Chapter 9

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=x+y\)

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 \(P_{1}\) 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 \(f_{v}(3,1)\).

The fallout, \(V\) (in kilograms per square a 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.

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?

In the 1940 s the quantity, \(q\), of beer sold each year in Britain was found to depend on \(I\) (the aggregate personal income, adjusted for taxes and inflation), \(p_{1}\) (the average price of beer), and \(p_{2}\) (the average price of all other goods and services). Would you expect \(\partial q / \partial I, \partial q / \partial p_{1}, \partial q / \partial p_{2}\) to be positive or negative? Give reasons for your answers.

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=3 x+3 y\)

The fallout, \(V\) (in kilograms per square a 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 \(d=0\), \(d=1\), and \(d=2 .\) As time passes since the explosion, how does the fallout change? Look at the relationship between the graphs: how does fallout change as a function of distance? Explain your answers in terms of volcanoes.

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=x+y+1\)

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.

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.

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=2 x-y\)

The quantity of a product demanded by consumers is a function of its price. The quantity of one product demanded may also depend on the price of other products. For example, the demand for tea is affected by the price of coffee; the demand for cars is affected by the price of gas. The quantities demanded, \(q_{1}\) and \(q_{2}\), of two products depend on their prices, \(p_{1}\) and \(p_{2}\), as follows: $$ \begin{array}{l} q_{1}=150-2 p_{1}-p_{2} \\ q_{2}=200-p_{1}-3 p_{2} \end{array} $$ (a) What does the fact that the coefficients of \(p_{1}\) and \(p_{2}\) are negative tell you? Give an example of two products that might be related this way. (b) If one manufacturer sells both products, how should the prices be set to generate the maximum possible revenue? What is that maximum possible revenue?

A company has the production function \(P(x, y)\), which gives the number of units that can be produced for given values of \(x\) and \(y\); the cost function \(C(x, y)\) gives the cost of production for given values of \(x\) and \(y\). (a) If the company wishes to maximize production at a cost of $$\$ 50,000,$$ what is the objective function \(f\) ? What is the constraint equation? What is the meaning of \(\lambda\) in this situation? (b) If instead the company wishes to minimize the costs at a fixed production level of 2000 units, what is the objective function \(f ?\) What is the constraint equation? What is the meaning of \(\lambda\) in this situation?

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=-x-y\)

A company's production output, \(P\), is given in tons, and is a function of the number of workers, \(N\), and the value of the equipment, \(V\), in units of $$\$ 25,000.$$The production function for the company is $$ P=f(N, V)=5 N^{0.75} V^{0.25} $$ The company currently employs 80 workers, and has equipment worth $$\$ 750,000.$$What are \(N\) and \(V\) ? Find the values of \(f, f_{N}\), and \(f_{V}\) at these values of \(N\) and \(V\). Give units and explain what each answer means in terms of production.

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 \(f(105,21)\)

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 the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=y-x^{2}\)

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?

For a function \(f(r, s)\), we are given \(f(50,100)=5.67\), and \(f_{r}(50,100)=0.60\), and \(f_{s}(50,100)=-0.15 .\) Estimate \(f(52,108)\).

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

The quantity, \(q\), of a product manufactured depends on the number of workers, \(W\), and the amount of capital invested, \(K\), and is given by the Cobb-Douglas function $$q=6 W^{3 / 4} K^{1 / 4}.$$ In addition, labor costs are $$\$ 10$$ per worker and capital costs are $$\$ 20$$ per unit and the budget is $$\$ 3000.$$ (a) What are the optimum number of workers and the optimum number of units of capital? (b) Recompute the optimum values of \(W\) and \(K\) when the budget is increased by $$\$ 1.$$ Check that increasing the budget by $$\$ 1$$ allows the production of \(\lambda\) extra units of the product, where \(\lambda\) is the Lagrange multiplier.

The cardiac output, represented by \(c\), is the volume of blood flowing through a person's heart, per unit time. The systemic vascular resistance (SVR), represented by \(s\), is the resistance to blood flowing through veins and arteries. Let \(p\) be a person's blood pressure. Then \(p\) is a function of \(c\) and \(s\), so \(p=f(c, s)\). (a) What does \(\partial p / \partial c\) represent? Suppose now that \(p=k c s\), where \(k\) is a constant. (b) Sketch the level curves of \(p\). What do they represent? Label your axes. (c) For a person with a weak heart, it is desirable to have the heart pumping against less resistance, while maintaining the same blood pressure. Such a person may be given the drug nitroglycerine to decrease the SVR and the drug Dopamine to increase the cardiac output. Represent this on a graph showing level curves. Put a point \(A\) on the graph representing the person's state before drugs are given and a point \(B\) for after. (d) Right after a heart attack, a patient's cardiac output drops, thereby causing the blood pressure to drop. A common mistake made by medical residents is to get the patient's blood pressure back to normal by using drugs to increase the SVR, rather than by increasing the cardiac output. On a graph of the level curves of \(p\), put a point \(D\) representing the patient before the heart attack, a point \(E\) representing the patient right after the heart attack, and a third point \(F\) representing the patient after the resident has given the drugs to increase the SVR.

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

In each case, give a possible contour diagram for the function \(f(x, y)\) if (a) \(f_{x}>0\) and \(f_{y}>0\) (b) \(f_{x}>0\) and \(f_{y}<0\) (c) \(f_{x}<0\) and \(f_{y}>0\) (d) \(f_{x}<0\) and \(f_{y}<0\)

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?

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

Each person tries to balance his or her time between leisure and work. The tradeoff is that as you work less your income falls. Therefore each person has indifference curves which connect the number of hours of leisure, \(l\), and income, \(s .\) If, for example, you are indifferent between 0 hours of leisure and an income of $$\$ 1125$$ a week on the one hand, and 10 hours of leisure and an income of $$\$ 750$$ a week on the other hand, then the points \(l=0\), \(s=1125\), and \(l=10, s=750\) both lie on the same indifference curve. Table \(9.11\) gives information on three indifference curves, I, II, and III. $$\begin{array}{l} \text { Weekly income } \quad \text { Weekly leisure hours }\\\ \begin{array}{c|c|c|c|c|c} \hline \text { I } & \text { II } & \text { III } & \text { I } & \text { II } & \text { III } \\ \hline 1125 & 1250 & 1375 & 0 & 20 & 40 \\ \hline 750 & 875 & 1000 & 10 & 30 & 50 \\ \hline 500 & 625 & 750 & 20 & 40 & 60 \\ \hline 375 & 500 & 625 & 30 & 50 & 70 \\ \hline 250 & 375 & 500 & 50 & 70 & 90 \\ \hline \end{array} \end{array}$$ (a) Graph the three indifference curves. (b) You have 100 hours a week available for work and leisure combined, and you earn $$\$ 10 /$$ hour. Write an equation in terms of \(l\) and \(s\) which represents this constraint. (c) On the same axes, graph this constraint. (d) Estimate from the graph what combination of leisure hours and income you would choose under these circumstances. Give the corresponding number of hours per week you would work.

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

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 $$\$ 6 .$$

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

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

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

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

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

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

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

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

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. $$ \partial M / \partial B $$