Exam 8: Functions of Several Variables

The amount, , of your paycheck is a function of your hourly wage, w, and the number of hours, h, you work. Would you expect f_h(w,h) to be positive or negative?

A television salesman earns a fixed salary of $9.00 per hour plus a $25 commission for each television he sells. If h is the number of hours worked and s is the number of televisions sold, find a formula for , the salesman's total earnings, and use it to calculate .

The table below gives values of a function f(x, y) near x = 1, y = 2. Estimate .

A company has two manufacturing plants which manufacture the same item. Suppose the cost function is given by where q₁ and q₂ are the quantities (measured in thousands) produced in each plant. The total demand q₁ + q₂ is related to the price, p, by How much should each plant produce in order to maximize the company's profit?

Let Which figure best represents the contours of this function?

Use the values of in the following table to estimate f_yx (3,2). Use the next higher point to make your estimate.

The contour diagram of f is shown below. Which of the points A, B, C, D, and E appear to be critical points? Select all that apply.

A manufacturer sells two products. The first sells for $7 and the second sells for $5. The manufacturer's costs are given by the equation , where and are the respective quantities produced of each product. Let represent the profit at a production level of and units. Find the value of .

A company sells two products. The fixed costs for the company are $3300, the variable costs for the first product are $8 per unit, and the variable costs for the second product are $10 per unit.. If the company produces units of the first product and units of the second product, and the total cost is given by , is C an increasing or decreasing function of ?

For , calculate s² .

A company's cost function to produce x units of one product and y units of a second product is given by . The minimum cost occurs when x = _____ and y = _____.

Suppose the Cobb-Douglas production function for a company is given by , where P is production in tons, L is the number of workers, and K is the capital investment, in thousands of dollars. Find L if the company has a capital investment of 25 thousand dollars and the company employs 15 workers. Round to the nearest tenth.

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. Is the heat index an increasing or decreasing function of temperature?

Given the following contour diagram of , estimate f_x (1,2). Use the next higher point to make your estimate.

Suppose that the price P (in dollars) to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years). So P = f(C,A). What is the sign of

The function has a critical point which is a local minimum when x = _____ and y = _____.

The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, . What is the revenue when ticket sales are 100 and food sales are 200?

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

The production function for a company is , where P is the amount produced given x units of labor and y units of equipment. Each unit of labor costs $1200 and each unit of equipment costs $350. Assuming the goal of the company is to maximize production given a fixed budget of $45,000, what are the objective and constraint functions?

The function has a local __________ (maximum / minimum / neither) at the critical point, where x = _____ and y = _____.