1. Using Derivatives for Curve Sketching
4.3
Using Derivatives for Curve Sketching
Yellowstone Falls, Yellowstone National Park
Created by Greg Kelly, Hanford High School, Richland, Washington
Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Photo by Vickie Kelly, 1995

2. In the past, one of the important uses of derivatives was as a critical aid in curve sketching. Now, we usually use a calculator or computer to draw complicated graphs, but it is still important to understand the relationships between derivatives and graphs.

3. First derivative:
is positive
y is rising.
is negative
y is falling.
is zero
POSSIBLE local max or min on y (check for sign change on each side).
Second derivative:
y is concave up.
is positive
is negative
y is concave down.
is zero
POSSIBLE inflection point on y
where concavity changes
(check for sign change on each side).

4. There are roots/x-intercepts at and .
Example:
Graph
There are roots/x-intercepts at and
First derivative test:
Set
to identify extrema
candidates:
y’ > 0
y’ < 0
y’ > 0
Possible extrema at

5. There are roots/x-intercepts at and .
Example:
Graph
There are roots/x-intercepts at and
First derivative test:
Set
to identify extrema
candidates:
y’ > 0
y’ < 0
y’ > 0
maximum at
minimum at

6. \ \ \ Graph There are roots at and . Possible extrema at .
Example:
Graph
There are roots at and
Possible extrema at
Or you could use the second derivative test:
Set
y’’ < 0 \
concave down \
maximum at
y’’ > 0
concave up \
minimum at

7. Possible inflection point at .
Example:
Graph
Last, to identify the location of inflection points, set the second derivative equal to zero.
Possible inflection point at
negative
positive
inflection point at

8. p Make a summary table: y rising , concave down y local max
, inflection point
y falling
y local min
, concave up
y rising
, concave up p