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Unformatted text preview: Lecture 18 — How Derivatives Shape a Curve Another application of the derivative is to help us to sketch the shapes of curves for which the graph is unfamiliar to us. The two main features of a curve are its orientation (i.e., going upward or down ward) and the way it bends. We already know from what our derivative repre sents in terms of slope (rate of change), that when the derivative is positive, the function values are trending upward at that point. It turns out that one can prove this fairly easily on intervals in general: If f > on an interval, then f is increasing on the interval. If f < on an interval, then f is decreasing on the interval. On what intervals are the following functions increasing/decreasing? F ( t ) = 3 t 1 3 t L ( x ) = x 2 ln  x  + 8 x Fermat’s Theorem told us that all relative extrema occur at critical numbers. However, we noted that not all critical numbers yield relative extrema. The following is a simple test for these critical numbers when f is continuous:...
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 Spring '08
 ALL
 Calculus, Derivative, critical numbers

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