91Ó°ÊÓ

The following table gives the cost per mile of operating a compact car, depending upon the number of miles driven per year. Find the least squares line for these data. Use your answer to predict the cost per mile for a car driven 25,000 miles annually \((x=5)\). $$ \begin{array}{rcc} \hline \begin{array}{c} \text { Annual } \\ \text { Mileage } \end{array} & \boldsymbol{x} & \begin{array}{c} \text { Cost per Mile } \\ \text { (cents) } \end{array} \\ 5000 & 1 & 50 \\ 10,000 & 2 & 35 \\ 15,000 & 3 & 27 \\ 20,000 & 4 & 25 \\ \hline \end{array} $$

A sociologist finds the following data for the number of felony arrests per year in a town. Find the least squares line. Then use it to predict the number of felony arrests in the next year. $$ \begin{array}{lcccc} \hline \text { Year } & 1 & 2 & 3 & 4 \\ \text { Arrests } & 120 & 110 & 90 & 100 \\ \hline \end{array} $$

For each function, find the partials a. \(f_{x}(x, y)\) and b. \(f_{y}(x, y)\). \(f(x, y)=2 x^{3} e^{-5 y}\)

Evaluate each iterated integral. $$ \int_{0}^{1} \int_{0}^{2} x^{3} y^{7} d x d y $$

Find the total differential of each function. $$ z=x e^{2 y} $$

Use Lagrange multipliers to maximize each function \(f(x, y)\) subject to the constraint. (The maximum values do exist.) \(f(x, y)=x^{2}+y^{2}, \quad 2 x+y=15\)

For each function, evaluate the given expression. $$ g(x, y)=\ln \left(x^{2}+y^{4}\right), \text { find } g(0, e) $$

Find the relative extreme values of each function. $$ f(x, y)=e^{\left(x^{2}+y^{2}\right) / 2} $$

For each function, find the partials a. \(f_{x}(x, y)\) and b. \(f_{y}(x, y)\). \(f(x, y)=e^{x+y}\)

Evaluate each iterated integral. $$ \int_{0}^{1} \int_{0}^{3} x^{8} y^{2} d y d x $$