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# Math M119 Ch 1 Sec 4 Notes M119

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This 7 page Bundle was uploaded by Meegan Voss on Monday January 25, 2016. The Bundle belongs to M119 at Indiana University taught by Gregory Kattner in Winter 2016. Since its upload, it has received 53 views. For similar materials see Brief Survey of Calculus in Mathematics (M) at Indiana University.

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Date Created: 01/25/16

Chapter 1 Section 4 Some applications of functions to economics q the quantity of some item to be manufactured or provided 19 the price of one item clq the total cost to manufacture or to provide q items qu the revenue which results from selling q items T q the pro t which results from selling q items The total cost to manufacture or to provide q items has two components i xed cost ii variable cost The total cost is the sum of the xed and the variable costs Rqpq quotRevenue is Price times Quantityquot Ti lqlqulClql quotPro t is Revenue minus Costquot In these rst simple economic models we will assume that cost revenue and pro t are linear functions of the quantity q Example Jessica plans to start a small business by offering her services as a math tutor Her xed costs for printing business cards starting a web site and advertising will be 4000 Thereafter she will incur variable costs for transportation and facilities that amount to 5 per hour of tutoring Write an equation to represent the total costs forJessica to run her business as a function of the number of hours spent tutoring q the total number of hours spent tutoring Cq the total cost in dollars to provide q hours of tutoring The requested equation is Cq405q Note that this is an increasing function since the line has a positive slope Marginal Cost MC is de ned as the change in cost that would result by producing one more item In most cases MC is a quantity that changes depending on the value of q However when the cost function is linear as in the previous example the marginal cost works out to be the slope of the line and is therefore constant with respect to q Marginal Revenue MR and Marginal Pro t MP are de ned similarly Example Suppose thatJessica will charge 10 per hour for her tutoring services Write an equation to represent the total revenue as a function of the number of hours spent tutoring q the total number of hours spent tutoring p the price charged for 1 hour of tutoring p10 dollars per hour Rq the revenue as a number ofdollars which results from q hours of tutoring The requested equation is Riqipq10q Note that in this example the price charged per hour of tutoring is constant with respect to q No matter how many hours of tutoring are involved the cost per hour is always 10 However in many of the problems we ll consider in the future we ll nd that price and quantity uctuate depending on each other s values When the cost and the revenue are equal then an important breakeven point has been reached The value of q at this point represents an important quantity If q is increased in value above the breakeven value then a pro t will result In general a pro t results at any production level q at which the revenue is greater than the cost We can nd the breakeven point by drawing a graph of the cost and the revenue function in the same picture and then looking to see where the two graphs intersect each other Another way to nd the breakeven point is to set the cost function equal to the revenue function and then let the equation tell us the value of q at the breakeven point Examples Work through questions 9 and 10 on page 35 of the text book As another application for the mathematical idea of a function comes from the analysis of the producers supply ofquot and quotconsumers demand forquot a given item Each of these describes a relationship between the price p and the quantity q of the item which can be modeled by a function In our early examples we will assume that the functions involved are linear functions The Supply Curve is a function that shows a relationship between the price p that the item of interest can sell for and the quantity q of that item that the producer are consequently willing to make It would be expected that when the price increases then producers are willing to make still more of the item in an effort to make as much money as possible The supply curve will always be an increasing function irrespective if whether we decide to express price as a function of quantity or if we instead decide to write quantity as a function of price The Demand Curve is a function that shows a relationship between the price p that the item of interest can sell for and the quantity q of that item that consumers are willing to purchase It would be expected that as the price increases then consumers will be willing to purchase less and less of the item in an effort to spend as little money as possible The demand curve will always be a decreasing function irrespective if whether we decide to express price as a function of quantity or if we instead decide to write quantity as a function of price The price and quantity at the point where the graphs of the supply and the demand curve intersect are of special interest This point of intersection is called the equilibrium point The coordinates are called the equilibrium quantity q and the equilibrium price p In a simple economic model we suppose that normal market forces will always work to drive the price and quantity to the equilibrium values In many of our problems involving supply and demand curves when a graph is drawn we traditionally let the vertical axis represent the price p while the horizontal axis represents the quantity q Example Suppose that a supply curve is given by the equation qSp2p30 and a demand curve is given by the equation qDp 3p270 Find the equilibrium point Interpret the intercepts for each of these curves Example Suppose that a speci c tax of 2 per item is levied on the producers of this product The producers now realize 2 less in revenue from each sale and that will have an impact on the quantity they are driven to produce Find the new equilibrium point resulting from this speci c tax Example Work question 38 in the text book on page 38 The last topics that provide an application for our idea of function are in the form of Budget Constraints and Depreciation Functions Example Work question 20 in the text book on page 36 about depreciation and question 32 on page 38 concerning a budget constraint

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