Exam 8: Calculus of Several Variables

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 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 .

The annual sales (in billions of dollars) of global positioning systems (GPS) equipment from 2000 through 2006 follow (sales in 2004 through 2006 are projections).Here, corresponds to 2000. Find an equation of the least-squares line for these data.Please round coefficients to the nearest hundredth if necessary.Use the equation found to estimate the annual sales of GPS equipment for 2005.

Find the average value of the given function f(x, y) over the plane region R. ; R is the region bounded by the graph of y = 6x and y = 0 from x = 1 to x = 5.

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.

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

Maximize the function subject to the constraints .

Determine 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 true. If satisfies at a critical point of , then cannot have a relative extremum at .

The total daily revenue (in dollars) that Weston Publishing realizes in publishing and selling its English-language dictionaries is given by where x denotes the number of deluxe copies and y denotes the number of standard copies published and sold daily.The total daily cost of publishing these dictionaries is given by dollars.Determine how many deluxe copies and how many standard copies Weston should publish each day to maximize its profits.Please round the answers to the nearest integer.What is the maximum profit realizable? Please round the answer to the nearest dollar.

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.

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 = $__________

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 second-order partial derivatives of the function.

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.

An open rectangular box is to be constructed from material that costs for the bottom and for its sides.Find the dimensions of the box of greatest volume that can be constructed for .

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?

The following data, compiled by the superintendent of schools in a large metropolitan area, shows the average SAT verbal scores of high school seniors during the 5 years since the district implemented its "back-to-basics" program. Determine the equation of the least-squares line for these data.Use the result obtained to predict the average SAT verbal score of high school seniors 2 years from now (x = 7).Please round your answer to the nearest whole number.

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

The estimated number of computers (in millions) connected to the Internet in Europe from 1998 through 2002 is summarized in the accompanying table: (Here, x = 0 corresponds to the beginning of 1998.) Find an equation of the least-squares line for these data.Use the result to estimate the projected number of computers connected to the Internet in Europe at the beginning of 2004, assuming the trend continues.

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 three decimal places.