Chapter 13

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. $$ (-2,2,4),(-2,2,6),(-2,4,8) $$

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

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=x^{3}+y^{3}-3 x^{2}+6 y^{2}+3 x+12 y+7 $$

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

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 \(a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0\). $$ 2 x-z=1,4 x+y+8 z=10 $$

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. $$ (5,0,0),(0,2,0),(0,0,-3) $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x\\\ &R \text { : rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$

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

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) Line: \(x+y=6,(0,0)\) Minimize \(d^{2}=x^{2}+y^{2}\)

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=x^{2 / 3}+y^{2 / 3} $$

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=x y z $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x y\\\ &R: \text { rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$

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. $$ \int_{0}^{4} \int_{\sqrt{x}}^{2} d y d x $$

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) Circle: \((x-4)^{2}+y^{2}=4,(0,10)\) Minimize \(d^{2}=x^{2}+(y-10)^{2}\)

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=\left(x^{2}+y^{2}\right)^{2 / 3} $$

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=x^{2}-3 x y+4 y z+z^{3} $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x^{2}+y^{2}\\\ &R: \text { square with vertices }(0,0),(2,0),(2,2),(0,2) \end{aligned} $$

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. $$ \int_{0}^{1} \int_{y^{2}}^{\sqrt[3]{y}} d x d y $$

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) Plane: \(x+y+z=1,(2,1,1)\) Minimize \(d^{2}=(x-2)^{2}+(y-1)^{2}+(z-1)^{2}\)

Find the critical points of the function and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point. $$ f(x, y, z)=(x-1)^{2}+(y+3)^{2}+z^{2} $$

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

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=e^{x+y}\\\ &R: \text { triangle with vertices }(0,0),(0,1),(1,1) \end{aligned} $$

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. $$ \int_{-2}^{2} \int_{0}^{4-y^{2}} d x d y $$

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) Cone: \(z=\sqrt{x^{2}+y^{2}},(4,0,0)\) Minimize \(d^{2}=(x-4)^{2}+y^{2}+z^{2}\)

Find the critical points of the function and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point. $$ f(x, y, z)=6-[x(y+2)(z-1)]^{2} $$

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

A company sells two products whose demand functions are given by \(x_{1}=500-3 p_{1}\) and \(x_{2}=750-2.4 p_{2}\) So, the total revenue is given by \(R=x_{1} p_{1}+x_{2} p_{2}\) Estimate the average revenue if the price \(p_{1}\) varies between \(\$ 50\) and \(\$ 75\) and the price \(p_{2}\) varies between \(\$ 100\) and \(\$ 150\).

Evaluate the double integral. Note that it is necessary to change the order of integration. $$ \int_{0}^{3} \int_{y}^{3} e^{x^{2}} d x d y $$

The revenues \(y\) (in millions of dollars) for Earthlink from 2000 through 2006 are shown in the table. $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 \\ \hline \text { Revenue, } y & 986.6 & 1244.9 & 1357.4 & 1401.9 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2004 & 2005 & 2006 \\ \hline \text { Revenue, } y & 1382.2 & 1290.1 & 1301.3 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility or a spreadsheet to create a scatter plot of the data. Let \(t=0\) represent the year 2000 . (b) Use the regression capabilities of a graphing utility or a spreadsheet to find an appropriate model for the data. (c) Explain why you chose the type of model that you created in part (b).

Find the dimensions of the rectangular package of largest volume subject to the constraint that the sum of the length and the girth cannot exceed 108 inches (see figure). (Hint: Maximize \(V=x y z\) subject to the constraint \(x+2 y+2 z=108\).)

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 30 and the product is a maximum.

Evaluate \(w_{x}, w_{y}\), and \(w_{z}\) at the point. $$ w=\sqrt{x^{2}+y^{2}+z^{2}} \quad(2,-1,2) $$

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=x+y \quad c=-1,0,2,4 $$

Evaluate the double integral. Note that it is necessary to change the order of integration. $$ \int_{0}^{2} \int_{x}^{2} e^{-y^{2}} d y d x $$

The global numbers of personal computers \(x\) (in millions) and Internet users \(y\) (in millions) from 1999 through 2005 are shown in the table. $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Personal computers, } x & 394.1 & 465.4 & 526.7 & 575.5 \\ \hline \text { Internet users, } y & 275.5 & 390.3 & 489.9 & 618.4 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Personal computers, } x & 636.6 & 776.6 & 808.7 \\ \hline \text { Internet users, } y & 718.8 & 851.8 & 982.5 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility or a spreadsheet to create a scatter plot of the data. (b) Use the regression capabilities of a graphing utility or a spreadsheet to find an appropriate model for the data. (c) Explain why you chose the type of model that you created in part (b).

Evaluate \(w_{x}, w_{y}\), and \(w_{z}\) at the point. $$ w=\frac{x y}{x+y+z} \quad(1,2,0) $$

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=6-2 x-3 y \quad c=0,2,4,6,8,10 $$

A firm's weekly profit in marketing two products is given by \(P=192 x_{1}+576 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-5000\) where \(x_{1}\) and \(x_{2}\) represent the numbers of units of each product sold weekly. Estimate the average weekly profit if \(x_{1}\) varies between 40 and 50 units and \(x_{2}\) varies between 45 and 50 units.

Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ \begin{aligned} &(1,13), \quad(2,16.5),(4,24),(5,28),(8,39),(11,50.25) \\ &(17,72),(20,85) \end{aligned} $$

A cargo container (in the shape of a rectangular solid) must have a volume of 480 cubic feet. Use Lagrange multipliers to find the dimensions of the container of this size that has a minimum cost, if the bottom will cost \(\$ 5\) per square foot to construct and the sides and top will cost \(\$ 3\) per square foot to construct.

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 30 and the sum of the squares is a minimum.

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=\sqrt{25-x^{2}-y^{2}} \quad c=0,1,2,3,4,5 $$

Find the standard equation of the sphere. Center: \((1,1,5) ;\) radius: 3

After a change in marketing, the weekly profit of the firm in Exercise 35 is given by \(P=200 x_{1}+580 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-7500\) Estimate the average weekly profit if \(x_{1}\) varies between 55 and 65 units and \(x_{2}\) varies between 50 and 60 units.

Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ (1,5.5),(3,7.75),(6,15.2),(8,23.5),(11,46),(15,110) $$

Find three positive numbers \(x, y\), and \(z\) that satisfy the given conditions. The sum is 1 and the sum of the squares is a minimum.

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ f(x, y)=x^{2}+y^{2} \quad c=0,2,4,6,8 $$

Find the standard equation of the sphere. Center: \((4,-1,1) ;\) radius: 5

The Cobb-Douglas production function for an automobile manufacturer is \(f(x, y)=100 x^{0.6} y^{0.4}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Estimate the average production level if the number of units of labor \(x\) varies between 200 and 250 and the number of units of capital \(y\) varies between 300 and 325 .

Use the regression capabilities of \(a\) graphing utility or a spreadsheet to find any model that best fits the data points. $$ (1,1.5),(2.5,8.5),(5,13.5),(8,16.7),(9,18),(20,22) $$