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

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

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).

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

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

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

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

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

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

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