← Back to course

Average Rate of Change

You’ll be able to

One number for a whole interval

The average rate of change of f on the interval from a to b is the amount f changed divided by the amount x changed: [f(b) − f(a)]/(b − a). Geometrically it is the slope of the secant line — the straight line drawn through the two points (a, f(a)) and (b, f(b)). It is a single number summarizing an entire interval, which means it deliberately throws away everything that happened in between.

Average rate of change
AROC on [a, b] = [f(b) − f(a)] / (b − a)
Units are (units of f) per (unit of x). Writing the units is half the point on an AP free-response question.

Why the units carry the meaning

If P(t) is a population in thousands and t is hours, then the average rate of change has units of thousand people per hour. That phrase is what makes the number interpretable — and it is the difference between a scoring point and a lost one. A bare "278.7" answers nothing; "the population grew by an average of about 278.7 thousand cells per hour over the first five hours" answers the question.

Worked example

For f(x) = x³ − 4x, find the average rate of change on [1, 3] and on [1, 1.5].

  1. 1.On [1, 3]: f(1) = 1 − 4 = −3 and f(3) = 27 − 12 = 15.
  2. 2.AROC = (15 − (−3))/(3 − 1) = 18/2 = 9.
  3. 3.On [1, 1.5]: f(1.5) = 3.375 − 6 = −2.625, and f(1) = −3.
  4. 4.AROC = (−2.625 − (−3))/(1.5 − 1) = 0.375/0.5 = 0.75.
  5. 5.The two answers differ enormously, and neither is wrong — they describe different intervals.
Answer: On [1, 3] the average rate of change is 9; on [1, 1.5] it is 0.75. The same function can have wildly different average rates of change depending on the interval chosen, because the secant line through the endpoints changes when an endpoint moves.
Watch out

Average rate of change is not the average of the rates, and it is not f(b) − f(a). Dividing by (b − a) is not optional decoration — leave it off and the number has the wrong units and the wrong size.

Checkpoint

A tank holds V(t) = 200 − 3t² liters of water t minutes after a valve opens. What is the average rate of change of V on [0, 4]?

Shrinking the interval

Keep a fixed and slide b closer to it. The secant line pivots, and its slope settles down toward a limiting value: the slope of the tangent line at a — the instantaneous rate of change. Precalculus stops just short of computing that limit; calculus calls it the derivative. But the idea is fully available now, and the AP Precalculus exam expects you to describe it: as the interval shrinks, the average rate of change approaches the rate of change at the point.

Worked example

For f(x) = x², compute the average rate of change on [2, 2 + h] for h = 1, 0.1 and 0.01. What value is it approaching?

  1. 1.AROC = [(2 + h)² − 4]/h = [4 + 4h + h² − 4]/h = (4h + h²)/h = 4 + h, for h ≠ 0.
  2. 2.h = 1 gives 5. h = 0.1 gives 4.1. h = 0.01 gives 4.01.
  3. 3.The algebra shows the pattern exactly: the answer is always 4 + h.
Answer: The average rates are 5, 4.1 and 4.01, approaching 4. The simplification to 4 + h makes it visible that the limiting slope at x = 2 is exactly 4, with no numerical guesswork.
On the exam

When a free-response question says "average rate of change," it wants a computation and units. When it says "describe the rate of change," it usually wants increasing or decreasing, and whether that rate is itself increasing or decreasing — a different question, taken up in the next lesson.

Checkpoint

For a linear function f(x) = mx + b, what is the average rate of change on any interval [a, b]?

Answer the 2 checkpoints as you read.

Sign in to save your progress