ACOW RATES OF CHANGE AND DERIVATIVES MODULE Updated 5/28/2016 Page 1 of 2 Activity D1 – Marginal Analysis Suppose the Cotton Candy company has a cost function, in dollars, given by 0.4 x 2 6 , where x is the number of items produced. If the company currently C ( x) x 10 produces 4 items, how much more would it cost the company to produce one more (i.e. the 5th) item? To find this answer we need to calculate the following C(5) – C(4) = 1.3076 – 1.0333 = 0.2743 Thus, it will cost the company $0.27 more to produce the 5th item. In economics the word “marginal” refers to an instantaneous rate of change. Thus, the Cotton Candy’s marginal cost function is given by 1 x 10 (0.8 x) 0.4 x 2 6 x 1/ 2 2 C '( x) x 10 2 and it represents the rate of change of cost, in dollars per item. We can use the marginal cost function C’(x) to approximate how the cost changes in response to one unit change in production. Thus, to approximate how much more it would cost to produce one more item if the company’s current production level 4 units, we need to evaluate C '(4) as shown below. C '(4) 12 (3.2) 12.4 0.25 2 12 35.3 144 0.2451 Thus, at a production level of 4 units is will cost the company approximately $0.25 to produce the next (5th) item. Comparing this to the actual cost of producing the 5th item from above, we see a two cent difference. Because the difference is small, company’s tend to use marginal cost, marginal revenue, and marginal profit functions. A summary of marginal analysis(RS1) is given at right. Solve the following. 1. a) A company’s cost function, in dollars, is given by 2 x3 C ( x) 2 0.2 x 2 x 1 where x is the number of items produced. Find (to the nearest cent) the approximate cost of producing the 40th item. b) Find the company’s actual cost to produce the 40th item. ACOW RATES OF CHANGE AND DERIVATIVES MODULE Updated 5/28/2016 Page 2 of 2 2. A company’s cost function, in dollars, is given by 1 C ( x) x 2 x and its revenue function, in dollars, is given by R( x) 5 x3 ( x 4 3x 1) . Find the company’s marginal profit function. 3. If a company’s revenue function, in hundreds of dollars, is given by 10 x5 R( x) 2 x 2 where x is the number of thousands of items produced find R '(4) . 4. Using the revenue equation from question #3, which of the following best interprets R '(4) ? a) The company’s revenue will be $45,827 when they sell 4000 items. b) At a production level of 4000 items, the company will obtain approximately $45,827 in revenue when the next 1000 items are produced. c) At a production level of 4 items, the company will obtain approximately $458.27 in revenue when the 5th item is produced. d) At a production level of 4000 items, the company will obtain approximately $458.27 in revenue when the next 1000 items are produced.