Exam 8: Calculus of Several Variables

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 °F and S is the wind speed in mph. ​ What is the wind chill temperature when the actual air temperature is 40 °F and the wind speed is 15 mph (T​₁)? Please round the answer to the nearest °F ​ If the temperature is 40 °F, by how much approximately will the wind chill temperature change if the wind speed increases from 15 mph to 16 mph (T​₂)? Please round the answer to the nearest tenth. ​

Find the equation of the least-squares line for the given data. Draw a scatter diagram for the given data and graph the least-squares line. x 1 3 5 7 9 Y 9 7 6 3 2 ​ ​

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

The Country Workshop's total weekly profit (in dollars) realized in manufacturing and selling its rolltop desks is given by ​ ​ where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold per week. Currently, the weekly output is 145 finished and 130 unfinished units. Determine the approximate change in the total weekly profit if the sole proprietor of the Country Workshop decides to increase the number of finished units to 155 per week and decrease the number of unfinished units to 125 per week. ​ $__________ per week

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 $15/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. ​ ​

Sketch the level curves of the function corresponding to each value of z. ​ ; ​

The total daily revenue (in dollars) that Weston Publishing realizes in publishing and selling its English-language dictionaries is given by ​ ​ Where x denotes the number of deluxe copies and y denotes the number of standard copies published and sold daily. The total daily cost of publishing these dictionaries is given by ​ ​ Dollars. Determine how many deluxe copies and how many standard copies Weston should publish each day to maximize its profits. Please round the answers to the nearest integer.What is the maximum profit realizable? Please round the answer to the nearest dollar. ​

Find the second-order partial derivatives of the given function. Show that the mixed partial derivatives f_xy and f_yx are equal. ​ ​

Find the first partial derivatives of the function. ​ ​

Evaluate the double integral ​ ​ for the given function f(x, y) and the region R. Enter your answer as an expression. ​ ; R is bounded by and y = x.

​Find the approximate change in z when the point changes from to . ​ ; from to ​​ __________

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 total weekly profit (in dollars) realized by Country Workshop in manufacturing and selling its rolltop desks is given by the profit function ​ ​ Where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The company's management decides to restrict the manufacture of these desks to a total of exactly 100 units/week. How many finished and how many unfinished units should be manufactured each week to maximize the company's weekly profit? ​

Find the total differential of the function. ​ ​

Maximize the function ​ ​ Subject to the constraint . ​

Evaluate the double integral ​ ​ For the function and the region R. ​ and R is bounded by , , and . ​

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 double integral ​ for the given function f(x, y) and the region R. Enter your answer as a fraction. ; R is the rectangle defined by and .

Find the average value of the given function over the plane region R. ​ ; R is the triangle with vertices (0, 0), (2, 0) and (2, 2). ​

The total weekly revenue (in dollars) of Country Workshop associated with manufacturing and selling their rolltop desks is given by the function ​ ​ Where x denotes the number of finished units and y denotes the number of unfinished units manufactured and sold each week. Compute and when x = 200 and y = 180. Interpret your results. ​