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, 2007 Yellowstone Falls, Yellowstone National Park

4.3 Using Derivatives for Curve Sketching Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Mammoth Hot Springs, Yellowstone National Park

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.

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. We can use a chart to organize our thoughts.

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

Example: Graph First derivative test: NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! 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.

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

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

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