P12 - t days after the rumor begins to circulate ( t = 0 )...

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Project 12 MAC 2233 1. Examine the function f ( x ) = x 3 - 3 x . From precalculus, what should be the two distinct possibilities for the number of turning points of any cubic (degree 3) polynomial function? In order for a smooth graph to “turn around”, what value would the slope of the curve need to be at the turning point? (Illustrate with a small picture.) Is the reverse true (can a graph have that slope without turning around)? Find the values of x at which f ( x ) has horizontal tangent lines. Find the zeros ( x -intercepts) of f ( x ) . Calculate the slope of f ( x ) at EACH of its zeros. How many horizontal tangent lines does the function g ( x ) = x 3 + 3 x have? Can it turn around? How many zeros does it have? Sketch the functions f ( x ) and g ( x ) , incorporating the above information and labelling the points where the horizontal tangent lines occur. Based on your discoveries so far, how does the derivative of a cubic function determine the number of turning points that it has?
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2. A sociologist determines that the percentage of a population that has heard a rumor
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Unformatted text preview: t days after the rumor begins to circulate ( t = 0 ) is given by R ( t ) = 100 t 2 20 + t + t 2 What is the domain of R ( t ) in this context? Calculate R (0) and lim t →∞ R ( t ) . Do your answers make sense? Calculate dR dt , simplify, and factor. What is the initial rate of change of R ( t ) with respect to t ? (Include units.) How can this be? What percentage of the population has heard the rumor after one day? What is the rate of change of R ( t ) with respect to t at that time? (Include units; round to the nearest tenth.) How many days pass before forty percent of the population has heard the rumor? fifty per-cent? What is the rate of change of R ( t ) with respect to t at those time? (Include units.) Of the times that you examined above, what was the general trend (up and down) of the rate at which the rumor is spread? Briefly explain why you think this might be true about the spread of a rumor....
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This note was uploaded on 12/27/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.

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P12 - t days after the rumor begins to circulate ( t = 0 )...

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