Chapter 13: Problem 25
Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=(x y)^{2} $$
Chapter 13: Problem 25
Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=(x y)^{2} $$
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Get started for freeSketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{-2}^{2} \int_{0}^{4-y^{2}} d x d y $$
After a change in marketing, the weekly profit of the firm in Exercise 35 is given by \(P=200 x_{1}+580 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-7500\) Estimate the average weekly profit if \(x_{1}\) varies between 55 and 65 units and \(x_{2}\) varies between 50 and 60 units.
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).
Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-5,1),(1,3),(2,3),(2,5) $$
Plot the points and determine whether the data have positive, negative, or no linear correlation (see figures below). Then use a graphing utility to find the value of \(r\) and confirm your result. The number \(r\) is called the correlation coefficient. It is a measure of how well the model fits the data. Correlation coefficients vary between \(-1\) and 1, and the closer \(|r|\) is to 1, the better the model. $$ (0.5,2),(0.75,1.75),(1,3),(1.5,3.2),(2,3.7),(2.6,4) $$
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