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.

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.

The population density of a certain city is given by the function where the origin (0, 0) gives the location of the government center.Find the population inside the rectangular area described by

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 triangle with vertices (0, 0), (5, 0) and (0, 5).

Find the volume of the solid bounded above by the surface and below by the plane region . and is the region bounded by the graphs of and .

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 volume of a cylindrical tank of radius r and height h is given by Find the volume of a cylindrical tank of radius 2 ft and height 8 ft.

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.

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

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

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

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

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 .

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. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

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 = 2x and y = 0 from x = 1 to x = 3.

Maximize the function subject to the constraints .

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.

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