Chapter 13

Evaluate the double integral. $$ \int_{0}^{1} \int_{0}^{x} \sqrt{1-x^{2}} d y d x $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-10,10),(-5,8),(3,6),(7,4),(5,0) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Maximize \(f(x, y, z)=x y z\) Constraint: \(x+y+z-6=0\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{2}-3 x y-y^{2} $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ g(x, y)=\ln \left(x^{2}+y^{2}\right) $$

Find the function values. \(f(x, y)=3 x y+y^{2}\) (a) \(f(x+\Delta x, y)\) (b) \(\frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\)

Find the distance between the point and the plane (see figure). The distance \(D\) between \(\mathrm{a}\) point \(\left(x_{0}, y_{0}, z_{0}\right)\) and the plane \(a x+b y+c z+d=0\) is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (0,0,0), 8 x-4 y+z=8 $$

Find the distance between the two points. $$ (-4,-1,1),(2,-1,5) $$

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

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (0,0),(1,1),(3,4),(4,2),(5,5) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Minimize \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) Constraint: \(x+y+z=1\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=\frac{1}{2} x y $$

Let \(f(x, y)=3 x^{2} y e^{x-y}\) and \(g(x, y)=\) \(3 x y^{2} e^{y-x}\). Find each of the following. $$ f_{x}(x, y) $$

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function, and find the range of the function. $$ f(x, y)=\sqrt{16-x^{2}-y^{2}} $$

Find the distance between the point and the plane (see figure). The distance \(D\) between \(\mathrm{a}\) point \(\left(x_{0}, y_{0}, z_{0}\right)\) and the plane \(a x+b y+c z+d=0\) is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (1,5,-4), 3 x-y+2 z=6 $$

Find the distance between the two points. $$ (-1,-5,7),(-3,4,-4) $$

Evaluate the double integral. $$ \int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y d x d y $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (1,0),(3,3),(5,6) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Minimize \(f(x, y)=x^{2}-8 x+y^{2}-12 y+48\) Constraint: \(x+y=8\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=x+y+2 x y-x^{2}-y^{2} $$

Let \(f(x, y)=3 x^{2} y e^{x-y}\) and \(g(x, y)=\) \(3 x y^{2} e^{y-x}\). Find each of the following. $$ f_{y}(x, y) $$

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function, and find the range of the function. $$ f(x, y)=x^{2}+y^{2}-1 $$

Find the distance between the point and the plane (see figure). The distance \(D\) between \(\mathrm{a}\) point \(\left(x_{0}, y_{0}, z_{0}\right)\) and the plane \(a x+b y+c z+d=0\) is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (3,2,1), x-y+2 z=4 $$

Find the distance between the two points. $$ (8,-2,2),(8,-2,4) $$

Evaluate the double integral. $$ \int_{1}^{2} \int_{0}^{4}\left(3 x^{2}-2 y^{2}+1\right) d x d y $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (0,6),(4,3),(5,0),(8,-4),(10,-5) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Maximize \(f(x, y, z)=x+y+z\) Constraint: \(x^{2}+y^{2}+z^{2}=1\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=(x+y) e^{1-x^{2}-y^{2}} $$

Let \(f(x, y)=3 x^{2} y e^{x-y}\) and \(g(x, y)=\) \(3 x y^{2} e^{y-x}\). Find each of the following. $$ g_{x}(x, y) $$

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function, and find the range of the function. $$ f(x, y)=e^{x / y} $$

Find the distance between the point and the plane (see figure). The distance \(D\) between \(\mathrm{a}\) point \(\left(x_{0}, y_{0}, z_{0}\right)\) and the plane \(a x+b y+c z+d=0\) is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (1,0,-1), 2 x-4 y+3 z=12 $$

Find the coordinates of the midpoint of the line segment joining the two points. $$ (6,-9,1),(-2,-1,5) $$

Evaluate the double integral. $$ \int_{0}^{1} \int_{y}^{2 y}\left(1+2 x^{2}+2 y^{2}\right) d x d y $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (6,4),(1,2),(3,3),(8,6),(11,8),(13,8) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Maximize \(f(x, y, z)=x^{2} y^{2} z^{2}\) Constraint: \(x^{2}+y^{2}+z^{2}=1\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=3 e^{-\left(x^{2}+y^{2}\right)} $$

Let \(f(x, y)=3 x^{2} y e^{x-y}\) and \(g(x, y)=\) \(3 x y^{2} e^{y-x}\). Find each of the following. $$ g_{y}(x, y) $$

Find the distance between the point and the plane (see figure). The distance \(D\) between \(\mathrm{a}\) point \(\left(x_{0}, y_{0}, z_{0}\right)\) and the plane \(a x+b y+c z+d=0\) is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (2,-1,0), 3 x+3 y+2 z=6 $$

Find the coordinates of the midpoint of the line segment joining the two points. $$ (4,0,-6),(8,8,20) $$

Evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{\sqrt{1-y^{2}}}-5 x y d x d y $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression quadratic for the given points. Then plot the points and graph the least squares regression quadratic. $$ (-2,0),(-1,0),(0,1),(1,2),(2,5) $$

Use Lagrange multipliers to find the given extremum of \(f\) subject to two constraints. In each case, assume that \(x, y\), and \(z\) are nonnegative. Maximize \(f(x, y, z)=x y z\) Constraints: \(x+y+z=32, x-y+z=0\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=4 e^{x y} $$

Let \(f(x, y)=3 x^{2} y e^{x-y}\) and \(g(x, y)=\) \(3 x y^{2} e^{y-x}\). Find each of the following. $$ f_{x}(1,1) $$

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ z=\sqrt{4-x^{2}-y^{2}} $$

Find the distance between the point and the plane (see figure). The distance \(D\) between \(\mathrm{a}\) point \(\left(x_{0}, y_{0}, z_{0}\right)\) and the plane \(a x+b y+c z+d=0\) is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (3,2,-1), 2 x-3 y+4 z=24 $$

Find the coordinates of the midpoint of the line segment joining the two points. $$ (-5,-2,5),(6,3,-7) $$

Evaluate the double integral. $$ \int_{0}^{4} \int_{0}^{x} \frac{2}{x^{2}+1} d y d x $$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression quadratic for the given points. Then plot the points and graph the least squares regression quadratic. $$ (-4,5),(-2,6),(2,6),(4,2) $$

Use Lagrange multipliers to find the given extremum of \(f\) subject to two constraints. In each case, assume that \(x, y\), and \(z\) are nonnegative. Minimize \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) Constraints: \(x+2 z=6, x+y=12\)