Chapter 8

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. $$f_{x} \text { and } f_{y} \text { if } f(x, y)=x^{2}+2 x y+y^{3}$$

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. $$f(x, y)=x^{2}+4 x y, \quad x+y=100$$

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

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. (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. $$f(x, y)=x y, \quad 5 x+2 y=100$$

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$$

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. $$f_{u} \text { and } f_{v} \text { if } f(u, v)=u^{2}+5 u v+v^{2}$$

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. $$\frac{\partial z}{\partial x} \text { if } z=x^{2} e^{y}$$

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: \(f(3,2)=780 \quad f_{c}(3,2)=-60 \mathrm{f}_{2}(3,2)=20\)

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

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

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

Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$\frac{\partial Q}{\partial p} \text { if } Q=5 a^{2} p-3 a p^{3}$$

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 \(x\) mm in the direction of the blood flow measured from the point of injection and at time \(t\) sec. onds 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) \(\partial c / \partial t\)

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

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

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

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

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 \(\partial Q / \partial c ?\)

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

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 all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{2}+y^{2}+6 x-10 y+8$$

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

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

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

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=y^{3}-3 x y+6 x$$

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

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

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

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}+y^{2}-3 x^{2}+10 y+6$$

The monthly mortgage payment in dollars, \(P\), for a house is a function of three variables: $$ P=f(\boldsymbol{A}, \boldsymbol{r}, \boldsymbol{N}) $$ 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. $$z_{x} \text { if } z=x^{2} y+2 x^{5} y$$

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

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}+y^{3}-6 y^{2}-3 x+9$$

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.

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

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. $$f(x, y)=x^{2}+y^{2}$$

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}+y^{3}-3 x^{2}-3 y+10$$

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

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

The quantity, \(Q,\) of a good produced depends on the quantities \(x_{1}\) and \(x_{2}\) of two raw materials used: $$ Q=x_{1}^{0.3} x_{2}^{0.7} $$ A unit of \(x_{1}\) costs \(\$ 10,\) and a unit of \(x_{2}\) costs \(\$ 25 .\) We want to maximize production with a budget of \(\$ 50\) thousand for raw materials. (a) What is the objective function? (b) What is the constraint?

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, a saddle point, or none of these. $$f(x, y)=x^{2}-2 x y+3 y^{2}-8 y$$

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

Draw a contour diagram for \(C(d, m)=40 d+0.15 m\) Include contours for \(C=50,100,150,200\)