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Chapter 13: Problem 19
Find the coordinates of the midpoint of the line segment joining the two points. $$ (-5,-2,5),(6,3,-7) $$
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Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ (1,5.5),(3,7.75),(6,15.2),(8,23.5),(11,46),(15,110) $$
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. $$ (1,7.5),(2,7),(3,7),(4,6),(5,5),(6,4.9) $$
Sketch 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. $$ \int_{0}^{1} \int_{0}^{2} d y d x $$
Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x+v, x^{2}+v^{2}=4 \text { (first octant) } $$
Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x d A\\\ &R: \text { semicircle bounded by } y=\sqrt{25-x^{2}} \text { and } y=0 \end{aligned} $$
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