Chapter 12

Sketch the level curves of the function corresponding to each value of \(z\). \(f(x, y)=-x^{2}+y ; z=-2,-1,0,1,2\)

The total weekly revenue (in dollars) of the Country Workshop realized in manufacturing and selling its rolltop desks is given by $$R(x, y)=-0.2 x^{2}-0.25 y^{2}-0.2 x y+200 x+160 y$$ where \(x\) denotes the number of finished units and \(y\) denotes the number of unfinished units manufactured and sold each week. The total weekly cost attributable to the manufacture of these desks is given by $$C(x, y)=100 x+70 y+4000$$ dollars. Determine how many finished units and how many unfinished units the company should manufacture each week in order to maximize its profit. What is the maximum profit realizable?

Find the first partial derivatives of the function. \(f(x, y, z)=x y z+x y^{2}+y z^{2}+z x^{2}\)

The total daily revenue (in dollars) that Weston Publishing realizes in publishing and selling its English-language dictionaries is given by $$\begin{aligned}R(x, y)=&-0.005 x^{2}-0.003 y^{2}-0.002 x y \\\&+20 x+15 y\end{aligned}$$ where \(x\) denotes the number of deluxe copies and \(y\) denotes the number of standard copies published and sold daily. The total daily cost of publishing these dictionaries is given by $$C(x, y)=6 x+3 y+200$$ dollars. Determine how many deluxe copies and how many standard copies Weston should publish each day to maximize its profits. What is the maximum profit realizable?

Find the first partial derivatives of the function. \(g(u, v, w)=\frac{2 u w w}{u^{2}+v^{2}+w^{2}}\)

Sketch the level curves of the function corresponding to each value of \(z\). \(f(x, y)=x y ; z=-4,-2,2,4\)

Find the first partial derivatives of the function. \(h(r, s, t)=e^{r s t}\)

Sketch the level curves of the function corresponding to each value of \(z\). \(f(x, y)=\sqrt{16-x^{2}-y^{2}} ; z=0,1,2,3,4\)

C\&G Imports imports two brands of white wine, one from Germany and the other from Italy. The German wine costs $$\$ 4 /$$ bottle, and the Italian wine costs $$\$ 3 /$$ bottle. It has been estimated that if the German wine retails at \(p\) dollars/bottle and the Italian wine is sold for \(q\) dollars/bottle, then $$2000-150 p+100 q$$ bottles of the German wine and $$1000+80 p-120 q$$ bottles of the Italian wine will be sold each week. Determine the unit price for each brand that will allow \(\mathrm{C} \& \mathrm{G}\) to realize the largest possible weekly profit.

Find the first partial derivatives of the function. \(f(x, y, z)=x e^{y / z}\)

Sketch the level curves of the function corresponding to each value of \(z\). \(f(x, y)=e^{x}-y ; z=-2,-1,0,1,2\)

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=x^{2} y+x y^{2} ;(1,2)\)

Find an equation of the level curve of \(f(x, y)=\sqrt{x^{2}+y^{2}}\) that contains the point \((3,4)\).

An open rectangular box having a volume of 108 in. \(^{3}\) is to be constructed from a tin sheet. Find the dimensions of such a box if the amount of material used in its construction is to be minimal. Hint: Let the dimensions of the box be \(x^{\prime \prime}\) by \(y^{\prime \prime}\) by \(z^{\prime \prime}\). Then, \(x y z=108\) and the amount of material used is given by \(S=x y+\) \(2 y z+2 x z .\) Show that $$S=f(x, y)=x y+\frac{216}{x}+\frac{216}{y}$$ Minimize \(f(x, y)\)

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=x^{2}+x y+y^{2}+2 x-y ;(-1,2)\)

Find an equation of the level surface of \(f(x, y, z)=2 x^{2}+\) \(3 y^{2}-z\) that contains the point \((-1,2,-3) \)

An open rectangular box having a surface area of \(300 \mathrm{in} .^{2}\) is to be constructed from a tin sheet. Find the dimensions of the box if the volume of the box is to be as large as possible. What is the maximum volume?

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=x \sqrt{y}+y^{2} ;(2,1)\)

The volume of a cylindrical tank of radius \(r\) and height \(h\) is given by $$V=f(r, h)=\pi r^{2} h$$ Find the volume of a cylindrical tank of radius \(1.5 \mathrm{ft}\) and height \(4 \mathrm{ft}\).

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 130 in. Find the dimensions of the rectangular package that would have the greatest possible volume under these regulations. Hint: Let the dimensions of the box be \(x^{\prime \prime}\) by \(y^{\prime \prime}\) by \(z^{\prime \prime}\) (see the figure below). Then, \(2 x+2 z+y=130\), and the volume \(V=x y z\). Show that $$V=f(x, z)=130 x z-2 x^{2} z-2 x z^{2}$$ Maximize \(f(x, z)\)

Evaluate the first partial derivatives of the function at the given point. \(g(x, y)=\sqrt{x^{2}+y^{2}} ;(3,4)\)

The IQ (intelligence quotient) of a person whose mental age is \(m\) yr and whose chronological age is \(c\) yr is defined as $$f(m, c)=\frac{100 m}{c}$$ What is the IQ of a 9 -yr-old child who has a mental age of \(13.5\) yr?

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=\frac{x}{y} ;(1,2)\)

The body mass index (BMI) is used to identify, evaluate, and treat overweight and obese adults. The BMI value for an adult of weight \(w\) (in kilograms) and height \(h\) (in meters) is defined to be $$M=f(w, h)=\frac{w}{h^{2}}$$ According to federal guidelines, an adult is overweight if he or she has a BMI value between 25 and \(29.9\) and is "obese" if the value is greater than or equal to 30 . a. What is the BMI of an adult who weighs in at \(80 \mathrm{~kg}\) and stands \(1.8 \mathrm{~m}\) tall? b. What is the maximum weight for an adult of height \(1.8 \mathrm{~m}\), who is not classified as overweight or obese?

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=\frac{x+y}{x-y} ;(1,-2)\)

Poiseuille's law states that the resistance \(R\), measured in dynes, of blood flowing in a blood vessel of length \(l\) and radius \(r\) (both in centimeters) is given by $$R=f(l, r)=\frac{k l}{r^{4}}$$ where \(k\) is the viscosity of blood (in dyne-sec/cm \(^{2}\) ). What is the resistance, in terms of \(k\), of blood flowing through an arteriole \(4 \mathrm{~cm}\) long and of radius \(0.1 \mathrm{~cm} ?\)

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=e^{x y} ;(1,1)\)

Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week of its rolltop desks in the finished and unfinished versions are \(x\) and \(y\) units when the corresponding unit prices are $$\begin{array}{l}p=200-\frac{1}{5} x-\frac{1}{10} y \\\q=160-\frac{1}{10} x-\frac{1}{4} y \end{array}$$ dollars, respectively. a. What is the weekly total revenue function \(R(x, y)\) ? b. Find the domain of the function \(R\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \((a, b)\) is a critical point of \(f\) and both the conditions \(f_{x x}(a, b)<0\) and \(f_{y y}(a, b)<0\) hold, then \(f\) has a relative maximum at \((a, b)\).

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=e^{x} \ln y ;(0, e)\)

Evaluate the first partial derivatives of the function at the given point. \(f(x, y, z)=x^{2} y z^{3} ;(1,0,2)\)

Weston Publishing publishes a deluxe edition and a standard edition of its English language dictionary. Weston's management estimates that the number of deluxe editions demanded is \(x\) copies/day and the number of standard editions demanded is \(y\) copies/day when the unit prices are $$\begin{array}{l}p=20-0.005 x-0.001 y \\\q=15-0.001 x-0.003 y\end{array}$$ dollars, respectively. a. Find the daily total revenue function \(R(x, y)\). b. Find the domain of the function \(R\).

Evaluate the first partial derivatives of the function at the given point. \(f(x, y, z)=x^{2} y^{2}+z^{2} ;(1,1,2)\)

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=x^{2} y+x y^{3}\)

The volume of a certain mass of gas is related to its pressure and temperature by the formula $$V=\frac{30.9 T}{P}$$ where the volume \(V\) is measured in liters, the temperature \(T\) is measured in degrees Kelvin (obtained by adding \(273^{\circ}\) to the Celsius temperature), and the pressure \(P\) is measured in millimeters of mercury pressure. a. Find the domain of the function \(V\). b. Calculate the volume of the gas at standard temperature and pressure- that is, when \(T=273 \mathrm{~K}\) and \(P=760 \mathrm{~mm}\) of mercury.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Suppose \(h(x, y)=f(x)+g(y)\), where \(f\) and \(g\) have continuous second derivatives near \(a\) and \(b\), respectively. If \(a\) is a critical number of \(f, b\) is a critical number of \(g\), and \(f^{\prime \prime}(a) g^{\prime \prime}(b)>0\), then \(h\) has a relative extremum at \((a, b)\).

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=x^{3}+x^{2} y+x+4\)

Boor An empirical formula by E. F. Dubois relates the surface area \(S\) of a human body (in square meters) to its weight \(W\) (in kilograms) and its height \(H\) (in centimeters). The formula, given by $$S=0.007184 W^{0.425} H^{0.725}$$ is used by physiologists in metabolism studies. a. Find the domain of the function \(S\). b. What is the surface area of a human body that weighs \(70 \mathrm{~kg}\) and has a height of \(178 \mathrm{~cm}\) ?

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=x^{2}-2 x y+2 y^{2}+x-2 y\)

Suppose the output of a certain country is given by $$f(x, y)=100 x^{3 / 5} y^{2 / 5}$$ billion dollars if \(x\) billion dollars are spent for labor and \(y\) billion dollars are spent on capital. Find the output if the country spent $$\$ 32$$ billion on labor and \(\$ 243\) billion on capital.

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=x^{3}+x^{2} y^{2}+y^{3}+x+y\)

Economists have found that the output of a finished product, \(f(x, y)\), is sometimes described by the function $$f(x, y)=a x^{b} y^{1-b}$$ where \(x\) stands for the amount of money expended for labor, \(y\) stands for the amount expended on capital, and \(a\) and \(b\) are positive constants with \(0

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=\sqrt{x^{2}+y^{2}}\)

A study of arson for profit was conducted by a team of paid civilian experts and police detectives appointed by the mayor of a large city. It was found that the number of suspicious fires in that city in 2006 was very closely related to the concentration of tenants in the city's public housing and to the level of reinvestment in the area in conventional mortgages by the ten largest banks. In fact, the number of fires was closely approximated by the formula $$\begin{aligned}N(x, y) &=\frac{100\left(1000+0.03 x^{2} y\right)^{1 / 2}}{(5+0.2 y)^{2}} \\ (0&\leq x \leq 150 ; 5 \leq y \leq 35)\end{aligned}$$ where \(x\) denotes the number of persons/census tract and \(y\) denotes the level of reinvestment in the area in cents/dollar deposited. Using this formula, estimate the total number of suspicious fires in the districts of the city where the concentration of public housing tenants was \(100 / \mathrm{census}\) tract and the level of reinvestment was 20 cents/dollar deposited.

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=x \sqrt{y}+y \sqrt{x}\)

If a principal of \(P\) dollars is deposited in an account earning interest at the rate of \(r /\) year compounded continuously, then the accumulated amount at the end of \(t\) yr is given by $$A=f(P, r, t)=P e^{r t}$$ dollars. Find the accumulated amount at the end of 3 yr if a sum of $$\$ 10,000$$ is deposited in an account earning interest at the rate of $$6 % /$$ year.

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=e^{-x / y}\)

The monthly payment that amortizes a loan of \(A\) dollars in \(t\) yr when the interest rate is \(r\) per year, compounded monthly, is given by $$P=f(A, r, t)=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]}$$ a. What is the monthly payment for a home mortgage of $$\$ 300,000$$ that will be amortized over \(30 \mathrm{yr}\) with an interest rate of \(6% /\) year? An interest rate of \(8 \% /\) year? b. Find the monthly payment for a home mortgage of $$\$ 300,000$$ that will be amortized over 20 yr with an interest rate of \(8 \% /\) year.

Find the second-order partial derivatives of the function. In each case, show that the mixed partial derivatives \(f_{x y}\) and \(f_{y x}\) are equal. \(f(x, y)=\ln \left(1+x^{2} y^{2}\right)\)

The productivity of a South American country is given by the function $$f(x, y)=20 x^{3 / 4} y^{1 / 4}$$ when \(x\) units of labor and \(y\) units of capital are used. a. What is the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 256 units and 16 units, respectively? b. Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity?