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Chapter 13: Problem 46
Identify the quadric surface. $$ z=4 x^{2}+y^{2} $$
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Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x\\\ &R \text { : rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$
Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (0,0),(1,1),(3,4),(4,2),(5,5) $$
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. $$ (1,4),(2,6),(3,8),(4,11),(5,13),(6,15) $$
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.
Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} d y d x $$
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