1.2 Rates of Change

Average Rate of Change

The average rate of change of a function over an interval [a, b] is (f(b) − f(a)) / (b − a). It represents the slope of the secant line connecting the two points on the graph. This measures how much the output changes per unit input over that interval, not at a single instant.

Rates of Change Can Vary

For nonlinear functions, the average rate of change differs across intervals — this is why curves bend. Computing rates over smaller and smaller intervals lets us estimate the instantaneous rate of change at a point, which is the slope of the tangent line there.

Concavity and Rate of Change

If average rate of change is increasing over consecutive equal-width intervals, the function is concave up; if decreasing, it's concave down. This connects rates of change to the shape of a graph, helping predict whether a function's growth is speeding up or slowing down.

Key terms

Average rate of change formula

(f(b) − f(a)) / (b − a), the slope of the secant line through two points on the graph.

Secant line

A line connecting two points on a curve; its slope equals the average rate of change over that interval.

Instantaneous rate of change

The rate of change at a single point, approximated by shrinking the interval used for average rate of change.

Concave up

Graph shape where average rate of change increases over consecutive equal-width intervals.

Concave down

Graph shape where average rate of change decreases over consecutive equal-width intervals.

Try an AP-style question

What is the average rate of change of f(x) = x² on [1, 4]?

1. 15
2. 3
3. 5
4. 8