Chapter 13

Examine the function for relative extrema and saddle points. $$ f(x, y)=-\frac{3}{x^{2}+y^{2}+1} $$

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. $$ g_{x}(-2,-2) $$

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

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 $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (-2,1,0), 2 x+5 y-z=20 $$

Find the coordinates of the midpoint of the line segment joining the two points. $$ (0,-2,5),(4,2,7) $$

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

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. $$ (0,0),(2,2),(3,6),(4,12) $$

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: \(x^{2}+z^{2}=5, x-2 y=0\)

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right)\). $$ f_{x x}\left(x_{0}, y_{0}\right)=9, f_{y y}\left(x_{0}, y_{0}\right)=4, f_{x y}\left(x_{0}, y_{0}\right)=6 $$

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

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 5 x-3 y+z=4, x+4 y+7 z=1 $$

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

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. $$ (0,10),(1,9),(2,6),(3,0) $$

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 \(f(x, y, z)=x y+y z\) Constraints: \(x+2 y=6, x-3 z=0\)

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right)\). $$ f_{x x}\left(x_{0}, y_{0}\right)=-3, f_{y y}\left(x_{0}, y_{0}\right)=-8, f_{x y}\left(x_{0}, y_{0}\right)=2 $$

Evaluate \(f_{x}\) and \(f_{u}\) at the point. $$ f(x, y)=x^{2}-3 x y+y^{2} \quad(1,-1) $$

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 3 x+y-4 z=3,-9 x-3 y+12 z=4 $$

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x y, z=0, y=0, y=4, x=0, x=1 $$

Evaluate the double integral. $$ \int_{0}^{\infty} \int_{0}^{\infty} e^{-(x+y) / 2} d y d x $$

Use the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (-4,1),(-3,2),(-2,2),(-1,4),(0,6),(1,8),(2,9) $$

Use a spreadsheet to find the given extremum. In each case, assume that \(x, y\), and \(z\) are nonnegative. Maximize \(f(x, y, z)=x y z\) Constraints: \(x+3 y=6, x-2 z=0\)

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right)\). $$ f_{x x}\left(x_{0}, y_{0}\right)=-9, f_{y y}\left(x_{0}, y_{0}\right)=6, f_{x y}\left(x_{0}, y_{0}\right)=10 $$

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x-5 y-z=1,5 x-25 y-5 z=-3 $$

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x, z=0, y=x, y=0, x=0, x=4 $$

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

Use a spreadsheet to find the given extremum. In each case, assume that \(x, y\), and \(z\) are nonnegative. Minimize \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) Constraints: \(x+2 y=8, x+z=4\)

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right)\). $$ f_{x x}\left(x_{0}, y_{0}\right)=25, f_{v v}\left(x_{0}, y_{0}\right)=8, f_{x v}\left(x_{0}, y_{0}\right)=10 $$

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\frac{1}{x-y} $$

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+3 y+2 z=6,4 x-12 y+8 z=24 $$

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x^{2}, z=0, x=0, x=2, y=0, y=4 $$

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 the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (0,769),(1,677),(2,601),(3,543),(4,489),(5,411) $$

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 120 and the product is maximum.

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=(x y)^{2} $$

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

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+2 y=3,4 x+8 y=5 $$

Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these. $$ (0,0,0),(2,2,1),(2,-4,4) $$

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x+v, x^{2}+v^{2}=4 \text { (first octant) } $$

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

Use the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (1,10.3),(2,14.2),(3,18.9),(4,23.7),(5,29.1),(6,35) $$

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 120 and the sum of the squares is minimum.

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

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

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ x+3 y+z=7, x-5 z=0 $$

Find the lengths of the sides of the triangle with the given vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither of these. $$ (5,3,4),(7,1,3),(3,5,3) $$

The population density (in people per square mile) for a coastal town can be modeled by \(f(x, y)=\frac{120,000}{(2+x+y)^{3}}\) where \(x\) and \(y\) are measured in miles. What is the population inside the rectangular area defined by the vertices \((0,0)\), \((2,0),(0,2)\), and \((2,2) ?\)

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. $$ \int_{0}^{1} \int_{2 y}^{2} d x d y $$

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=x^{3}+y^{3} $$

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ g(x, y)=\ln (4-x-y) $$

Determine whether the planes \(a_{1} x+b_{1} y+c_{1} z=d_{1}\) and \(a_{2} x+b_{2} y+c_{2} z=d_{2}\) are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant \(k\) such that \(a_{1}=k a_{2}, b_{1}=k b_{2}\), and \(c_{1}=k c_{2}\), and are perpendicular if \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 2 x+y=3,3 x-5 z=0 $$