dy = f 0 (x)dx Let s look at an example:

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1 Exam # Material We previously looked at using di erentials to estimate the change in a function s output. This is done a lot in business and economics to project or estimate changes in cost, revenue, and profit. Remember that the formula for the di erential of y = f(x) is: Let s look at an example: dy = f 0 (x)dx Example. The Sunglass Hut Company has a profit function given by P (x) = 0.0x +x where x is the number of thousands of pairs of sunglasses produced and sold, and P (x) isthe total profit, in thousands of dollars, from producing and selling x thousand pairs of sunglasses. The company currently sells and produces,000 pairs of sunglasses, and they are thinking about increasing their production and sales to 0,000. What would be the projected change in profit from this increase? Now, let s look at the same example and just analyze what would happen if we increased the number of sales/production by one unit (in the context of this problem, one unit would be producing and selling one thousand more pairs of sunglasses). Is the computation of the di erential any easier? Example. The Sunglass Hut Company has a profit function given by P (x) = 0.0x +x where x is the number of thousands of pairs of sunglasses produced and sold, and P (x) isthe total profit, in thousands of dollars, from producing and selling x thousand pairs of sunglasses. The company currently sells and produces,000 pairs of sunglasses, and they are thinking about increasing their production and sales to,000. What would be the projected change in profit from this increase? Marginal analysis is an examination of the additional benefits, like revenue or profit, associated with an activity incurred by the addition of unit of production/cost/etc. Here is an example to help us think about marginal analysis:

2 If you have decided to go out on a weekend, how many drinks do you buy? This is a decision where we actually use marginal analysis without thinking about it! Marginal analysis is the process of breaking down a decision into a series of yes or no decisions, where each decision only involves adding one more unit or the activity or product. To make a decision using marginal analysis, we need to know the willingness to pay for each level of the activity. As mentioned, this is also known as the marginal benefit from an action. To decide how many drinks to buy, you have to make a series of yes or no decisions on whether to buy an additional drink or not. In the table below, you can see that the marginal benefit is diminishing. This means that you are willing to pay more for the st drink than the next. Your friends are all drinking, so you are likely willing to pay quite a lot for your st drink. By the th, you may feel as though you do not need another. So how many drinks will you buy if the cost is $? To make this decision, we must use marginal analysis for each level. This means comparing our marginal benefit with marginal cost of an additional unit of activity. In this case marginal cost is just equal to $. Marginal Benefit First Drink $0 Second Drink $ Third Drink $ Fourth Drink $ Thinking about this in terms of marginal analysis: For the first drink, the marginal benefit is $0 and the marginal cost is $ (so, buy the drink!) For the second drink, the marginal benefit is $ and the marginal cost is $ (so, buy the drink!) For the third drink, the marginal benefit is $ and the marginal cost is $ (so, you should not buy the drink!) For the fourth drink, the marginal benefit is $ and the marginal cost is $ (so, you should definitely not buy the drink!) Okay, so we can see that consuming a third drink would be an awful idea... but let s look at the net benefit of consuming just the two drinks: Total Benefit: Total Cost: Net Benefit: What would the net benefit be for buying and consuming three drinks? What would the net benefit be for buying and consuming three drinks?

3 If f(x) isacost(orrevenueorprofit)functionthenthemarginalcost(orrevenue or profit) is the acquired cost (or revenue or profit) from producing one more unit of product or service, and it is given by: dy = f 0 (x) Marginal cost (or revenue or profit) is the instantaneous rate of change of cost (or revenue or profit) relative to production at a given production level. Example. Acompanymanufacturesfueltanksforcars. Thetotalweeklycost(indollars)of producing x tanks is given by () Find the marginal cost function. C(x) = x 0.0x. () Find the marginal cost at a production level of 00 tanks per week. () Write an explanation about what the answer in the previous part means. () Calculate the exact cost of producing the 0 st item.

4 8 Example. The total cost of producing x bicycles is given by the function C(x) =0000+0x 0.x. () Find the exact cost of producing the st bicycle. () Use marginal analysis to approximate the cost of producing the st bicycle. Example. The total profit (in dollars) from the sale of x sweatshirts is P (x) =x 0.00x 0 0 apple x apple 000. () Find the exact profit from producing the 00 th sweatshirt. () Use marginal analysis to approximate the profit from producing the 00 th sweatshirt.

5 So, with marginal analysis, why do we use the number that is one less than the number we want to estimate? 9 8 y = f(x) f 0 () y Example. Let C(x) =0000+0x 0.x be the total cost (in dollars) of producing x bicycles. () Find the total cost of producing 00 bicycles. () Find the exact cost of producing the 00th bicycle. () Use marginal analysis to estimate the cost of producing the 00th bicycle. () Find the average cost per bicycle of producing 00 bicycles.

6 0 Example. Acompany smarketresearchdepartmentrecommendsthemanufactureandmarketing of a new headphone. After suitable test marketing, the research department presents the following price-demand equation: x = p. In the price-demand equation, the demand x is given as a function of the price, p. Additionally, the financial department provided the cost function C(x) =000+x where $,000 is the estimate of fixed costs (tools and overhead) and $ is the estimate of variable costs per headphone (materials, labor, marketing, transportation, storage, etc.). () Find the marginal cost function, and write a sentence about what it means. () Find the revenue function as a function of x. () Find the marginal revenue when x =000,andwriteasentenceaboutwhatitmeans. () Graph the cost function and the revenue function on the same coordinate axes on your calculator. Find the intersection points of these two graphs and interpret the results. () Find the profit function. () Find the marginal profit when x =000,andwriteasentenceaboutwhatitmeans.

7 Example. The price-demand equation and the cost function for the production of table saws are given, respectively by x =000 0p and C(x) =000+0x where x is the number of saws that can be sold at a price of $p per saw and C(x) isthetotal cost (in dollars) of producing x saws. () Find the marginal cost, and write a sentence about what it means. () Find the revenue function. () Find the marginal revenue function. () Find R 0 (00), and write a sentence about what it means. () Find the break-even points. () Find the profit function. () Find P 0 (00), and write a sentence about what it means.

8 Example. A small machine shop manufactures drill bits used in the petroleum industry. The manager estimates that the total daily cost (in dollars) of producing x bits is () Find the average cost function, C(x). C(x) =000+x 0.x () Determine C(0), and write a sentence about what it means. () Find the marginal average cost function, C 0 (x). () Determine C 0 (0), and write a sentence about what it means.

9 Example. For some practice on finding functions used in these business and economics applications, consider the following cost and revenue functions: C(x) =0+.x and R(x) =x 0.0x. () Find the profit function. () Find the average cost function. () Find the average revenue function. () Find the average profit function. () Find the marginal profit function. () Find the marginal average cost function. () Find the marginal average revenue function. (8) Find the marginal average profit function.

10 Example. The total cost (in dollars) of printing x board games is C(x) =0, x () Find the average cost per unit if,000 board games are produced. () Find the marginal average cost at a production level of,000 units, and write a sentence about what it means. () Use the results from the previous two parts to estimate the average cost per board game if,00 board games are produced.

11 Chapter : Elasticity of Demand (pages 8-) Next, we want to look at another application of derivatives to business and economics known as elasticity of demand. This concept can be used to answer the question When does an increase in price lead to an increase in revenue? Let s take a look at the formula and how this works. Definition. If the price and demand are related by x = f(p), then the elasticity of demand is given by E(p) = relative rate of change of demand relative rate of change of price = pf 0 (p) f(p) Example. The price p and the demand x for a product are related by the price-demand equation x +00p =0, 000. () Find the elasticity of demand function. () Find E(), and write a sentence about what it means. () Find E(), and write a sentence about what it means. () Find E(0), and write a sentence about what it means.

12 The previous example illustrates the three di erent cases: E(p) Demand Interpretation Revenue 0 <E(p) < Inelastic Demand is not sensitive to changes in price; that is, percentage change in price produces a smaller percentage change in demand. E(p) > Elastic Demand is sensitive to changes in price; that is, a percentage change in price produces a larger percentage change in demand. E(p) = Unit Apercentagechangeinprice produces the same percentage change in demand. Apriceincreasewill increase revenue Apriceincreasewill decrease revenue. Example. A manufacturer of sunglasses currently sells one type for $ a pair. The price p and the demand x for these glasses are related by x =9, 00 0p. If the current price is increased, will revenue increase or decrease? What if the current price for sunglasses is $ a pair?

13 Chapter : Graphing and Optimization (pages -) Recall the following definitions from Math 0 (College Algebra): Definition. () Afunctionf is called increasing on an open interval I, ifforalla, b I, a<bimplies f(a) <f(b). (Rises as you go to the right) () Afunctionf is called decreasing on an open interval I, ifforalla, b I, a<bimplies f(a) >f(b). (Falls as you go to the right) () Afunctionf is called constant on I, iff(a) =f(b) foralla, b I. Example. Consider the function f(x) =x x: Where is f(x) increasing?decreasing?constant? Calculate f 0 (x) anddoasignchartforf 0 (x). Theorem. Consider the function f(x). For the interval (a, b), if f 0 (x) > 0, then f(x) isincreasing.... f 0 (x) < 0, then f(x) isdecreasing.

14 8 Example. Consider the function f(x) =x x +0. () Which values of x correspond to horizontal tangent lines? () For which values of x is f(x) increasing?decreasing? () Sketch a graph of y = f(x) Definition. Arealnumberx in the domain of a function f(x) suchthatf 0 (x) =0orf 0 (x) does not exist is called a critical number of f(x).

15 9 Example. Complete problems 9- on page -. Example. Consider the function f(x) = x. () Find the critical numbers of f(x). () Find the intervals on which f(x) isincreasing?decreasing?

16 0 Example. Consider the function f(x) = x 9x +x +0. () Find the critical numbers of f(x). () Find the intervals on which f(x) isincreasing?decreasing? Example. Consider the function f(x) = x +8x. x () Find the critical numbers of f(x). () Find the intervals on which f(x) isincreasing?decreasing?

17 0 9 8 f 0 () = < f 0 () = > 0 Example. Determine the sign chart for f 0 (x) giventhegraphofy = f(x) below

18 Example. Determine the intervals where the graph of f(x) isincreasinganddecreasingfor f(x) = x + x + 0 x + Example. Determine the intervals where the graph of f(x) isincreasinganddecreasingfor f(x) = x +x +

19 Definition. Consider the graph of a function y = f(x). Let c be a number in the domain of f(x). () We call f(c) alocal maximum if there exists an interval (m, n) containingc such that f(x) apple f(c) for all x in (m, n). () We call f(c) alocal minimum if there exists an interval (m, n) containingc such that for all x in (m, n). f(x) f(c) Theorem. If f(c) isalocalextremumofthefunctionf, thenc is a critical number of f(x). First-Derivative Test for Local Extrema: Letc be a critical number of f (i.e. f(c)isdefined and either f 0 (c) =0orf 0 (c) doesnotexist). Constructasignchartforf 0 (x) closetoandon either side of c, then:

20 Example. Find the local maxima and minima for f(x) =x information to sketch a graph of y = f(x) x +9x +. Then use this Example. Find the local maxima and minima for f(x) =x 9x +x 0. Then use this information to sketch a graph of y = f(x)

21 Example. An o ce supply company sells x permanent markers per year at $p per marker. The price-demand equation for these markers is p = x. What price should the company charge for the markers to maximize revenue? What is the maximum revenue? Example. The total annual cost of manufacturing x permanent markers for the o company from the previous example is given by: C(x) =000+x ce supply What is the company s maximum profit? How many markers should be produced? What should the company charge for each marker?

22 Example. Given the graph of f 0 (x), determine the intervals of increase and decrease for f(x) as well as the location of any local extrema. Graph of y = f 0 (x) Graph of y = f 0 (x) Interval of Increase for f(x): Interval of Decrease for f(x): Local Minimum for f(x) at: Interval of Increase for f(x): Interval of Decrease for f(x): Local Minimum forf(x) at: Local Maximum for f(x) at: Local Maximum for f(x) at: Example. Find the intervals of increase and decrease, along with any local extrema for f(x) =x 8x + Interval of Increase for f(x): Interval of Decrease for f(x): Local Minimum for f(x): Local Maximum for f(x):

23 Example. A company manufactures and sells x television sets per month. The monthly cost and price-demand equations are (respectively): C(x) =, x x p = 00 for0 apple x apple, Find the maximum revenue. The weekly price- Example. Acompanymanufacturesandsellsx smartphones per week. demand and cost equations are (respectively): p =00 0.x and C(x) =0, x What is the maximum weekly profit? How much should the company charge for the smartphones to reach this maximum weekly profit? How many smartphones should be produced to reach this maximum weekly profit?