on a closed interval, yes. That is the Mean Value Theorem for integrals.

In the CED: Unit 8: Applications of Integration