Exam 8: Calculus of Several Variables

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 management of UNICO Department Store decides to enclose an area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls of the store. Two sides of the enclosure will be made of pine board, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places.

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.

Because of new, lower standards, experts in a study conducted in early 2000 projected a rise in the market for cholesterol-reducing drugs. The any market (in billions of dollars) for such drugs from 1999 through 2004 is given in the following table ( represents 1999): Find an equation of the least-squares line for these data. Please round coefficients to the nearest hundredth if necessary. Estimate the market for cholesterol-reducing drugs in 2005, assuming the trend continued.

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?

Minimize the function subject to the constraint .

Evaluate the double integral for the function and the region R. and R is the rectangle defined by 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 gives rise to a (constrained) relative extremum of subject to the constraint , then , and , simultaneously.

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 first partial derivatives of the following function at the given point.

Let Compute the following: g(1, 3), g(3, 1)

Find the domain of the function. f(m, n) = ln(m + n - 8)

Minimize the function subject to the constraint .

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.

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 132 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 by by (see the figure below). Then, , and the volume . So that . Maximize .) x = __________ inches y = __________ inches z = __________ inches V = __________ cubic inches

Minimize the function subject to the constraint .

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

Sketch the level curves of the function corresponding to the given values of z. f(x, y) = 2x + 6y; z = - 2, - 1, 0, 1, 2

Let f(x, y) = 3x + 5y - 6. Compute the following. f(0, 0) = __________ f(1, 0) = __________ f(0, 1) = __________ f(1, 2) = __________ f(2, - 1) = __________

The volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 1.5 ft and height 4 ft.