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. ​ ​

Find the volume of the solid bounded above by the surface ​ ​ and below by the plane region R. Enter your answer as a formula. ​ and R is the region bounded by the graphs of and .

A building in the shape of a rectangular box is to have a volume of 21,000ft³ (see the figure). It is estimated that the annual heating and cooling costs will be $2/square foot for the top, $7/square foot for the front and back, and $3/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)? ​ ​ x = __________ ​ y = __________ ​ z = __________ ​ C = __________

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 radius and height of a right circular cylinder are measured with a maximum error of 0.1 cm in each measurement. Approximate the maximum error in calculating the volume of the cylinder if the measured dimensions r = 5 cm and h = 24 cm are used. ​

Find the average value of the given function ​ f(x, y) ​ over the plane region R. Enter your answer as an expression. ​ ; R is the region bounded by the graph of y = 2x and y = 0 from x = 1 to x = 7.

Find the maximum and minimum values of the function ​ ​ subject to the constraint . ​ Find the maximum. ​ Find the minimum.

Find the average value of the function over the plane region R. ​ and ​R is the triangle with vertices (0, 0), (0, 1) and (1, 1).

The Ross-Simons Company has a monthly advertising budget of $20,000. 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. ​

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, $4/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)? ​ ​

Sketch the domain of the function. ​ ​

Let . Compute . ​

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

Use a double integral to find the volume of the solid shown in the figure. Enter your answer as an expression. ​

Find the average value of the given function ​ f(x, y) ​ over the plane region R. Enter your answer as a fraction. ​ ; R is the triangle with vertices (0, 0), (5, 0) and (5, 5).

Find the total differential of the function. ​

If it is true, explain why it is true. If it is false, give an example to show why it is false. ​If and , then f must have a critical point at (a, b).

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. ​ ​

Maximize the function ​ ​ subject to the constraint .