4 3 Using Derivatives for Curve Sketching In
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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).
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
Homework: 4. 3 a 4. 3 p 215 1, 7, 18, 27 4. 2 p 202 6, 15, 24, 33 3. 3 p 124 46 4. 3 b 4. 3 p 215 3, 9, 16, 21, 29, 30, 35, 46 3. 4 p 135 10, 19