Chapter 6

Evaluate. $$ \int_{0}^{4} \int_{0}^{3} 3 x d x d y $$

Find \(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y},\left.\frac{\partial z}{\partial x}\right|_{(-2,-3)},\) and \(\left.\frac{\partial z}{\partial y}\right|_{(0,-5)}\) $$z=7 x-5 y$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=x y ; 3 x+y=10 $$

Find the regression line for each data set. $$ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 4 & 5 \\ \hline y & 1 & 3 & 3 & 4 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=x^{2}+x y+3 y^{2}+11 x $$

For \(f(x, y)=x^{2}-3 x y,\) find \(f(0,-2), f(2,3),\) and \(f(10,-5)\).

Evaluate. $$ \int_{0}^{3} \int_{0}^{2} 2 y d x d y $$

Find \(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y},\left.\frac{\partial z}{\partial x}\right|_{(-2,-3)},\) and \(\left.\frac{\partial z}{\partial y}\right|_{(0,-5)}\) $$z=2 x-3 y$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=2 x y ; 4 x+y=16 $$

Find the regression line for each data set. $$ \begin{array}{|c|c|c|c|} \hline x & 1 & 3 & 5 \\ \hline y & 2 & 4 & 7 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=x^{2}+x y+y^{2}-5 y $$

For \(f(x, y)=\left(y^{2}+2 x y\right)^{3},\) find \(f(-2,0), f(3,2),\) and \(f(-5,10)\).

Evaluate. $$ \int_{-1}^{3} \int_{1}^{2} x^{2} y d y d x $$

Find \(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y},\left.\frac{\partial z}{\partial x}\right|_{(-2,-3)},\) and \(\left.\frac{\partial z}{\partial y}\right|_{(0,-5)}\) $$z=3 x^{2}-2 x y+y$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=x^{2}+y^{2} ; 2 x+y=10 $$

Find the regression line for each data set. $$ \begin{array}{|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 5 \\ \hline y & 0 & 1 & 3 & 4 \\ \hline \end{array} $$

For \(f(x, y)=3^{x}+7 x y,\) find \(f(0,-2), f(-2,1),\) and \(f(2,1)\).

Evaluate. $$ \int_{1}^{4} \int_{-2}^{1} x^{3} y d y d x $$

Find \(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y},\left.\frac{\partial z}{\partial x}\right|_{(-2,-3)},\) and \(\left.\frac{\partial z}{\partial y}\right|_{(0,-5)}\) $$z=2 x^{3}+3 x y-x$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=x^{2}+y^{2} ; x+4 y=17 $$

Find the regression line for each data set. $$ \begin{array}{|c|c|c|c|} \hline x & 1 & 2 & 4 \\ \hline y & 3 & 5 & 8 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=4 x y-x^{3}-2 y^{2} $$

For \(f(x, y)=\log _{10}(x+y)+3 x^{2},\) find \(f(3,7), f(1,99),\) and \(f(2,-1)\).

Evaluate. $$ \int_{-4}^{-1} \int_{1}^{3}(x+5 y) d x d y $$

Find \(f_{x}(x, y), f_{y}(x, y), f_{x}(-2,4),\) and \(f_{y}(4,-3)\) $$f(x, y)=2 x-5 x y$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=4-x^{2}-y^{2} ; x+2 y=10 $$

Find an exponential regression curve for each data set. $$ \begin{array}{|c|c|c|c|} \hline x & 0 & 1 & 2 \\ \hline y & 10 & 19 & 42 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=x^{3}+y^{3}-6 x y $$

For \(f(x, y)=\ln x+y^{3},\) find \(f(e, 2), f\left(e^{2}, 4\right),\) and \(f\left(e^{3}, 5\right)\).

Evaluate. $$ \int_{0}^{5} \int_{-2}^{-1}(3 x+y) d x d y $$

Find \(f_{x}(x, y), f_{y}(x, y), f_{x}(-2,4),\) and \(f_{y}(4,-3)\) $$f(x, y)=5 x+7 y$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=3-x^{2}-y^{2} ; x+6 y=37 $$

Find an exponential regression curve for each data set. $$ \begin{array}{|r|r|r|r|r|} \hline x & 1 & 2 & 3 & 4 \\ \hline y & 8 & 25 & 72 & 225 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=x^{3}+y^{3}-3 x y $$

For \(f(x, y)=2^{x}-3^{y},\) find \(f(0,2), f(3,1),\) and \(f(2,3)\).

Evaluate. $$ \int_{0}^{1} \int_{x}^{1} x y d y d x $$

Find \(f_{x}, f_{y}, f_{x}(-2,1),\) and \(f_{y}(-3,-2)\). $$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=2 y^{2}-6 x^{2} ; 2 x+y=4 $$

Find an exponential regression curve for each data set. $$ \begin{array}{|c|c|c|c|} \hline x & 1 & 3 & 7 \\ \hline y & 8 & 4 & 1.5 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=x^{2}+y^{2}-4 x+2 y-5 $$

For \(f(x, y, z)=x^{2}-y^{2}+z^{2},\) find \(f(-1,2,3)\) and \(f(2,-1,3)\).

Evaluate. $$ \int_{-1}^{1} \int_{x}^{2}\left(x^{2}+y\right) d y d x $$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y)=2 x^{2}+y^{2}-x y ; x+y=8 $$

Find an exponential regression curve for each data set. $$ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 4 & 6 & 8 \\ \hline y & 13 & 7 & 3.7 & 1 \\ \hline \end{array} $$

Find the relative maximum and minimum values. $$ f(x, y)=x^{2}+2 x y+2 y^{2}-6 y+2 $$

For \(f(x, y, z)=2^{x}+5 z y-x,\) find \(f(0,1,-3)\) and \(f(1,0,-3)\).

Evaluate. $$ \int_{0}^{1} \int_{x^{2}}^{x}(x+y) d y d x $$

Find \(f_{x}\) and \(f_{y}\). $$f(x, y)=e^{3 x-2 y}$$

Find the extremum of \(f(x, y)\) subject to the given constraint, and state whether it is a maximum or a minimum. $$ f(x, y, z)=x^{2}+y^{2}+z^{2} ; y+2 x-z=3 $$

The minimum hourly wage in the United States has grown over the years, as shown in the table below. $$ \begin{array}{c|c} \begin{array}{c} \text { NUMBER OF YEARS, } x, \\ a) For the data in the table, find the regression line, \(y=m x+b\) b) Use the regression line to predict the minimum hourly wage in 2020 and \(2025 .\) \text { SINCE } 1997 \end{array} & \begin{array}{c} \text { MINIMUM HOURLY WAGE } \\ \text { (dollars) } \end{array} \\ \hline 0 & \$ 5.15 \\ 10 & 5.85 \\ 11 & 6.55 \\ 12 & 7.25 \\ 18 & 10.10 \end{array} $$