Chapter 8

The quantity, \(Q,\) of a certain product manufactured depends on the quantity of labor, \(L,\) and of capital, \(K,\) used according to the function $$Q=900 L^{1 / 2} K^{2 / 3}$$ Labor costs \(\$ 100\) per unit and capital costs \(\$ 200\) per unit. What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}-3 x+y^{3}-3 y$$

$$\text { If } f(x, y)=x^{3}+3 y^{2}, \text { find } f(1,2), f_{x}(1,2), f_{y}(1,2)$$

Maple syrup production is highest when the nights are cold and the days are warm. Make a possible contour diagram for maple syrup production as a function of the high (daytime) temperature and the low (nighttime) temperature. Label the contours with \(10,20,30,\) and 40 (in liters of maple syrup).

The Cobb-Douglas production function for a product is $$ P=5 L^{0.8} K^{0.2} $$ where \(P\) is the quantity produced, \(L\) is the size of the labor force, and \(K\) is the amount of total equipment. Each unit of labor costs \(\$ 300,\) each unit of equipment costs \(\$ 100,\) and the total budget is \(\$ 15,000\) (a) Make a table of \(L\) and \(K\) values which exhaust the budget. Find the production level, \(P\), for each. (b) Use the method of Lagrange multipliers to find the optimal way to spend the budget.

The fallout, \(V\) (in kilograms per square kilometer), from a volcanic explosion depends on the distance, \(d,\) from the volcano and the time, \(t,\) since the explosion: $$V=f(d, t)=(\sqrt{t}) e^{-d}$$. On the same axes, graph cross-sections of \(f\) with \(t=1\) and \(t=2 .\) As distance from the volcano increases, how does the fallout change? Look at the relationship between the graphs: how does the fallout change as time passes? Explain your answers in terms of volcanoes.

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=400-3 x^{2}-4 x+2 x y-5 y^{2}+48 y$$

People commuting to a city can choose to go either by bus or by train. The number of people who choose either method depends in part upon the price of each. Let \(f\left(P_{1}, P_{2}\right)\) be the number of people who take the bus when \(P\) is the price of a bus ride and \(P_{2}\) is the price of a train ride. What can you say about the signs of \(\partial f / \partial P_{1}\) and \(\partial f / \partial P_{2}\) ? Explain your answers.

If \(f(u, v)=5 u v^{2},\) find \(f(3,1), f_{u}(3,1),\) and \(f_{v}(3,1)\)

Hiking on a level trail going due east, you decide to leave the trail and climb toward the mountain on your left. The farther you go along the trail before turning off, the gentler the climb. Sketch a possible topographical map showing the elevation contours.

A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities, \(q_{1}\) and \(q_{2},\) supplied by each factory, and is expressed by the joint cost function, $$C=f\left(q_{1}, q_{2}\right)=2 q_{1}^{2}+q_{1} q_{2}+q_{2}^{2}+500$$ The company's objective is to produce 200 units, while minimizing production costs. How many units should be supplied by each factory?

Suppose that \(x\) is the price of one brand of gasoline and \(y\) is the price of a competing brand. Then \(q_{1},\) the quantity of the first brand sold in a fixed time period, depends on both \(x\) and \(y,\) so \(q_{1}=f(x, y) .\) Similarly, if \(q_{2}\) is the quantity of the second brand sold during the same period, \(q_{2}=g(x, y) .\) What do you expect the signs of the following quantities to be? Explain. (a) \(\partial q_{1} / \partial x\) and \(\partial q_{2} / \partial y\) (b) \(\partial q_{1} / \partial y\) and \(\partial q_{2} / \partial x\)

For \(f(x, y)=A-\left(x^{2}+B x+y^{2}+C y\right),\) what values of \(A, B,\) and \(C\) give \(f\) a local maximum value of 15 at the point (-2,1)\(?\)

The amount of money, \(\$ B,\) in a bank account earning interest at a continuous rate, \(r,\) depends on the amount deposited, \(\$ P,\) and the time, \(t,\) it has been in the bank, where $$ B=P e^{r t} $$ Find \(\partial B / \partial t, \partial B / \partial r\) and \(\partial B / \partial P\) and interpret each in financial terms.

A missile has a guidance device which is sensitive to both temperature, \(t^{\circ} \mathrm{C},\) and humidity, \(h .\) The range in \(\mathrm{km}\) over which the missile can be controlled is given by Range \(=27,800-5 t^{2}-6 h t-3 h^{2}+400 t+300 h\) What are the optimal atmospheric conditions for controlling the missile?

For a function \(f(x, y),\) we are given \(f(100,20)=2750\) and \(f_{x}(100,20)=4,\) and \(f_{y}(100,20)=7 .\) Estimate \(f(105,21)\)

A manufacturer sells two products, one at a price of \(\$ 4000\) a unit and the other at a price of \(\$ 13,000\) a unit. A quantity \(q_{1}\) of the first product and \(q_{2}\) of the second product are sold at a total cost of \(\$\left(4000+q_{1}+q_{2}\right)\) to the manufacturer. (a) Express the manufacturer's profit, \(\pi\), as a function of \(q_{1}\) and \(q_{2}\) (b) Sketch contours of \(\pi\) for \(\pi=10,000, \pi=20,000\) and \(\pi=30,000\) and the break-even curve \(\pi=0\)

You have set aside 20 hours to work on two class projects. You want to maximize your grade (measured in points), which depends on how you divide your time between the two projects. (a) What is the objective function for this optimization problem and what are its units? (b) What is the constraint? (c) Suppose you solve the problem by the method of Lagrange multipliers. What are the units for \(\lambda ?\) (d) What is the practical meaning of the statement \(\lambda=5 ?\)

The cost of renting a car from a certain company is \(\$ 40\) per day plus 15 cents per mile, and so we have $$ C=40 d+0.15 m $$ Find \(\partial C / \partial d\) and \(\partial C / \partial m .\) Give units and explain why your answers make sense.

A steel manufacturer can produce \(P(K, L)\) tons of steel using \(K\) units of capital and \(L\) units of labor, with production costs \(C(K, L)\) dollars. With a budget of \(\$ 600,000,\) the maximum production is 2,500,000 tons, using \(\$ 400,000\) of capital and \(\$ 200,000\) of labor. The Lagrange multiplier is \(\lambda=3.17\) (a) What is the objective function? (b) What is the constraint? (c) What are the units for \(\lambda ?\) (d) What is the practical meaning of the statement \(\lambda=\) \(3.17 ?\)

A company operates two plants which manufacture the same item and whose total cost functions are $$C_{1}=8.5+0.03 q_{1}^{2} \quad \text { and } \quad C_{2}=5.2+0.04 q_{2}^{2}$$ where \(q_{1}\) and \(q_{2}\) are the quantities produced by each plant. The company is a monopoly. The total quantity demanded, \(q=q_{1}+q_{2},\) is related to the price, \(p,\) by $$p=60-0.04 q$$ How much should each plant produce in order to maximize the company's profit?\(^{11}\)

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, y)=x^{2} y$$

Sketch a contour diagram for \(z=y-\sin x .\) Include at least four labeled contours. Describe the contours in words and how they are spaced.

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, y)=x e^{y}$$

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, y)=x^{2}+2 x y+y^{2}$$

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, y)=\frac{2 x}{y}, \quad y \neq 0$$

If \(x_{1}\) and \(x_{2}\) are the number of items of two goods bought, a customer's utility is $$U\left(x_{1}, x_{2}\right)=2 x_{1} x_{2}+3 x_{1}$$ The unit cost is \(\$ 1\) for the first good and \(\$ 3\) for the second. Use Lagrange multipliers to find the maximum value of \(U\) if the consumer's disposable income is \(\$ 100 .\) Estimate the new optimal utility if the consumer's disposable income increases by \(\$ 6\)

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f=5+x^{2} y^{2}$$

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f=e^{x y}$$

In a small printing business, \(P=2 N^{0.6} V^{0.4},\) where \(N\) is the number of workers. \(V\) is the value of the equipment, and \(P\) is production, in thousands of pages per day. (a) If this company has a labor force of 300 workers and 200 units worth of equipment, what is production? (b) If the labor force is doubled (to 600 workers), how does production change? (c) If the company purchases enough equipment to double the value of its equipment (to 400 units), how does production change? (d) If both \(N\) and \(V\) are doubled from the values given in part (a), how does production change?

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$B=5 x e^{-2 t}$$

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, t)=t^{3}-4 x^{2} t$$

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f=100 e^{r t}$$

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$Q=5 p_{1}^{2} p_{2}^{-1}, \quad p_{2} \neq 0$$

The power \(P\) produced by a windmill is proportional to the square of the diameter \(d\) of the windmill and to the cube of the speed \(v\) of the wind. \(^{5}\) (a) Write a formula for \(P\) as a function of \(d\) and \(v\) (b) A windmill generates \(100 \mathrm{kW}\) of power at a certain wind speed. If a second windmill is built having twice the diameter of the original, what fraction of the original wind speed is needed by the second windmill to produce \(100 \mathrm{kW} ?\) (c) Sketch a contour diagram for \(P\).

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$P=2 K L^{2}$$

Is there a function \(f\) which has the following partial derivatives? If so what is it? Are there any others? $$ \begin{array}{l} f_{x}(x, y)=4 x^{3} y^{2}-3 y^{4} \\ f_{y}(x, y)=2 x^{4} y-12 x y^{3} \end{array} $$

Each contour diagram (a) and (b) shows satisfaction with quantities of two items \(X\) and \(Y\) combined. Match (a) and (b) with the items in (I) and (II). (1) \(X:\) Income; \(Y:\) Leisure time (II) \(X:\) Income; \(Y:\) Hours worked

Show that the Cobb-Douglas function $$ Q=b K^{\alpha} L^{1-\alpha} \text { where } \quad 0 < \alpha < 1 $$ satisfies the equation $$ K \frac{\partial Q}{\partial K}+L \frac{\partial Q}{\partial L}=Q $$

Problems are about the money supply, \(M,\) which is the total value of all the cash and checking account balances in an economy. It is determined by the value of all the cash, \(B\), the ratio, \(c,\) of cash to checking deposits, and the fraction, \(r,\) of checking account deposits that banks hold as cash: $$ M=\frac{c+1}{c+r} B $$ (a) Find the partial derivative. (b) Give its sign. (c) Explain the significance of the sign in practical terms. $$\partial M / \partial B$$

A shopper buys \(x\) units of item \(A\) and \(y\) units of item \(B,\) obtaining satisfaction \(s(x, y)\) from the purchase. (Satisfaction is called utility by economists.) The contours \(s(x, y)=x y=c\) are called indifference curves because they show pairs of purchases that give the shopper the same satisfaction. (a) A shopper buys 8 units of \(A\) and 2 units of \(B\). What is the equation of the indifference curve showing the other purchases that give the shopper the same satisfaction? Sketch this curve. (b) After buying 4 units of item \(A\), how many units of \(B\) must the shopper buy to obtain the same satisfaction as obtained from buying 8 units of \(A\) and 2 units of \(B ?\) (c) The shopper reduces the purchase of item \(A\) by \(k,\) a fixed number of units, while increasing the purchase of \(B\) to maintain satisfaction. In which of the following cases is the increase in \(B\) largest? Initial purchase of \(A\) is 6 units Initial purchase of \(A\) is 8 units

Problems are about the money supply, \(M,\) which is the total value of all the cash and checking account balances in an economy. It is determined by the value of all the cash, \(B\), the ratio, \(c,\) of cash to checking deposits, and the fraction, \(r,\) of checking account deposits that banks hold as cash: $$ M=\frac{c+1}{c+r} B $$ (a) Find the partial derivative. (b) Give its sign. (c) Explain the significance of the sign in practical terms. $$\partial M / \partial r$$

Problems are about the money supply, \(M,\) which is the total value of all the cash and checking account balances in an economy. It is determined by the value of all the cash, \(B\), the ratio, \(c,\) of cash to checking deposits, and the fraction, \(r,\) of checking account deposits that banks hold as cash: $$ M=\frac{c+1}{c+r} B $$ (a) Find the partial derivative. (b) Give its sign. (c) Explain the significance of the sign in practical terms. $$\partial M / \partial c$$