Question

In business, we are often interested in knowing when the rate of change in sales reaches a maximum valuc. Thc point at which this occurs is called the point of diminishing returns. If the sales are defined by S(t), then in terms of Calculus, we want to know when the derivative S"() attains a maximum. In this assignment we are going to use the graphing calculator or Microsoft Excel to fit the logistic model to the Nintendo DS game console dataset. Lcarning objectives and cxpectations associated with this assignment includo: (i) Demonstrate knowledge of simple regression analysis by creating a scatterplot, and fitting a logistic regression curve using the least-squares method, and ii) Use Calculus to forecast future results and make informed decisions. Table 1: Nintendo DS Cumulative Sales (Source:Statista.com) Year Cumulative Sales (millions) 2005 5.27 2006 16.73 2007 40.29 70.60 2008 2009 101.78 2010 128.89 146.41 2011 2012 151.51 2013 2014 153.86 153.99 2015 154.01 (a) Use the logistic regression to model the cumulative sales S(t) as a function of time t since 2005. Include a sketch of the scatterplot and the regression curve. (Round all coefficients to three significant digits.) b) Calculate lim S(0) and interpret the result. (c) Find S'(t). (d) Find the point of diminishing returns for Nintendo DS game console cumulative salds by first graphing S'(t) on the graphing calculator and finding the maximum point. (e) Using the result from part (d), determine the cumulative sales and the marginal cumulative sales at the point of diminishing returns.

Answer 1

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