1. 4.3
Using Derivatives for Curve Sketching

2. In the past, one of the important uses of derivatives was as an aid in curve sketching. Even though we usually use a calculator or 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

10. Homework:
4.3a 4.3 p215 1,7,18,27
4.2 p202 6,15,24,33
3.3 p124 46
4.3b 4.3 p215 3,9,16,21,29,30,35,46
3.4 p135 10,19