4 3 Using Derivatives for Curve Sketching Old

4. 3 Using Derivatives for Curve Sketching Old Faithful Geyser, Yellowstone National Park Photo by Vickie Kelly, 1995 Greg Kelly, Hanford High School, Richland, Washington

4. 3 Using Derivatives for Curve Sketching Yellowstone Falls, Yellowstone National Park Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington

4. 3 Using Derivatives for Curve Sketching Mammoth Hot Springs, Yellowstone National Park Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington

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 . Possible extreme at Set . We can use a chart to organize our thoughts. First derivative test: negative positive

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

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 and for all x in (0, 2). There is a local minimum at (2, 0) because (0, 2) and for all x in

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.

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

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