Reading a Derivative Graph

What f prime says about f.

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

NameDate

Figure — use these to answer the problems

Move the point and read the tangent slope. Where the slope is zero, the curve is momentarily flat.

y = 1x³ − 3x + 0

b-3

Point of tangency1

Figure 1

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

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
A maximum of f prime corresponds to what in f? 1 a maximum, 2 an inflection point.
Answer
f(x) = x³ − 3x, so f prime = 3x² − 3. How many critical points?
Answer

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

4 problems

9
Match each fact about f prime with what it says about f.
Draw a line from each item on the left to its match on the right.
- f prime positive
- f prime zero with a sign change
- f prime turning
- f is increasing
- f has a local extremum
- f has an inflection point
f double prime is negative. Concave up or down? 1 up, 2 down.
Answer
The Sign
f prime is negative on an interval. Is f increasing or decreasing? 1 increasing, 2 decreasing.
Answer
The Turning Point
f prime changes from positive to negative at x = 4. Maximum or minimum? 1 maximum, 2 minimum.
Answer