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Derivative_Sketch

# Derivative_Sketch - WHAT DOES f SAY ABOUT f Since f(x...

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WHAT DOES f 0 SAY ABOUT f Since f 0 ( x ) represents the slope of the curve y = f ( x ), we can find the direction in which the curve proceed at each point. Thus, we can find some information about f ( x ) from information about f 0 ( x ). In particular, we will see how the derivative of f can tell us where f is increasing or decreasing. (1) If f 0 ( x ) > 0 on an interval, then f is increasing on that interval. (2) If f 0 ( x ) < 0 on an interval, then f is decreasing on that interval. Example 1. The graph of a function f 0 is given. What can we say about f ? Solution From Figure 1, we can see that f 0 ( x ) is negative when 0 < x < 1. So the original function f must be decreasing on the interval (0 , 1). Similarly, f 0 ( x ) is positive for x < 0 and x > 1, so f is increasing on the intervals ( -∞ , 0) and (0 , ). Also note that f has horizontal tangents when x = 0 and x = 1, since f 0 (0) = f 0 (1) = 0. We say that f in the above example has a local maximum at 0 and local minimum at 1. Since f 00 is the derivative of f 0 , if f 00 is positive, then f 0 is an increasing function.

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