Exam 8: Calculus of Several Variables

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 700 copies/day. How many deluxe copies and how many standard copies should be published each day to maximize Weston's daily profit? ​

Find the average value of the given function ​ F(x, y) ​ Over the plane region R. ​ ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2). ​

A closed rectangular box having a volume of is to be constructed. If the material for the sides costs and the material for the top and bottom costs , find the dimensions of the box that can be constructed with minimum cost.

Find the average value of the function ​ ​ over the plane region R. Enter your answer as an expression. ​ and R is the triangle with vertices , and .

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 . ​

Let . Compute the following: ​ ​

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. ​ X 1 2 3 4 Y 4 6 8 9 ​ ​

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 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 betweenand 100 and 120 per week. Please round your answer to the nearest dollar, if necessary. ​

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.

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 gives rise to a (constrained) relative extremum of subject to the constraint , then , and , simultaneously. ​

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 , , and . ​

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

Find the total differential of the function. ​

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

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

Find the volume of the solid bounded above by the surfac and below by the plane region R. ; R is the triangle with vertices (0, 0), (2, 0) and (0, 2).

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 .

Minimize the function ​ ​ subject to the constraint .

Find the domain of the function. ​

Find the average value of the given function ​ ​ Over the plane region R. ​ ; R is the region bounded by the graph of y = 8x and y = 0 from x = 1 to x = 7. ​