1. Using Derivatives for Curve Sketching
4.3
Using Derivatives for Curve Sketching
Old Faithful Geyser, Yellowstone National Park
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1995

2. Using Derivatives for Curve Sketching
4.3
Using Derivatives for Curve Sketching
Yellowstone Falls, Yellowstone National Park
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1995

3. 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.

4. 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).

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

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

7. Or you could use the second derivative test:
Example:
Graph
There are roots at and
Possible extreme at
Set
Or you could use the second derivative test:
negative
concave down
local maximum
positive
concave up
local minimum
maximum at
minimum at

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

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

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

11. Mathematicians never die - they only loose some of their functions.