4.3 Using Derivatives for Curve Sketching.

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Presentation transcript:

4.3 Using Derivatives for Curve Sketching

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).

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

First derivative test: 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 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 .

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.

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.

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

Homework: 4.3a 4.3 p215 1,7,18,27 4.2 p202 6,15,24,33 3.3 p124 46 4.3b 4.3 p215 3,9,16,21,29,30,35,46 3.4 p135 10,19