Exam 8: Calculus of Several Variables

The following table gives the total sales of drugs (in billions of dollars) in one country from 1999 through 2003: Find an equation of the least-square line for these data.

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

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. then f must have a critical point at (a, b).

Minimize the function subject to the constraint .

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 .

Find the second-order partial derivatives of the given function. Show that the mixed partial derivatives and are equal.

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?

Find the first partial derivative of the following function.

Evaluate the double integral for the function and the region . and is bounded by , , and .

Let . Compute .

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?

Minimize the function subject to the constraint .

With an aging population, the demand for health care, as measured by outpatient visits, is steadily growing. The number of outpatient visits (in million) from 1991 through 2001 is recorded in the following table ( corresponds to 1991): Find an equation on the least-squares line for these data. Please round coefficients to the nearest hundredth if necessary.

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.

The projections of male life expectancy at age 65 in the United States are summarized in the following table. (Here, x = 0 corresponds to the beginning of 2000.) Find the equation of the least-squares line for these data. Use the equation found to estimate the life expectancy at 65 of a male in 2050.

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

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

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 6x + 6y; R is bounded by x = 0, , y = 0 and y = 4.

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

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 5 ; R is bounded by the lines x = 1, y = 0 and y = x.