# P19 - Now piece your shapes together to draw the function h...

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Project 19 MAC 2311 1. Examine the function h ( x ) = 3 ln | x | x . What is its domain h ( x ) ? Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h ( x ) . Does the function have any removable discontinuities? What are the intercepts for the graph of h ( x ) ? Does it display any symmetry? If so, what kind? Calculate h 0 ( x ) , and use a number line to determine the interval(s) on which the function is increasing/decreasing. What are the local extrema? Are they cusps or horizontal tangent lines? Calculate h 00 ( x ) . Use a number line to determine the interval(s) on which the function is concave up/down. What are the points of inﬂection? Separate a number line into the intervals on which h ( x ) is: both increasing AND concave up both increasing AND concave down both decreasing AND concave up both decreasing AND concave down and draw the basic shape of the curve on each interval.

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Unformatted text preview: Now piece your shapes together to draw the function h ( x ) below. Label all intercepts, local extrema, inﬂection points, removable discontinuities, and asymptotes. 2. If a metal cable of radius r is covered by insulation so that the distance from the center of the cable to the exterior is R , the velocity of an electrical impulse through the cable is: ν =-cx ln( x ) for some positive constant c , where x is the ratio of r to R (i.e., x = r R ). Use some of the techniques learned so far to sketch a graph of the function ν ( x ) . (To evaluate lim x → + ν ( x ) , use L’Hˆopital’s Rule; if you have trouble with the c , take c = 1 .) Looking at your graph, what ratio x yields the maximum velocity for the impulse? How can you tell this just from your number line for the derivative ?...
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## This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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P19 - Now piece your shapes together to draw the function h...

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