Chapter 6

Use a 3D graphics program to graph each of the following functions. Then estimate any relative extrema. $$ f(x, y)=\frac{y+x^{2} y^{2}-8 x}{x y} $$

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

Find the absolute maximum and minimum values of each function, subject to the given constraints. $$ \begin{array}{l} k(x, y)=-x^{2}-y^{2}+4 x+4 y ; \quad 0 \leq x \leq 3, y \geq 0 \\ \text { and } x+y \leq 6 \end{array} $$

Because wind speed enhances the loss of heat from the skin, we feel colder when there is wind than when there is not. The wind chill temperature is what the temperature would have to be with no wind in order to give the same chilling effect. The wind chill temperature, \(W\), is given by \(\mathrm{W}(v, T)=91.4-\frac{(10.45+6.68 \sqrt{\mathrm{v}}-0.447 \mathrm{v})(457-5 \mathrm{~T})}{110}\)where \(T\) is the temperature measured by a thermometer, in degrees Fahrenheit, and \(v\) is the speed of the wind, in miles per hour. Find the wind chill temperature in each case. Round to the nearest degree. $$ T=20^{\circ} \mathrm{F}, v=40 \mathrm{mph} $$

Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=3 x+5 y$$

Farmer Frank grows two crops: celery and lettuce. He has determined that the cost of planting these crops is modeled by $$ C(x, y)=x^{2}+3 x y+3.5 y^{2}-775 x-1600 y+250,000 $$ where \(x\) is the number of acres of celery and \(y\) is the number of acres of lettuce. Suppose Farmer Frank has 300 acres available for planting and must plant more acres of lettuce than of celery. Find the number of acres of celery and of lettuce he should plant to minimize the cost, and state the cost.

Because wind speed enhances the loss of heat from the skin, we feel colder when there is wind than when there is not. The wind chill temperature is what the temperature would have to be with no wind in order to give the same chilling effect. The wind chill temperature, \(W\), is given by \(\mathrm{W}(v, T)=91.4-\frac{(10.45+6.68 \sqrt{\mathrm{v}}-0.447 \mathrm{v})(457-5 \mathrm{~T})}{110}\)where \(T\) is the temperature measured by a thermometer, in degrees Fahrenheit, and \(v\) is the speed of the wind, in miles per hour. Find the wind chill temperature in each case. Round to the nearest degree. $$ T=-10^{\circ} \mathrm{F}, v=30 \mathrm{mph} $$

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

A manufacturer of decorative end tables produces two models, basic and large. Its weekly profit function is modeled by $$ P(x, y)=-x^{2}-2 y^{2}-x y+140 x+210 y-4300 $$ where \(x\) is the number of basic models sold each week and \(y\) is the number of large models sold each week. The warehouse can hold at most 90 tables. Assume that \(x\) and \(y\) must be nonnegative. How many of each model should be produced to maximize weekly profit, and what will the maximum profit be?

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

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Minimum: \(f(x, y)=x y ; x^{2}+y^{2}=9\)

Use a 3D graphics program to generate the graph of each function. $$ f(x, y)=y^{2} $$

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

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Minimum: \(f(x, y)=2 x^{2}+y^{2}+2 x y+3 x+2 y ;\) \(y^{2}=x+1\)

Use a 3D graphics program to generate the graph of each function. $$ f(x, y)=x^{2}+y^{2} $$

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

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Maximum: \(f(x, y, z)=x+y+z ; x^{2}+y^{2}+z^{2}=1\)

Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=y \ln x$$

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Maximum: \(f(x, y, z)=x^{2} y^{2} z^{2} ; x^{2}+y^{2}+z^{2}=2\)

Use a 3D graphics program to generate the graph of each function. $$ f(x, y)=4\left(x^{2}+y^{2}\right)-\left(x^{2}+y^{2}\right)^{2} $$

Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=x \ln y$$

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Maximum: \(f(x, y, z)=x+2 y-2 z ; x^{2}+y^{2}+z^{2}=4\)

Use a 3D graphics program to generate the graph of each function. $$ f(x, y)=x^{3}-3 x y^{2} $$

Riverside Appliances has the following production function for a certain product: $$p(x, y)=1800 x^{0.621} y^{0.379}$$ where \(p\) is the number of units produced with \(x\) units of labor and y units of capital. a) Find the number of units produced with 2500 units of labor and 1700 units of capital. b) Find the marginal productivities. c) Evaluate the marginal productivities at \(x=2500\) and \(y=1700\) d) Interpret the meanings of the marginal productivities found in part (c).

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Maximum: \(f(x, y, z, t)=x+y+z+t ;\) \(x^{2}+y^{2}+z^{2}+t^{2}=1\)

Use a 3D graphics program to generate the graph of each function. $$ f(x, y)=\frac{1}{x^{2}+4 y^{2}} $$

Lincolnville Sporting Goods has the following production function for a certain product: $$p(x, y)=2400 x^{2 / 5} y^{3 / 5}$$ where \(p\) is the number of units produced with \(x\) units of labor and \(y\) units of capital. a) Find the number of units produced with 32 units of labor and 1024 units of capital. b) Find the marginal productivities. c) Evaluate the marginal productivities at \(x=32\) and \(y=1024\) d) Interpret the meanings of the marginal productivities found in part (c),

Find the indicated maximum or minimum values of \(f(x, y)\) subject to the given constraint. Minimum: \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; x-2 y+5 z=1\)

A computer company has the following Cobb-Douglas production function for a certain product: $$ p(x, y)=800 x^{3 / 4} y^{1 / 4} $$ where \(x\) is labor and \(y\) is capital, both measured in dollars. Suppose the company can make a total investment in labor and capital of \(\$ 1,000,000 .\) How should it allocate the investment between labor and capital in order to maximize production?

Find the point on the parabola \(y=x^{2}+2 x-5\) that is closest to the origin.

Find the point on the circle \(x^{2}+y^{2}=1\) that is closest to the point (2,1) .

The Mosteller formula for approximating the surface area, \(S,\) in \(\mathrm{m}^{2},\) of a human is \(s=\frac{\sqrt{h w}}{60}\) where \(h\) is the person's height in centimeters and \(w\) is the person's weight in kilograms. (Source: www.halls.md.) a) Compute \(\frac{\partial S}{\partial h}\). b) Compute \(\frac{\partial S}{\partial w}\). c) The change in \(S\) due to a change in \(w\) when \(h\) is constant is approximately \(\Delta S \approx \frac{\partial S}{\partial w} \Delta w\) Use this formula to approximate the change in someone's surface area given that the person is \(170 \mathrm{~cm}\) tall, weighs \(80 \mathrm{~kg}\), and loses \(2 \mathrm{~kg}\).

The Haycock formula for approximating the surface area, \(S,\) in \(\mathrm{m}^{2},\) of a human is $$S=0.024265 h^{0.3964} w^{0.5378}$$ where \(h\) is the person's height in centimeters and \(w\) is the person's weight in kilograms. (Source: www.halls.md.) a) Compute \(\frac{\partial S}{\partial h}\) b) Compute \(\frac{\partial S}{\partial w}\) c) The change in \(S\) due to a change in \(w\) when \(h\) is constant is approximately $$\Delta S \approx \frac{\partial S}{\partial w} \Delta w$$ Use this formula to approximate the change in someone's surface area given that the person is \(170 \mathrm{~cm}\) tall weighs \(80 \mathrm{~kg},\) and loses \(2 \mathrm{~kg}\).

The following formula is used by psychologists and educators to predict the reading ease, \(E,\) of a passage of words: $$E=206.835-0.846 w-1.015 s$$ where \(w\) is the number of syllables in a 100 -word section and s is the average number of words per sentence. $$\text { Find } E \text { when } w=146 \text { and } s=5$$

The following formula is used by psychologists and educators to predict the reading ease, \(E,\) of a passage of words: $$E=206.835-0.846 w-1.015 s$$ where \(w\) is the number of syllables in a 100 -word section and s is the average number of words per sentence. $$\text { Find } E \text { when } w=180 \text { and } s=6$$

The following formula is used by psychologists and educators to predict the reading ease, \(E,\) of a passage of words: $$E=206.835-0.846 w-1.015 s$$ where \(w\) is the number of syllables in a 100 -word section and s is the average number of words per sentence. $$\text { Find } \frac{\partial E}{\partial w}$$

The following formula is used by psychologists and educators to predict the reading ease, \(E,\) of a passage of words: $$E=206.835-0.846 w-1.015 s$$ where \(w\) is the number of syllables in a 100 -word section and s is the average number of words per sentence. $$\text { Find } \frac{\partial E}{\partial s}$$

Find \(f_{x}\) and \(f_{t}\). $$f(x, t)=\frac{x^{2}+t^{2}}{x^{2}-t^{2}}$$

Find \(f_{x}\) and \(f_{t}\). $$f(x, t)=\frac{x^{2}-t}{x^{3}+t}$$

Find \(f_{x}\) and \(f_{t}\). $$f(x, t)=\frac{2 \sqrt{x}-2 \sqrt{t}}{1+2 \sqrt{t}}$$

Find \(f_{x}\) and \(f_{t}\). $$f(x, t)=\sqrt[4]{x^{3} t^{5}}$$

Find \(f_{x}\) and \(f_{t}\). $$f(x, t)=6 x^{2 / 3}-8 x^{1 / 4} t^{1 / 2}-12 x^{-1 / 2} t^{3 / 2}$$

Find \(f_{x}\) and \(f_{t}\). $$f(x, t)=\left(\frac{x^{2}+t^{2}}{x^{2}-t^{2}}\right)^{5}$$

find \(f_{x x} f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=\frac{x}{y^{2}}-\frac{y}{x^{2}}$$

find \(f_{x x} f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=\frac{x y}{x-y}$$

Do some research on the Cobb-Douglas production function, and explain how it was developed.

Explain the meaning of the first partial derivatives of a function of two variables in terms of slopes of tangent lines.

Consider \(f(x, y)=\ln \left(x^{2}+y^{2}\right) .\) Show that \(f\) is a solution of the partial differential equation \(\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0\).

Consider \(f(x, y)=x^{3}-5 x y^{2}\). Show that \(f\) is a solution of the partial differential equation $$x f_{x y}-f_{y}=0$$

Consider the function \(f\) defined as follows: \(f(x, y)=\left\\{\begin{array}{ll}\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}, & \text { for }(x, y) \neq(0,0), \\\ 0, & \text { for }(x, y)=(0,0)\end{array}\right.\) a) Find \(f_{x}(0, y)\) by evaluating the limit $$ \lim _{h \rightarrow 0} \frac{f(h, y)-f(0, y)}{h} $$ b) Find \(f_{y}(x, 0)\) by evaluating the limit $$ \lim _{h \rightarrow 0} \frac{f(x, h)-f(x, 0)}{h} $$ c) Now find and compare \(f_{y x}(0,0)\) and \(f_{x y}(0,0)\)