Exam 8: Calculus of Several Variables

Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than inches.Find the diamension of a cylendrical package of greatest volume that can be sent throught the mail. What is the volume of such a package? Hint: The length plus the grith is .

A level curves of a function of two variables, ,

Maximize the function subject to the constraint .

Evaluate the double integral for the function and the region . and is bounded by , , and .

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.

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. __________

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.

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 also gives rise to the unconstrained relative extremum of .

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)? x = __________ y = __________ z = __________ C = __________

Maximize the function subject to the constraint .

Find the first partial derivatives of the function.

The IQ (itelligence quotient) of a person whose mental age is years and whose chronological age is years is defined as What is the IQ of a 10-years-old child who has a mental age of 19 years?

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

Let f(x, y) = 2x + 3y - 4. Compute the following. f(1, 0), f(0, 1), f(1, 2)

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

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

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

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. Please round the coefficients in your equation to three decimal places.

Postal regulations specify that the combined length and girth of a parcel sent by parcel post may not exceed 108 in. Find the dimensions of the rectangular package that would have the greatest possible volume under these regulations. (Hint: Let the dimensions of the box be by by (see the figure below). Then, , and the volume . So that . Maximize .)