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.

4.3 Connecting f ’ and f ” with the graph of f

Slide 4- 3 What you’ll learn about First Derivative Test for Local Extrema Concavity Points of Inflection Second Derivative Test for Local Extrema Learning about Functions from Derivatives …and why Differential calculus is a powerful problem-solving tool precisely because of its usefulness for analyzing functions.

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.

Slide 4- 9 Second Derivative Test for Local Extrema

Slide Concavity

Slide Concavity

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

Slide Point of Inflection

Slide Concavity Test

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

Slide Learning about Functions from Derivatives

Slide First Derivative Test for Local Extrema

Slide First Derivative Test for Local Extrema