AP Calculus AB and BC glossary

Average rate of change

The average rate of change of a function over an interval is the change in output divided by the change in input, which is the slope of the secant line joining the endpoints. It needs no calculus, unlike the instantaneous rate of change.

f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}

Average rate of change is a two-point calculation over an interval. Instantaneous rate of change is a one-point calculation and requires a limit. Confusing the two is the most frequent error on rate-interpretation questions.

The Mean Value Theorem connects them: on a suitable interval, some interior point has an instantaneous rate exactly equal to the average rate across the whole interval.

For a rate function

If you are given a rate r(t)r(t) and asked for its average value, that is 1b−a∫abr(t) dt\frac{1}{b-a}\int_a^b r(t)\,dt, which is a different quantity from the average rate of change of rr.

Appears in: Unit 2: Defining the Derivative, Unit 4: Contextual Applications