Chapter 9

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x^{2}+4 x y, \quad x+y=100 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{x}\) and \(f_{y}\) if \(f(x, y)=2 x^{2}+3 y^{2}\)

Concern the cost, \(C\), of renting a car from a company which charges $$\$ 40$$ a day and 15 cents a mile, so \(C=f(d, m)=40 d+0.15 m\), where \(d\) is the number of days, and \(m\) is the number of miles. Make a table of values for \(C\), using \(d=1,2,3,4\) and \(m=100,200,300,400\). You should have 16 values in your table.

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x y, \quad 5 x+2 y=100 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{x}\) and \(f_{y}\) if \(f(x, y)=100 x^{2} y\)

Concern the cost, \(C\), of renting a car from a company which charges $$\$ 40$$ a day and 15 cents a mile, so \(C=f(d, m)=40 d+0.15 m\), where \(d\) is the number of days, and \(m\) is the number of miles. (a) Find \(f(3,200)\) and interpret it. (b) Explain the significance of \(f(3, m)\) in terms of rental car costs. Graph this function, with \(C\) as a function of \(m\). (c) Explain the significance of \(f(d, 100)\) in terms of rental car costs. Graph this function, with \(C\) as a function of \(d\).

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x^{2}+3 y^{2}+100, \quad 8 x+6 y=88 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{x}\) and \(f_{y}\) if \(f(x, y)=x^{2}+2 x y+y^{3}\)

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=5 x y, \quad x+3 y=24 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(z_{x}\) if \(z=x^{2} y+2 x^{5} y\)

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x+y, \quad x^{2}+y^{2}=1 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{u}\) and \(f_{v}\) if \(f(u, v)=u^{2}+5 u v+v^{2}\)

The demand for coffee, \(Q\), in pounds sold per week, is a function of the price of coffee, \(c\), in dollars per pound and the price of tea, \(t\), in dollars per pound, so \(Q=f(c, t)\). (a) Do you expect \(f_{c}\) to be positive or negative? What about \(f_{t} ?\) Explain. (b) Interpret each of the following statements in terms of the demand for coffee: \(f(3,2)=780 \quad f_{c}(3,2)=-60 \quad f_{t}(3,2)=20\)

Draw a contour diagram for the function \(C=40 d+\) \(0.15 \mathrm{~m} .\) Include contours for \(C=50,100,150,200\).

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=x^{2}+4 x+y^{2} $$

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x^{2}+y^{2}, \quad 4 x-2 y=15 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(\frac{\partial z}{\partial x}\) if \(z=x^{2} e^{y}\)

The quantity \(Q\) (in pounds) of beef that a certain community buys during a week is a function \(Q=f(b, c)\) of the prices of beef, \(b\), and chicken, \(c\), during the week. Do you expect \(\partial Q / \partial b\) to be positive or negative? What about \(\partial Q / \partial c\) ?

Refer to Table \(9.2\), which shows \({ }^{1}\) the weekly beef consumption, \(C\), (in lbs) of an average household as a function of \(p\), the price of beef (in $$\$ / \mathrm{lb}$$ ) and \(I\), annual household income (in $$\$ 1000 \mathrm{~s}$$ ). $$\begin{array}{|r|c|c|c|c|} \hline & 3.00 & 3.50 & 4.00 & 4.50 \\ \hline 20 & 2.65 & 2.59 & 2.51 & 2.43 \\ \hline 40 & 4.14 & 4.05 & 3.94 & 3.88 \\ \hline 60 & 5.11 & 5.00 & 4.97 & 4.84 \\ \hline 80 & 5.35 & 5.29 & 5.19 & 5.07 \\ \hline 100 & 5.79 & 5.77 & 5.60 & 5.53 \\ \hline \end{array}$$ Make a table of the proportion, \(P\), of household income spent on beef per week as a function of price and income. (Note that \(P\) is the fraction of income spent on beef.)

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=x^{2}+x y+3 y $$

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=3 x-2 y, \quad x^{2}+2 y^{2}=44 $$

A drug is injected into a patient's blood vessel. The function \(c=f(x, t)\) represents the concentration of the drug at a distance \(x \mathrm{~mm}\) in the direction of the blood flow measured from the point of injection and at time \(t\) seconds since the injection. What are the units of the following partial derivatives? What are their practical interpretations? What do you expect their signs to be? (a) \(\partial c / \partial x\) (b) \(\partial c / \partial t\)

Refer to Table \(9.2\), which shows \({ }^{1}\) the weekly beef consumption, \(C\), (in lbs) of an average household as a function of \(p\), the price of beef (in $$\$ / \mathrm{lb}$$ ) and \(I\), annual household income (in $$\$ 1000 \mathrm{~s}$$ ). $$\begin{array}{|r|c|c|c|c|} \hline & 3.00 & 3.50 & 4.00 & 4.50 \\ \hline 20 & 2.65 & 2.59 & 2.51 & 2.43 \\ \hline 40 & 4.14 & 4.05 & 3.94 & 3.88 \\ \hline 60 & 5.11 & 5.00 & 4.97 & 4.84 \\ \hline 80 & 5.35 & 5.29 & 5.19 & 5.07 \\ \hline 100 & 5.79 & 5.77 & 5.60 & 5.53 \\ \hline \end{array}$$ Express \(P\), the proportion of household income spent on beef per week, in terms of the original function \(f(I, p)\) which gave consumption as a function of \(p\) and \(I\).

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)-x^{2}+y^{2}+6 x-10 y+8 $$

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x^{2}+y, \quad x^{2}-y^{2}=1 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{t}\) if \(f(t, a)=5 a^{2} t^{3}\)

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=y^{3}-3 x y+6 x $$

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x y, \quad 4 x^{2}+y^{2}=8 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{x}\) and \(f_{y}\) if \(f(x, y)=5 x^{2} y^{3}+8 x y^{2}-3 x^{2}\)

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 total sales of a product, \(S\), can be expressed as a function of the price \(p\) charged for the product and the amount, \(a\), spent on advertising, so \(S=f(p, a)\). Do you expect \(f\) to be an increasing or decreasing function of \(p ?\) Do you expect \(f\) to be an increasing or decreasing function of \(a\) ? Why?

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)-x^{2}-2 x y+3 y^{2}-8 y $$

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$ f(x, y)=x^{2}+y^{2}, \quad x^{4}+y^{4}=2 $$

The monthly mortgage payment in dollars, \(P\), for a house is a function of three variables: $$ P=f(A, r, N), $$ where \(A\) is the amount borrowed in dollars, \(r\) is the interest rate, and \(N\) is the number of years before the mortgage is paid off. (a) \(f(92000,14,30)=1090.08\). What does this tell you, in financial terms? (b) \(\left.\frac{\partial P}{\partial r}\right|_{(92000,14,30)}=72.82 .\) What is the financial significance of the number \(72.82 ?\) (c) Would you expect \(\partial P / \partial A\) to be positive or negative? Why? (d) Would you expect \(\partial P / \partial N\) to be positive or negative? Why?

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{x}\) and \(f_{y}\) if \(f(x, y)=10 x^{2} e^{3 y}\)

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=x^{3}-3 x+y^{3}-3 y $$

The sales of a product, \(S=f(p, a)\), are a function of the price, \(p\), of the product (in dollars per unit) and the amount, \(a\), spent on advertising (in thousands of dollars). (a) Do you expect \(f_{p}\) to be positive or negative? Why? (b) Explain the meaning of the statement \(f_{a}(8,12)=\) 150 in terms of sales.

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.

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(\frac{\partial P}{\partial r}\) if \(P=100 e^{r t}\)

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=x^{3}+y^{2}-3 x^{2}+10 y+6 $$

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(\frac{\partial A}{\partial h}\) if \(A=\frac{1}{2}(a+b) h\)

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=x^{3}+y^{3}-6 y^{2}-3 x+9 $$

The quantity, \(Q\), of a good produced depends on the quantities \(x_{1}\) and \(x_{2}\) of two raw materials used: $$Q=x_{1}^{0.6} x_{2}^{0.4} .$$ A unit of \(x_{1}\) costs $$\$ 127,$$ and a unit of \(x_{2}\) costs $$\$ 92.$$We want to minimize the cost, \(C\), of producing 500 units of the good. (a) What is the objective function? (b) What is the constraint?

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(\frac{\partial}{\partial m}\left(\frac{1}{2} m v^{2}\right)\)

The monthly payments, \(P\) dollars, on a mortgage in which \(A\) dollars were borrowed at an annual interest rate of \(r \%\) for \(t\) years is given by \(P=f(A, r, t)\). Is \(f\) an increasing or decreasing function of \(A\) ? Of \(r\) ? Of \(t\) ?

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=x^{3}+y^{3}-3 x^{2}-3 y+10 $$

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?

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

You are planning a trip whose principal cost is gasoline. (a) Make a table showing how the daily fuel cost varies as a function of the price of gasoline (in dollars per gallon) and the number of gallons you buy each day. (b) If your car goes 30 miles on each gallon of gasoline, make a table showing how your daily fuel cost varies as a function of your daily travel distance and the price of gas.

Find all the critical points and determine whether each is a local maximum, local minimum, or neither. $$ f(x, y)=400-3 x^{2}-4 x+2 x y-5 y^{2}+48 y $$