08.Concavity

# 08.Concavity - dollars each month The graph of R is given...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Name: Two points . Professor : Stephen Suen Section: Peer Leaders : Hana Aboul-Hosn, Toni Jung, Renan Mendonca 8. Concavity and points of diminishing returns . For a di±erentiable function f , we say that f is con- cave up on an interval I if f is increasing on the interval, and f is concave down on an interval I if f is decreasing on the interval. Concavity of functions is related to points of diminishing returns. We did not go through points of diminishing returns in class, and the purpose of this worksheet is to discuss this topic in details. Example . (Point of diminishing returns.) Company XYZ wants to investigate the e±ect of advertise- ment on the revenue of a certain product. The following function is used to model the situation: R = - 1 10 x 3 + 2 x 2 + 4 x + 60 , 0 x 14 . where R is the monthly revenue in thousands of dollars, and x is the amount of advertising in thousand

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Unformatted text preview: dollars each month. The graph of R is given below. We shall follow the steps below to investigate the e±ect of the amount of advertising x on the revenue R . (a) Evaluate dR dx when x = 10. (b) What does the answer in (a) tell us about the advertising dollars on revenue when x = 10. (c) Complete the following table for dR dx at diFerent values of x . x 2 3 4 5 6 7 8 9 10 11 12 13 dR dx 10.8 13.3 16.5 17.2 17.3 15.7 14.0 11.7 8.8 5.3 (d) What does the table tell us? (e) Use calculus to ±nd the point of diminishing returns. Note. Investments beyond the point of diminishing returns is usually considered not a good way to allocate limited resources. Thus, the company in our example would want to consider spending about 6.67 thousand dollars, boosting revenue from 60 thousand dollars (when x = 0) to about 146 thousand dollars....
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## This note was uploaded on 01/02/2012 for the course MAC 2233 taught by Professor Danielyan during the Fall '08 term at University of South Florida.

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08.Concavity - dollars each month The graph of R is given...

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