Exam 8: Calculus of Several Variables

An auxiliary electric power station will serve three communities, A, B, and C, whose relative locations are shown in the accompanying figure. Determine where the power station should be located if the sum of the squares of the distances from each community to the site is minimized. Please round the answer to the nearest hundredth. ​

Find the volume of the solid bounded above by the surface ​ z = f(x, y) ​ and below by the plane region R. Enter your answer as an expression. ​ ; R is the region bounded by , and .

Maximize the function ​ ​ subject to the constraint .

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

The Company requires that its corned beef hash containers have a capacity of , be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs and the metal for the pull-off lid costs . ​

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 120 in. Find the dimensions of the rectangular package that would have the greatest possible volume under these regulations. (Hint: Let the dimensions of the box be x'' by y'' by z'' (see the figure below). Then, 2x + 2x + y= 120 , and the volume V=xyz. So that . Maximize f(x,z) .)

Find an equation of the least-squares line for the data. Please round the coefficients to the nearest hundredth. x 1 1 2 3 4 4 5 Y 2 3 5 6 6 6)5 8)5 ​

Find the first partial derivatives of the function. ​ ​

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

Find the total differential of the function. ​

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 108 in. Find the dimensions of the rectangular package that would have the greatest possible volume under these regulations.

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. ​ ​

Sketch the level curves of the function corresponding to the given values of z. ​ ; ​

Find the total differential of the function. ​ ​

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 population density of a certain city is given by the function ​ ​ Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by ​ ​

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

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 the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.

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.