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Unformatted text preview: 220 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION 4.3 Derivatives and the Shape of Graphs The goal of this this section is to use the derivative of a function to help us sketch the graph of a function. We could use a graphing calculator to sketch the graph, but we are going to try to sketch the graphs by hand as an exercise and as part of a broader study of the behavior of functions. The figure below summarizes how derivative and slope of the tangent line are related. When function is increasing when the derivative is positive; the function is decreasing when the derivative is negative. If the derivative exists at a local max or a local min, then the derivative equals zero. Above we are looking at the graph of a function and drawing conclusions about the derivative. In the next example, we start with a function, and we use derivatives to sketch the graph. Example 1 Let f ( x ) = 3 x 4 4 x 3 12 x 2 + 5. 1. Find the critical points of the function. Solution We find f ( x ) and then solve f ( x ) = 0. f ( x ) = 12 x 3 12 x 2 24 x 4.3. DERIVATIVES AND THE SHAPE OF GRAPHS 221 f ( x ) = 0 12 x 3 12 x 2 24 x = 0 12 x ( x 2 x 2) = 0 12 x ( x 2)( x + 1) = 0 x = 0 , x = 2 , x = 1 Just by looking at the critical points alone, there is no way of knowing whether the function has a local max or a local min at these critical points. The next step will tell us whether these points are local maxipoints....
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 Spring '10
 Koines
 Math, Derivative, lim, Mathematical analysis, Convex function, local min

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