Exam 8: Calculus of Several Variables

According to industry sources, online banking is expected to take off in the near future. The projected number of households (in millions) using this service is given in the following table: (Here, x = 0 corresponds to the beginning of 1997.) Find an equation of the least-squares line for these data. Please round the coefficients in your equation to three decimal places. Use the result to estimate the number of households using online banking at the beginning of 2009, assuming the projection is accurate.

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).

Minimize the function subject to the constraint .

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

Minimize the function subject to the constraints

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 .

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

The projected number of wireless subscribers y (in millions) from the year 2000 through 2006 is summarized in the accompanying table: (Here, x = 0 corresponds to the beginning of 2000.) Find an equation of the least-squares line for these data. Please round the coefficients in your equation to two decimal places. Use the result to estimate the projected number of wireless subscribers at the beginning of 2010.

Sketch the domain of the function.

Maximize the function subject to the constraints .

Maximize the function subject to the constraint .

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 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 $10/running foot and the steel fencing costs $5/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places.

A building in the shape of a rectangular box is to have a volume of (see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $5/square foot for the front and back, and $2/square foot for the sides. Find the dimensions of the building that will result in a minimal annual heating and cooling cost. What is the minimal annual heating and cooling cost(C)?

The Ross-Simons Company has a monthly advertising budget of $40,000. Their marketing department estimates that if they spend x dollars on newspaper advertising and y dollars on television advertising, then the monthly sales will be given by dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads each month to maximize its monthly sales.

The spending (in billions of dollars) by Medicaid, the national health-care plan for the poor, over the 5-year period from 1988 to 1992 is summarized in the following table. (Here, x = 0 represents the beginning of the year 1988.) Find an equation of the least-squares line for these data. Please round the coefficients in your equation to three decimal places. Use the result of to estimate Medicaid spending in Massachusetts for the year 2001, assuming the trend continued.

Find the first partial derivatives of the function.

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 Company requires that its corned beef hash containers have a capacity of , be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs and the metal for the pull-off lid costs .

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. Find the critical point(s) of the function. Find the point(s) of maximum. Find the point(s) of minimum. Find the relative extrema of the function.