Reading a Derivative Graph

What f prime says about f.

12 problems · about 22 minutes · FUN-4.A

What this lesson teaches

The student relates the graph of a derivative to the behaviour of the original function.

The sign of f prime gives increasing or decreasing.
A sign change of f prime marks a local extremum.
The sign of f double prime gives concavity, and its change marks an inflection point.

Straightforward practice. Get the method working first.

5 problems

f prime changes from positive to negative at x = 3. Maximum or minimum? 1 max, 2 min.
Answer 1
Why A maximum.
A graph shows f prime with a peak at x = 2. What does that say about f?
Answer f has an inflection point at x = 2.
Why An inflection point of f.
f prime is positive. Is f increasing or decreasing? 1 increasing, 2 decreasing.
Answer 1
Why A positive slope rises.
f prime changes from negative to positive at x = 1. Maximum or minimum? 1 max, 2 min.
Answer 2
Why Falling then rising.
f double prime is positive. Concave up or down? 1 up, 2 down.
Answer 1
Why Positive is up.

The same ideas with more to keep track of.

3 problems

f prime is zero at x = 0 for y = x³. Is it an extremum? 1 yes, 0 no.
Answer 0
Why The sign does not change.
A maximum of f prime corresponds to what in f? 1 a maximum, 2 an inflection point.
Answer 2
Why f prime turning means f double prime is zero.
f(x) = x³ − 3x, so f prime = 3x² − 3. How many critical points?
Answer 2
Why 3x² = 3.

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

4 problems

Match each fact about f prime with what it says about f.
Answer f prime positive → f is increasing; f prime zero with a sign change → f has a local extremum; f prime turning → f has an inflection point
Why A turning derivative means the concavity flips.
f double prime is negative. Concave up or down? 1 up, 2 down.
Answer 2
Why Negative is down.
The Sign: f prime is negative on an interval. Is f increasing or decreasing? 1 increasing, 2 decreasing.
Answer 2
Why Decreasing.
The Turning Point: f prime changes from positive to negative at x = 4. Maximum or minimum? 1 maximum, 2 minimum.
Answer 1
Why A maximum.