1. Using Derivatives for Curve Sketching
4.3
Using Derivatives for Curve Sketching
Yellowstone Falls, Yellowstone National Park

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

3. First derivative:
is positive
Curve is rising.
is negative
Curve is falling.
is zero
Possible local maximum or minimum.
Second derivative:
is positive
Curve is concave up.
is negative
Curve is concave down.
is zero
Possible inflection point
(where concavity changes).

4. We can use a chart to organize our thoughts.
Example:
Graph
There are roots at and
Possible extreme at
We can use a chart to organize our thoughts.
Set
First derivative test:
negative
positive
positive

5. First derivative test:
Example:
Graph
There are roots at and
Possible extreme at
Set
First derivative test:
maximum at
minimum at

6. Example:
Graph
NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation!
First derivative test:
There is a local maximum at (0,4) because for all x in
and for all x in (0,2) .
There is a local minimum at (2,0) because for all x in
(0,2) and for all x in

7. Example:
Graph
There are roots at and
Possible extreme at
Or you could use the second derivative test:
Because the second derivative at
x = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum.
Because the second derivative at
x = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.

8. Possible inflection point at .
Example:
Graph
We then look for inflection points by setting the second derivative equal to zero.
Possible inflection point at
negative
positive
inflection point at
There is an inflection point at x = 1 because the second derivative changes from negative to positive.

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