Exam 8: Calculus of Several Variables

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

A formula used by meteorologists to calculate the wind chill temperature (the temperature that you feel in still air that is the same as the actual temperature when the presence of wind is taken into consideration) is where t is the actual air temperature in and S is the wind speed in mph. What is the wind chill temperature when the actual air temperature is and the wind speed is 25 mph? Please round the answer to the nearest __________ If the temperature is , by how much approximately will the wind chill temperature change if the wind speed increases from 25 mph to 26 mph? Please round the answer to the nearest tenth. __________

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.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.

John Mills- the propietor of Mills Engine Company, a manufacturer of mdel airplane engine-finds that it takes units of labor and units of capital to produce units of the product.If a unit of labor costs , a unit of capital costs and is bugeted for production, determine how long units should be expended on on labour and how many units should be expended on capital in order to maximize production.

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)?

Find the first partial derivatives of the function.

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, $7/square foot for the front and back, and $6/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)?

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 Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by the profit function where stands for the number of finished units and stands for the number of unfinished units manufactured and sold each week.Find the average weekly profit if the number of finished units manufactured and sold varies between and and the number of unfinished units varies between and per week.Please round your answer to the nearest dollar, if necessary.

Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 2x + 2y; R is bounded by x = 0, , y = 0 and y = 4.

The following data, compiled by the admissions office at Faber College during the past 5 years, relate the number of college brochures and follow-up letters (x) sent to a preselected list of high school juniors who had taken the PSAT and the number of completed applications (y) received from these students (both measured in units of 1,000). Determine the equation of the least-squares line for these data.Use the result obtained to predict the number of completed applications that might be expected if 6,400 brochures and follow-up letters are sent out during the next year.

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

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.

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.

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 .

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?