Exam 8: Calculus of Several Variables

The productivity of a South American country is given by the function when x units of labor and y units of capital are used. What is the marginal productivity of labor and what is the marginal productivity of capital when the amounts expended on labor and capital are 288 units and 18 units, respectively? Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity? Please answer yes or no.

Find the second-order partial derivatives of the function.

Minimize the function subject to the constraint .

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line.

A formula used by meteorologists to calculate the wind chill temperature (the temperature that you feel in still air that is the same as the actual temperature when the presence of wind is taken into consideration) is where t is the actual air temperature in and S is the wind speed in mph. What is the wind chill temperature when the actual air temperature is and the wind speed is 15 mph ( )? Please round the answer to the nearest If the temperature is , by how much approximately will the wind chill temperature change if the wind speed increases from 15 mph to 16 mph ( )? Please round the answer to the nearest tenth.

An open rectangular box having a surface area of is to be constructed from a tin sheet.Find the diamensions of the box if the volume of the box is to be as large as possible. What is the maximum volume? Hint : Let the diamentions of the box be (see the figure as follows). Then the surface area is and its volume is .

Find the domain of the function. f(u, v) = ln(u + v - 6)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.

Evaluate the double integral for the given function f(x, y) and the region R. R is the rectangle defined by

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.

Find the domain of the function. f(m, n) = 2m + 9n

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 2y + x; R is the rectangle defined by

Evaluate the first partial derivatives of the function at the given point.

Find the volume of the solid bounded above by the surface z = f(x, y) and below by the plane region R. ; R is the region bounded by .

Find the maximum and minimum values of the function subject to the constraint .

Use a double integral to find the volume of the solid shown in the figure.

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. Find the critical point(s) of the function. Find the point(s) of maximum. Find the point(s) of minimum. Find the relative extrema of the function.