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

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

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.

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).

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

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

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

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

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

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