AP Calculus AB and BC

Average vs Instantaneous Rate of Change

Average rate of change measures over an interval and equals the slope of the secant line joining the endpoints. Instantaneous rate of change measures at a single point and equals the slope of the tangent line, which is the derivative.

Average rate of change

Use when: The question names two times or two inputs, or asks how much something changed per unit over a stretch.

Instantaneous rate of change

Use when: The question names one moment, or uses the words at the instant, at time equals, or how fast right now.

Side by side

	Average rate	Instantaneous rate
Formula	$\frac{f(b) - f(a)}{b - a}$	$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
Geometry	Secant slope	Tangent slope
Needs calculus	No	Yes
Inputs required	Two points	One point

The two are connected by a limit: shrink the interval and the average rate approaches the instantaneous rate. That is the entire construction of the derivative, and it is why the secant becomes the tangent as the second point slides in.

The Mean Value Theorem is the other bridge. On a suitable interval, some interior point has an instantaneous rate exactly equal to the average rate across the whole interval.

Watch for a rate function

If you are already given a rate $r(t)$ , then its average over an interval is $\frac{1}{b-a}\int_a^b r(t)\,dt$ , which is the average value of a function and not the average rate of change of $r$ .

Frequently asked questions

Is average rate of change the same as average value?

No. Average rate of change divides a change in output by a change in input. Average value integrates a function and divides by the interval length.

In the CED: Unit 2: Defining the Derivative, Unit 4: Contextual Applications