Chapter 13: Problem 24
Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\frac{1}{x-y} $$
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The Cobb-Douglas production function for an automobile manufacturer is \(f(x, y)=100 x^{0.6} y^{0.4}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Estimate the average production level if the number of units of labor \(x\) varies between 200 and 250 and the number of units of capital \(y\) varies between 300 and 325 .
Evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{6 x^{2}} x^{3} d y d x $$
Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x y, z=0, y=0, y=4, x=0, x=1 $$
Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ (0,0.5),(1,7.6),(3,60),(4.2,117),(5,170),(7.9,380) $$
The revenues \(y\) (in millions of dollars) for Earthlink from 2000 through 2006 are shown in the table. $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 \\ \hline \text { Revenue, } y & 986.6 & 1244.9 & 1357.4 & 1401.9 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2004 & 2005 & 2006 \\ \hline \text { Revenue, } y & 1382.2 & 1290.1 & 1301.3 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility or a spreadsheet to create a scatter plot of the data. Let \(t=0\) represent the year 2000 . (b) Use the regression capabilities of a graphing utility or a spreadsheet to find an appropriate model for the data. (c) Explain why you chose the type of model that you created in part (b).
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