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4.3 Using Derivatives for Curve Sketching

Discover the importance of derivatives in curve sketching and graph analysis. Learn how to utilize the first and second derivatives to identify critical points, inflection points, and concavity changes. Enhance your graphing skills with practical examples and detailed explanations.

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4.3 Using Derivatives for Curve Sketching

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  1. Photo by Vickie Kelly, 1995 Greg Kelly, Hanford High School, Richland, Washington 4.3 Using Derivatives for Curve Sketching Old Faithful Geyser, Yellowstone National Park

  2. Photo by Vickie Kelly, 1995 Greg Kelly, Hanford High School, Richland, Washington 4.3 Using Derivatives for Curve Sketching Yellowstone Falls, Yellowstone National Park

  3. In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs.

  4. is positive is negative is zero is positive is negative is zero First derivative: Curve is rising. Curve is falling. Possible local maximum or minimum. Second derivative: Curve is concave up. Curve is concave down. Possible inflection point (where concavity changes).

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

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

  7. 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 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:

  8. There are roots at and . Possible extreme at . 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 Or you could use the second derivative test:

  9. Possible inflection point at . There is an inflection point at x =1 because the second derivative changes from negative to positive. inflection point at Example: Graph We then look for inflection points by setting the second derivative equal to zero. negative positive

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

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