Concavity and the Rate of the Rate

How the rate of change is itself changing.

11 problems · about 22 minutes · AP Precalculus 1.3

What this lesson teaches

The student determines concavity from how average rates of change behave over successive intervals.

Concavity is what the average rates of change are doing over successive intervals.
Rising rates mean concave up; falling rates mean concave down.
Concavity and direction are independent — a function can fall while flattening.

Straightforward practice. Get the method working first.

4 problems

Rates over equal intervals are 1, 5, 11. Concavity? 1 for up, 2 for down.
Answer 1
Why Concave up.
Rates 2, 4, 6. Concavity? 1 for up, 2 for down.
Answer 1
Why Rising rates.
Rates 6, 3, 0. Concavity? 1 for up, 2 for down.
Answer 2
Why Falling rates.
Rates −6, −3, 0. Concavity? 1 for up, 2 for down.
Answer 1
Why −6 to −3 is an increase.

The same ideas with more to keep track of.

3 problems

Sort each description into the concavity it describes.
Answer Concave up: Rates increasing, Falling but flattening · Concave down: Rates decreasing, Rising but levelling off
Why Ask what the rates are doing, not what the function is doing.
f(x) = x² on [0,1], [1,2], [2,3]. What is the third rate?
Answer 5
Why (9 − 4) ÷ 1.
Can a function be decreasing and concave up at once?
Answer Yes — falling, but flattening out.
Why Direction and bend are separate questions.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Rates 4, 4, 4. Concavity? 0 for neither, 1 for up, 2 for down.
Answer 0
Why Constant rates mean a straight line.
A cubic changes concavity how many times?
Answer 1
Why Once, at its inflection point.
Successive Rates: Successive rates over equal intervals are 8, 5, 2. What is the concavity? Answer 1 for up, 2 for down.
Answer 2
Why Concave down.
Falling but Flattening: Rates over equal intervals are −9, −4, −1. Is the function increasing? Answer 1 for yes, 0 for no.
Answer 0
Why Decreasing, but concave up.