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A project description for a mini-project in integral calculus, focusing on finding total cost functions using marginal cost and deriving profit functions from marginal revenue. The project demonstrates the optimization of profit given marginal cost and marginal revenue functions. An example with given marginal cost and revenue functions, and the process of finding the constants of integration and the profit function.
Typology: Exams
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4 /22/
Gina Rablau
Gina Rablau
1
This project demonstrates the following concepts in integral calculus:
Indefinite integrals.
Project Description
Use integration to find total cost functions from information involving marginal cost (that is, the rate of change of cost) for a commodity. Use integration to derive profit functions from the marginal revenue functions. Optimize profit, given information regarding marginal cost and marginal revenue functions.
The marginal cost for a commodity is M C = C′(x), where C(x) is the total cost function. Thus if we have the marginal cost function, we can integrate to find the total cost. That is, C(x) =ᔖ ᠹᠩ ᡖᡶ.
The marginal revenue for a commodity is M R = R′(x), where R(x) is the total revenue function.
If, for example, the marginal cost is MC = 1.01(x + 190)0.01^ and
M R = ( 1 / 2 x + 1 ) + 2 , where x is the number of thousands of units and both revenue and
cost are in thousands of dollars. Suppose further that fixed costs are $100,236 and that production is limited to at most 180 thousand units.
C ( x )= (^) ∫ MCdx =∫1.01(x+190)0.01^ dx = ( x+ 190 )^1.^01 + K