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Lesson 4-3: Using Derivatives to Sketch Curves

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1 Lesson 4-3: Using Derivatives to Sketch Curves
AP Calculus Mrs. Mongold

2 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.

3 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).

4 We can use a chart to organize our thoughts.
Example: Graph There are roots at and Possible extreme at We can use a chart to organize our thoughts. Set First derivative test: negative positive positive

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

6 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 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

7 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.

8 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 There is an inflection point at x = 1 because the second derivative changes from negative to positive.

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

10 Homework Page 203-204/3-30 multi of 3


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