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## 4.3 Using Derivatives for Curve Sketching

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**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**Photo by Vickie Kelly, 2007**Greg Kelly, Hanford High School, Richland, Washington 4.3 Using Derivatives for Curve Sketching Yellowstone Falls, Yellowstone National Park**Photo by Vickie Kelly, 2007**Greg Kelly, Hanford High School, Richland, Washington 4.3 Using Derivatives for Curve Sketching Mammoth Hot Springs, Yellowstone National Park**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.**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).**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**There are roots at and .**Possible extreme at . Set maximum at minimum at Example: Graph 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 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 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:**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**rising, concave down**local max falling, inflection point local min rising, concave up Make a summary table: p