Concavity and the Rate of the Rate

How the rate of change is itself changing.

11 problems · about 22 minutes · AP Precalculus 1.3

NameDate

Figure — use these to answer the problems

A cubic changes concavity once. Move the points across that place.

y = 1x³ + 0x + 0

First point x-3

Second point x-1

Figure 1

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
Rates 2, 4, 6. Concavity? 1 for up, 2 for down.
Answer
Rates 6, 3, 0. Concavity? 1 for up, 2 for down.
Answer
Rates −6, −3, 0. Concavity? 1 for up, 2 for down.
Answer

The same ideas with more to keep track of.

3 problems

Sort each description into the concavity it describes.
Write each item under the heading it belongs to: Rates increasing · Rates decreasing · Falling but flattening · Rising but levelling off
Concave up
Concave down
f(x) = x² on [0,1], [1,2], [2,3]. What is the third rate?
Answer
7
Can a function be decreasing and concave up at once?
- Yes — falling, but flattening out.
- No.

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
A cubic changes concavity how many times?
Answer
Successive Rates
Successive rates over equal intervals are 8, 5, 2. What is the concavity? Answer 1 for up, 2 for down.
Answer
Falling but Flattening
Rates over equal intervals are −9, −4, −1. Is the function increasing? Answer 1 for yes, 0 for no.
Answer