1. 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.

2. 4.3 Connecting f ’ and f ” with the graph of f

3. 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.

4. 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).

5. 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.

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

7. 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.

8. 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.

9. Slide 4- 9 Second Derivative Test for Local Extrema

10. Slide 4- 10 Concavity

11. Slide 4- 11 Concavity

12. 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

13. Slide 4- 13 Point of Inflection

14. Slide 4- 14 Concavity Test

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

16. Slide 4- 16 Learning about Functions from Derivatives

17. Slide 4- 17 First Derivative Test for Local Extrema

18. Slide 4- 18 First Derivative Test for Local Extrema