Chapter 13

Find the function values. \(V(r, h)=\pi r^{2} h\) (a) \(V(3,10)\) (b) \(V(5,2)\)

Find the intercepts and sketch the graph of the plane. $$ z=8 $$

Find the coordinates of the point. The point is located three units behind the \(y z\) -plane, four units to the right of the \(x z\) -plane, and five units above the \(x y\) -plane.

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} d y d x $$

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

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=3 x+y+10 \quad x^{2} y=6 $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=-x^{2}-5 y^{2}+8 x-10 y-13 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ f(x, y)=\frac{x y}{x^{2}+y^{2}} $$

Find the function values. \(F(r, N)=500\left(1+\frac{r}{12}\right)^{N}\) (a) \(F(0.09,60)\) (b) \(F(0.14,240)\)

Find the intercepts and sketch the graph of the plane. $$ x=5 $$

Find the coordinates of the point. The point is located seven units in front of the \(y z\) -plane, two units to the left of the \(x z\) -plane, and one unit below the \(x y\) -plane.

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x y d A\\\ &R \text { : rectangle with vertices at }(0,0),(0,5),(3,5),(3,0) \end{aligned} $$

Evaluate the partial integral. $$ \int_{0}^{x} y e^{x y} d y $$

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

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Maximize } f(x, y)=\sqrt{6-x^{2}-y^{2}} \quad x+y-2=0 $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{2}+6 x y+10 y^{2}-4 y+4 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ z=x^{2} e^{2 y} $$

Find the function values. \(A(P, r, t)=P\left[\left(1+\frac{r}{12}\right)^{12 t}-1\right]\left(1+\frac{12}{r}\right)\) (a) \(A(100,0.10,10)\) (b) \(A(275,0.0925,40)\)

Find the coordinates of the point. The point is located on the \(x\) -axis, 10 units in front of the \(y z\) -plane.

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x d A\\\ &R: \text { semicircle bounded by } y=\sqrt{25-x^{2}} \text { and } y=0 \end{aligned} $$

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

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{2}+6 x y+10 y^{2}-4 y+4 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ z=x e^{x+y} $$

Find the function values. \(A(P, r, t)=P e^{r t}\) (a) \(A(500,0.10,5)\) (b) \(A(1500,0.12,20)\)

Find the coordinates of the point. The point is located in the \(y z\) -plane, three units to the right of the \(x z\) -plane, and two units above the \(x y\) -plane.

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=\sqrt{x^{2}+y^{2}} \quad 2 x+4 y-15=0 $$

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int \frac{y}{x^{2}+y^{2}} d A\\\ &R: \text { triangle bounded by } y=x, y=2 x, x=2 \end{aligned} $$

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

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $$ (-2,2),(2,6),(3,7) $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=3 x^{2}+2 y^{2}-12 x-4 y+7 $$

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

Find the function values. \(f(x, y)=\int_{x}^{y}(2 t-3) d t\) (a) \(f(1,2)\) (b) \(f(1,4)\)

Find the intercepts and sketch the graph of the plane. $$ x+y-z=0 $$

What is the z-coordinate of any point in the \(x y\) -plane?

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Maximize } f(x, y)=e^{x y} \quad x^{2}+y^{2}-8=0 $$

Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int \frac{y}{1+x^{2}} d A\\\ &R: \text { region bounded by } y=0, y=\sqrt{x}, x=4 \end{aligned} $$

Evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{2}\left(6-x^{2}\right) d y d x $$

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

Examine the function for relative extrema and saddle points. $$ f(x, y)=-3 x^{2}-2 y^{2}+3 x-4 y+5 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ g(x, y)=e^{x / y} $$

Find the function values. \(g(x, y)=\int_{x}^{y} \frac{1}{t} d t\) (a) \(g(4,1)\) (b) \(g(6,3)\)

What is the \(x\) -coordinate of any point in the \(y z\) -plane?

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=2 x+y \quad x y=32 $$

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

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{2}-y^{2}+4 x-4 y-8 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ z=\ln \frac{x+y}{x-y} $$

Find the function values. \(f(x, y)=x^{2}-2 y\) (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 $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$$. $$ (0,0,0), 2 x+3 y+z=12 $$

Find the distance between the two points. $$ (4,1,5),(8,2,6) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Minimize \(f(x, y, z)=2 x^{2}+3 y^{2}+2 z^{2}\) Constraint: \(x+y+z-24=0\)