Rates of change

The average rate of change of a function $f$ over the interval $[a, b]$ is

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

change in output divided by change in input. Geometrically it is the slope of the secant line, the straight line through the two endpoint points $(a, f(a))$ and $(b, f(b))$. Every wiggle between the endpoints is invisible to it: the average rate of change is a two-point summary, nothing more.

Units come along for free and are worth writing every time. If $d(t)$ is miles at $t$ hours, the average rate of change is in miles per hour. If the units of your answer are not output-per-input, you computed something else.

A function also has a rate of change at a single input value: how fast the output is responding right there. AP Precalculus does not compute this exactly; it estimates it, using the average rate of change over a small interval containing the point. The smaller the interval, the better the estimate, because there is less room for the rate to drift inside it.

From a table, that means: to estimate the rate at $x = 2$, prefer the rows closest to 2. An average over $[2, 2.1]$ beats an average over $[2, 3]$, because the wide interval blends in behavior far from the point you care about.

The single most tested distinction in this topic: the average rate of change over $[a, b]$ is generally not the rate at $a$, not the rate at $b$, and not the rate at any particular point you can name in advance. A car averaging 60 mph over an hour may have been doing 80 at the end and 30 in traffic at the start.

1. Asked for the rate over an interval? Use the two endpoints, nothing else.
2. Asked for the rate at a point? Build the smallest interval the data allows around that point.
3. Asked whether they are equal? Only a linear function guarantees it, because only a linear function has one rate everywhere.

The average also is not the average of the endpoint rates, and it is not the total change. Dividing by $b - a$ is what turns a change into a rate; skipping the division leaves you holding a change.

The sign of a rate of change tells direction: positive means the output rises as the input does, negative means it falls. The magnitude tells intensity: how much response per unit of input. These are separate dials. A rate of $-8$ describes a faster change than a rate of $+2$.

And neither dial is concavity. Successive average rates of $-8, -5, -2$ belong to a function that is falling (all negative) while its rate is climbing (concave up). Read the sign for direction, the trend of the rates for concavity, and never let the function's values stand in for either.