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MATH 150 Section 2.7 Marginal Analysis in Business and Economics Definition Marginal Cost, Revenue, and Profit If π₯ is the number of units of a product produced in some time interval, then total cost = πΆ(π₯) marginal cost = πΆβ(π₯) total revenue = π (π₯) marginal revenue = π β(π₯) total profit = π(π₯) = π (π₯) β πΆ(π₯) marginal profit = πβ²(π₯) = π β²(π₯) β πΆβ²(π₯) (marginal revenue β marginal cost) Example A company manufactures fuel tanks for cars. The total weekly cost (in dollars) of producing π₯ tanks is given by πΆ(π₯) = 10,000 + 90π₯ β 0.05π₯ 2 (a) Find the marginal cost function. (b) Find the marginal cost at a production level of 500 tanks per week. (c) Interpret the results of part (b) (d) Find the exact cost of producing the 501st item. Theorem If πΆ(π₯) is the total cost of producing π₯ items, then the marginal cost function approximates the exact cost of producing the (π₯ + 1)st item: πΆ β² (π₯) β πΆ(π₯ + 1) β πΆ(π₯) Similar statements can be made for total revenue functions and total profit functions. 1 Example The price-demand equation and the cost function for the production of HDTVs are given, respectively, by π₯ = 9,000 β 30π πππ πΆ(π₯) = 150,000 + 30π₯ where x is the number of HDTVs that can be sold at a price of $p per TV and C(x)is the total cost (in dollars) of producing π₯ TVs. (a) Express the price π as a function of the demand π₯, and find the domain of this function. (b) Find the marginal cost. (c) Find the revenue function and state its domain. (d) Find the marginal revenue. (e) Find π β(3000) and π β(6,000) and interpret these quantities. (f) Graph the cost function and the revenue function on the same coordinate system for 0 β€ π₯ β€ 9000. Find the break-even points and indicate regions of loss and profit. (g) Find the profit function in terms of π₯. (h) Find the marginal profit. (i) Find πβ(1500) and πβ(4500) and interpret these quantities. 2