Using Derivatives for Curve Sketching 导数的应用:画图.ppt

Using Derivatives for Curve Sketching 导数的应用:画图.ppt

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Using Derivatives for Curve Sketching 导数的应用:画图

4.3 Using Derivatives for Curve Sketching Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1995 Old Faithful Geyser, Yellowstone National Park 4.3 Using Derivatives for Curve Sketching Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2007 Yellowstone Falls, Yellowstone National Park 4.3 Using Derivatives for Curve Sketching Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2007 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. 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 . Set First derivative test: negative positive positive Possible extreme at . We can use a chart to organize our thoughts. Example: Graph There are roots at and . Set First derivative test: maximum at minimum at Possible extreme at . Example: Graph First derivative test: NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! 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 .

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