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Unformatted text preview: AEB 3510 APPLICATIONS OF PARTIAL DERIVATIVES 1. A production function shows the amount of output that can be produced with a given set of inputs by a firm. Suppose that a firm is facing the following production function: Output = f ( x, y ) = 20 x 2 + 3 xy where x represents the units of labor and y the units of capital used in production. ∂f ? ∂x ∂f ? (b) What is the marginal product of capital, ∂y
(a) What is the marginal product of labor, (c) Evaluate both marginal productivities when the amount of labor and capital used is x=250 and y=100 units, respectively. 2. The monthly profit (in dollars) of a Department Store depends on both the level of inventory x (in thousands of dollars) and the floor space y (in thousands of square feet) available for the display of the merchandise and is given by the following function: π ( x, y ) = xy + 3x + 25 y (a) What is the marginal profit associated with an additional unit (thousand dollars) of inventory, ∂π ? ∂x ∂π ? ∂y (b) What is the marginal profit associated with an additional unit (thousand square feet) of floor space, 3. The total weekly revenue (in dollars) of a store that sells TVs and computers is given by the function: TR( x, y ) = x 2 + xy + y 2 where x represents the number of TVs and y the number of computers sold every week. ∂TR ? ∂x ∂TR ? (b) What is the marginal revenue generated from the sale of an additional computer, ∂y
(a) What is the marginal revenue generated from the sale of an additional TV, (c) What is the marginal revenue generated from the sale of an additional TV when the level of sales is x=50 TVs and y= 100 computers? 4. The total weekly cost (in dollars) associated with manufacturing desk lamps and floor lamps is given by: TC ( x, y ) = 100 x + 70 y + 50 where x denotes the number of desk lamps and y denotes the number of floor lamps. (a) Find the marginal cost associated with the manufacturing of an additional desk lamp, ∂TC . ∂x
(b) Find the marginal cost associated with the manufacturing of an additional floor lamp, (c) Evaluate both marginal costs when the production level is x=150 and y=75 units. 5. The utility function U=f(x,y) is a measure of the utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is: ∂TC ∂y U ( x, y ) = xy − 5 x 2 − 3 y 2 (a) Determine the marginal utility of product x, ∂U . ∂x ∂U . (b) Determine the marginal utility of product y, ∂y (c) When x=2 and y=3, should a person consume one more unit of product x or one more unit of product y? Explain your reasoning. ...
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- Spring '11