Exam 8: Calculus of Several Variables

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 200 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit?

​Find the approximate change in z when the point changes from to . Round your answer to two decimal places. ​ ; from to ​​ __________

Find the average value of the function over the plane region R. and R is the triangle with vertices , and . ​

The total daily profit (in dollars) realized by Weston Publishing in publishing and selling its dictionaries is given by the profit function ​ ​ where x stands for the number of deluxe editions and y denotes the number of standard editions sold daily. Weston's management decides that publication of these dictionaries should be restricted to a total of exactly 400 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit? ​ Find the number of deluxe copies. ​ Find the number of standard copies.

Find the volume of the solid bounded above by the surface ​ ​ And below by the plane region R. ​ and R is the region bounded by the graphs of and . ​

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 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). ​

Evaluate the double integral ​ ​ For the given function f(x, y) and the region R. ​ F(x, y) = 6x + 6y; R is bounded by x = 0, , y = 0 and y = 4. ​

Evaluate the double integral for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 2y + 3x; R is the rectangle defined by and .

Sketch the domain of the function. ​ ​

Maximize the function ​ ​ Subject to the constraint ​

Find the first partial derivatives of the function. ​ ​

Maximize the function ​ ​ Subject to the constraint . ​

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 maximum and minimum values of the function ​ ​ Subject to the constraint . ​

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.

Evaluate the double integral ​ for the given function f(x, y) and the region R. Enter your answer as a fraction. f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

​Find the approximate change in z when the point changes from to . ​ ; from to ​​ __________

Evaluate the double integral ​ ​ for the given function f(x, y) and the region R. Enter your answer as a fraction. ​ f(x, y) = 0x + 0y; R is bounded by x = 0, , y = 0 and y = 4.

Find the volume of the solid bounded above by the surface and below by the plane region R. ​ and R is the region bounded by the graphs of . ​