2.1 Defining Average and Instantaneous Rate of Change at a Point

Average Rate of Change

The average rate of change of a function f over [a, b] is the slope of the secant line connecting (a, f(a)) and (b, f(b)): (f(b) - f(a)) / (b - a). It measures how much the output changes per unit input, on average, over an interval — not at a single instant.

Instantaneous Rate of Change

The instantaneous rate of change at a point x = a is the limit of the average rate of change as the interval shrinks to zero: lim(h→0) [f(a+h) - f(a)]/h. This is the derivative f'(a), representing the slope of the tangent line at that exact point.

Secant vs. Tangent Lines

A secant line passes through two points on a curve and gives average rate of change. A tangent line touches the curve at one point and gives instantaneous rate of change. As the second point on the secant approaches the first, the secant line's slope approaches the tangent line's slope.

Key terms

Average rate of change formula

(f(b) - f(a)) / (b - a) — the slope of the secant line over [a, b].

Instantaneous rate of change formula

lim(h→0) [f(a+h) - f(a)] / h — the derivative f'(a), slope of the tangent line at x = a.

Secant line

A line through two points on a curve; its slope gives the average rate of change.

Tangent line

A line touching the curve at one point; its slope gives the instantaneous rate of change.

Relationship between secant and tangent lines

As the interval between two points on a secant line shrinks to zero, the secant line's slope approaches the tangent line's slope (the derivative).

Try an AP-style question

For f(x) = x² − 3x, what is lim(h→0) [f(2 + h) − f(2)] / h?

1. 1
2. −2
3. 0
4. 4