Exam 8: Calculus of Several Variables

An open rectangular box having a volume of is to be constructed from a tin sheet. Find the dimensions of such a box if the amount of material used in its construction is to be minimal. Hint: Let the dimensions of the box be x by y by z. Then, xyz = 108 and the amount of material used is given by S = xy + 2yz + 2xz. Show that Minimize f(x, y)

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

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 first partial derivatives of the function.

Find the first partial derivatives of the function.

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.

Let . Compute the following.

Sketch the domain of the function.

Find the average value of the function over the plane region . and is the triangle with vertices , and .

Find the first partial derivatives of the function.

The Social Security (FICA) wage base (in thousands of dollars) from 1996 to 2001 is given in the following table: Find an equation of the least-squares line for these data. (Let x = 1 represent the year 1996.) Please round the coefficients in your equation to three decimal places. Use the result to estimate the FICA wage base in the year 2007.

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 .

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. The least-squares line must pass through at least one data point.

Maximize the function subject to the constraint .

Minimize the function subject to the constraint .

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 4x + 8y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

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

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 and denote the unit prices, respectively, and and 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.

The volume V (in liters) of a certain mass of gas is related to its pressure P (in millimeters of mercury) and its temperature T (in degrees Kelvin) by the law Compute and when T = 260 and P = 700.

Find the second order partial derivative of the following function. Show that the mixed partial derivative and are equal.