Exam 8: Calculus of Several Variables

In a survey it was determined that the demand equation for VCRs is given by ​ ​ The demand equation for blank VCR tapes is given by ​ ​ Where p and q denote the unit prices, respectively, and x and y denote the number of VCRs and the number of blank VCR tapes demanded each week. Determine whether these two products are substitute, complementary, or neither. ​

The Ross-Simons Company has a monthly advertising budget of $40,000. Their marketing department estimates that if they spend x dollars on newspaper advertising and y dollars on television advertising, then the monthly sales will be given by ​ ​ dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads each month to maximize its monthly sales.

Find the average value of the given function over the plane region R. ​ ; R is the triangle with vertices (0, 0), (1, 0) and (1, 1).

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

Find the first partial derivatives of the function. ​ ​

Minimize the function ​ ​ ​ subject to the constraint .

The volume of a cylindrical tank of radius r and height h is given by ​ ​ Find the volume of a cylindrical tank of radius 3 ft and height 8 ft. ​

The total weekly revenue (in dollars) of the Country Workshop 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? ​

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 and . ​

The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function ​ ​ Where x stands for the number of finished units and y 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 180 and 200 and the number of unfinished units varies between 100 and 120 per week. Please round your answer to the nearest dollar, if necessary. ​

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

Find the total differential of the function. ​ ​

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. ​

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 °F and S is the wind speed in mph. ​What is the wind chill temperature when the actual air temperature is 52 °F and the wind speed is 30 mph? Please round the answer to the nearest °F ​T=__________ °F ​If the temperature is 52 °F, by how much approximately will the wind chill temperature change if the wind speed increases from 30 mph to 31 mph? Please round the answer to the nearest tenth. ​T=__________ °F

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

Find the average value of the given function over the plane region R. ​ ; R is the region bounded by the graph of and from to .

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

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

Minimize 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. ​ ​