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Rates of Change, Concavity & Inflection

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The rate of change of the rate of change

A function can be increasing in two very different ways. A savings account earning simple interest increases by the same amount each year — the rate of change is constant. One earning compound interest increases by more each year — the rate of change is itself increasing. Both graphs go up; only the second one bends upward. That bending is concavity, and in AP Precalculus it is described entirely through rates of change, with no second derivative in sight.

Concavity from rates of change
rate of change increasing → concave up · rate of change decreasing → concave down
True regardless of whether the function itself is increasing or decreasing. A falling graph can be concave up (falling ever more gently) or concave down (falling ever faster).

Reading it off a table

Given equally spaced x values, compute the successive differences in f. If those differences are growing, the rate of change is increasing and the graph is concave up. If they are shrinking, it is concave down. If they are constant, the function is linear and has no concavity at all. This "differences of the differences" test is the table version of concavity.

Worked example

A table gives f(0) = 2, f(1) = 5, f(2) = 11, f(3) = 20, f(4) = 32. Describe the function's behavior.

  1. 1.First differences: 5 − 2 = 3, 11 − 5 = 6, 20 − 11 = 9, 32 − 20 = 12.
  2. 2.Since the x values are spaced 1 apart, these differences are the average rates of change on each subinterval: 3, 6, 9, 12.
  3. 3.All are positive, so f is increasing throughout.
  4. 4.The rates 3, 6, 9, 12 are themselves increasing, so the rate of change is increasing.
  5. 5.Second differences are 3, 3, 3 — constant, which is the signature of a quadratic.
Answer: f is increasing and concave up on [0, 4]: every average rate of change is positive, and those rates grow steadily from 3 to 12. The constant second differences of 3 additionally identify f as quadratic.
Checkpoint

A quantity is decreasing, and the amount it drops each year is getting smaller. What does the graph look like?

Watch out

Concave up does not mean "going up." A decay curve leveling off is concave up while decreasing the whole way. Judge concavity from the rate, never from whether the graph rises.

Points of inflection

A point of inflection is a point where concavity changes — up to down, or down to up. It is where the rate of change stops increasing and starts decreasing (or the reverse), so it is where the rate of change is momentarily at a maximum or minimum. On an S-shaped growth curve the inflection point is the moment of fastest growth: growth has been accelerating and is about to start slowing.

Worked example

For f(x) = x³ − 3x², use average rates of change over short intervals to locate the point of inflection.

  1. 1.Estimate the rate near x = 0 using [−0.1, 0.1]: f(0.1) = 0.001 − 0.03 = −0.029, f(−0.1) = −0.001 − 0.03 = −0.031, so AROC = 0.002/0.2 = 0.01.
  2. 2.Near x = 1 using [0.9, 1.1]: f(1.1) = 1.331 − 3.63 = −2.299, f(0.9) = 0.729 − 2.43 = −1.701, so AROC = −0.598/0.2 = −2.99.
  3. 3.Near x = 2 using [1.9, 2.1]: f(2.1) = 9.261 − 13.23 = −3.969, f(1.9) = 6.859 − 10.83 = −3.971, so AROC = 0.002/0.2 = 0.01.
  4. 4.The rates run 0.01 → −2.99 → 0.01: decreasing, then increasing. The turn happens near x = 1.
Answer: The rate of change decreases from about 0.01 down to about −2.99 and then rises back, so it bottoms out near x = 1. Concavity changes from down to up there, making x = 1 the point of inflection, at the point (1, −2).
On the exam

Free-response questions ask you to justify concavity, not just assert it. A justification cites evidence: "the average rates of change on successive intervals are 3, 6, 9, 12, which are increasing, so f is concave up." Naming the evidence is where the point is earned.

Checkpoint

A cubic function is concave down for x < 2 and concave up for x > 2. What can you conclude about x = 2?

Answer the 2 checkpoints as you read.

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