Exam 8: Calculus of Several Variables

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. then f must have a critical point at (a, b).

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

Let Compute the following. g(1, 1, 1) = __________ g(- 1, 0, 1) = __________ g(1, - 1, - 1) = __________

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 2x + 2y; R is bounded by x = 0, , y = 0 and y = 4.

Let Compute the following. g(1, 1, 1), g(- 1, 0, 1), g(- 1, 1, 1)

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 4x + 2y; R is bounded by x = 0, x = 4, y = 0 and y = x.

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

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 triangle with vertices (0, 0), (5, 0) and (0, 5).

Let Compute the following. f(1, 2) = __________ f(2, 1) = __________ f(- 1, 2) = __________ f(2, - 1) = __________

The total weekly revenue (in dollars) of the company realized in manufacturing and selling its rolltop desks is given by where x denotes the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The total weekly cost attributable to the manufacture of these desks is given by dollars. Determine how many finished units and how many unfinished units the company should manufacture each week in order to maximize its profit. What is the maximum profit(P) realizable? x = __________ y = __________ P = $__________

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.

The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week. Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week. Please round your answer to the nearest dollar, if necessary. $__________ per week.

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 equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. Please round the coefficients in your equation to two decimal places.

The volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 2 ft and height 8 ft.

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

The total weekly profit (in dollars) realized by Country Workshop in manufacturing and selling its rolltop desks is given by the profit function where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The company's management decides to restrict the manufacture of these desks to a total of exactly 100 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If h(x, y) = f(x)g(y), where f is continuous on [a, b] and g is continuous on [c, d], then where

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

Find the average value of the given function f(x, y) Over the plane region R. ; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).