Exam 8: Functions of Several Variables

The cost, C, of renting a car is $45 per day and $0.20 per mile. Write a cost function, , for the total cost of renting a car where d is the number of days and m is the number of miles.

A mop company can produce mops using x units of capital and y units of labor, with production costs dollars. With a budget of $500,000, the maximum production is 65,000, using $350,000 in capital and $150,000 in labor. The Lagrange multiplier is . What is the practical meaning of the statement ?

Find a and b so that has a critical point at (1, 6).

The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and fuel economy in miles per gallon, m, of the car according to the formula . What is the cost when p = $2.00 and m = 20?

Given the following table of values for , estimate f_y (10,0.2). Use the next higher point to make your estimate.

The following table gives values for a function . For and , there is a global minimum of at most _____ near the point (_____,_____).

Find if .

Use Lagrange multipliers to find the maximum or minimum values of , subject to the constraint . = _____ and the maximum/minimum value of is _____.

For a production function , the maximum production cost of $400 is given by units, with . Estimate the production if the budget cost is raised to $410.

A person's weight, w, is a function of the number of calories consumed, c, and the number of calories burned, b. Thus, . Would you expect f_c (c,b) to be positive or negative?

If , .

For a function , we are given , , and . Estimate .

The consumption of beef, C (in pounds per week per household) is given by the function C = f(I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect to be positive or negative?

The number of tons, N, of a product produced at a factory in a day is a function of the number of workers, w, and the number of hours worked, h. We have . Which of the following equations tells us that when the factory has 9 workers who work for 12 hours, adding one additional hour will cause the amount produced to go up by about 44 tons?

The following table gives the number f(x, y) of grape vines, in thousands, of age x in year y. In one year a fungal disease killed most of the older grapevines, and in the following year a long freeze killed most of the young vines. Which are these years?

The owner of a jewelry store has to decide how to allocate a budget of $420,000. He notices that the earnings of the company depend on investment in inventory x₁ (in thousands of dollars) and expenditure x₂ on advertising (in thousands of dollars) according to the function How should the owner allocate the $420,000 between inventory and advertising to maximize his earnings?

If , find f_x(1, 3).

The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Heat exhaustion is likely to occur when the heat index is 105 or higher. If the temperature is about 95 , at about what humidity level does heat exhaustion become likely?

If , find r t.

If $P is invested in a bank account earning r% interest a year, compounded continuously, the balance, $B, at the end of t years is given by Find