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# Class Note for MATH 1314 at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 61 views.

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Date Created: 02/06/15
Math 1314 Lesson 9 Marginal Functions in Economics Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level Sometimes the business owner will want to know how much it costs to produce one more unit of this product The cost of producing this additional item is called the marginal cost Example 1 Suppose the total cost in dollars per week by ABC Corporation for producing its bestselling product is given by Cx 10000 3000x 04x2 Find the actual cost of producing the 101st item The cost of producing the 101St item can be found by computing the average rate of C101 C100 chan e that is b com utin g y p g 101 100 C101 C100 Cx h Cx 101 100 xh x Note that where x 100 and h 1 The right hand side of this equation is the formula for average rate of change of the cost function This will give us the actual cost of producing the next item However it is often inconvenient to use For this reason marginal cost is usually approximated by the instantaneous rate of change of the total cost function evaluated at the specific point of interest That is to say we ll find the derivative and substitute in our point of interest Example 2 Suppose the total cost in dollars per week by ABC Corporation for producing its bestselling product is given by Cx 10000 3000x 04x2 Find C 100 and interpret the results Note that the answers for examples 1 and 2 are very close This shows you why we can work with the derivative of the cost function rather than the average rate of change The derivative will be much easier for us to work with So we ll define the marginal cost function as the derivative of the total cost function You will find that by a marginal function we mean the derivative of the function So the marginal cost function is the derivative of the cost function the marginal revenue function is the derivative of the revenue function etc Example 3 A company produces noisecanceling headphones Management of the company has determined that the total daily cost of producing x headsets can be modeled by the function Cx 00001x3 003x2 l35x 15000 Find the marginal cost function Use the marginal cost function to approximate the actual cost of producing the 2151 and 181st headsets Average Cost and Marginal Average Cost Suppose C x is the total cost function for producing x units of a certain product Ifwe divide this function by the number of units produced x we get the average cost function We denote this function by C x Then we can express the average cost function as C x 50c The derivative of the average cost function is called the marginal average cost Example 4 A company produces office furniture Its management estimates that the total annual cost for producing x of its top selling executive desks is given by the function C x 400x 500000 Find the average cost function What is the average cost of producing 3000 desks What happens to 5x when x is very large Marginal Revenue We are often interested in revenue functions as well The basic formula for a revenue function is given by Rx px where x is the number of units sold and p is the price per unit Often p is given in terms of a demand function in terms of x which we can then substitute into Rx The derivative of Rx is called the marginal revenue function Example 5 A company estimates that the demand for its product can be expressed as p 04x 800 where p denotes the unit price and x denotes the quantity demanded Find the revenue function Then nd the marginal revenue function Use the marginal revenue function to approximate the actual revenue realized on the sale of the 4001St item Marginal Pro t The final function of interest is the profit function This function can be expressed as Px Rx C x where Rxis the revenue function and C x is the cost function As before we will find the marginal function by taking the derivative of the function so the marginal pro t function is the derivative of Px This will give us a good approximation of the profit realized on the sale of the x lSt unit of the product Example 6 A company estimates that the cost to produce x of its products is given by the function Cx 0000003x3 008x2 500x 250000 and the demand function is given by p 600 08x Find the profit function Then find the marginal profit function Use the marginal profit function to compute the actual profit realized on the sale of the 51St unit Example 7 The weekly demand for a certain brand of DVD player is given by p 02x 300 0 S x S 15000 where p gives the wholesale unit price in dollars and x denotes the quantity demanded The weekly cost function associated with producing the DVD players is given by Cx 0000003x3 004x2 200x 70000 Compute C 3000 R 3000 and P 3000 Interpret your results From this lesson you should be able to Explain what a marginal cost revenue pro t function is and what it is used for Find a marginal cost function and use it to approximate the cost of producing the x lst item Find an average cost function Find lim C x and explain what it means Hm Find a revenue function Find a marginal revenue function and use it to approximate the revenue realized on the sale ofthe x lst item Find a profit function Find a marginal profit function and use it to approximate the profit realized on the sale ofthe x lst item

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