Read the document \"Rates of Change in the Social Sciences\" before answering question 15 -18. 15. A company determines that the cost, in dollars, of producing 1: items is C(x) = 55000 + 23x + 0.0122:2 a) Find the marginal cost function. b) Find the marginal cost at a production level of 100 items. c) Find the cost of producing the 1015t item. 16. A manufacturer determines that the revenue derived from selling x units of one of their products is 1200 = 30003:: 0.023:3 a) Find the marginal revenue function. b) Find the marginal revenue when 300 units are sold. c) Compare this to the actual gain in revenue when the 301\" unit is sold. 17. Sue's Submarines has determined that the monthly demand for their submarines is given by _ 30000 x p _ 10000 And the cost of making it submarines is C(x) = 6000 + 0.8x 3) Find the revenue function. h) Find the marginal revenue function. c) Find the marginal revenue when x = 1000. d) Find the profit function. e) FiNd the marginal profit function. f) Find the marginal profit when x = 10000. 18. A manufacturer of CD players estimates that the cost of making x machines is C(x) = 87000 + 122x And the demand function is given by 600000 x 1000 3] Find the marginal cost function bl Film the revenue function. d) Find the profit function p: c) Find the marginal revenue function e) Find the marginal prot function. RATES OF CHANGE IN THE SOCIAL SCIENCES Although calculus has been applied to the natural sciences for centuries, it has only been recently that the social sciences, such as psychology, calculus. sociology, urban geography, and economics, have been making use of Psychologists interested in learning theory study the so-called learn- ing curve, which graphs the performance level P(t) of someone learning a skill as a function of the training time t. Of particular interest is the rate at which performance improves as time passes, that is, the deriv- ative P' (t). Sociologists use calculus to analyze the spread of rumours (or inno- vations or fads or fashions). If f() is the fraction of the population that knows a rumour by time t, then the derivative f'(t) represents the rate of spread of the rumour. These applications to psychology and sociology will be explored in Chapter 8. In this section, we examine rates of change in business and economics. If it costs a company C(x) to produce x units of a certain commodity, then the function C is called a cost function. If the number of items produced is increased from x, to x2, the additional cost is AC = C(x2) - C(x,) and the average rate of change of the cost is AC C(x2) - C(x) C(x, + Ax) - C(x1) Ax X2 - X1 Ax The limit of this quantity as Ax -> 0, that is, the instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists: AC dc marginal cost = lim Ar-0 Ax dx Since x can usually take on only integer values, it may not make literal sense to let Ax approach 0, but we can always replace C(x) by a smooth approximationg function as we did for growth functions in Section 3.3. Thus, the marginal cost is the derivative of the cost function. To see how to interpret the rate of change in this situation, we recall the definition of a derivative at x = n: C(n + h) - C(n) C'(n) = lim h Taking h = 1 and n large, we see that C'(n) = C(n + 1) - C(n)136 CHAPTER 3 APPLICATIONS OF Therefore, the marginal cost of producing " units is approximate equal to the cost of producing one more unit, the (n + 1)st unit It is often appropriate to represent a cost function by a polynomial (( x) = a+ bx + cx2 + dx where a represents the fixed cost (rent, heat, maintenance) and the other terms represent the cost of raw materials, labour, and so on. (The cox of raw materials may be proportional to x, but labour costs might depend partly on higher powers of x because of overtime costs and inefficiencies involved in large-scale operations.) Example 1 Quinton Mills is a large producer of flour. Management estimates that the cost (in dollars) of producing x 5-kg bags of flour is C(x) = 140 000 + 0.43x + 0.000 001x2 (a) Find the marginal cost at a production level of x = 1000 bags. (b) Find the actual cost of producing the 1001st bag. Solution (a) The marginal cost function is C'(x) = 0.43 + 0.000 002x The marginal cost when x = 1000 is C'(1000) = 0.43 + (0.000 002)(1000) = $0.432/bag (b) The cost of producing the 1001st bag is C(1001) - C(1000) = [140 000 + (0.43)(1001) + (0.000 001)(1001):] -[140 000 + (0.43)(1000) + (0.000 001)(1000)] = $0.432 001 Of course a businessman is interested not only in costs but also in revenue and profit. Let p(x) be the price per unit that a company can charge if it sells x units. Then p is called the demand function (or price function). If x units are sold and the price per unit is p(x), then the total revenue is R(x ) = xp(x) and R is called the revenue function. The derivative R' of the revenue function is called the marginal revenue function and is the rate of change of revenue with respect to the number of units sold. Example 2 Howard's Hamburgers has taken a market survey and has found this the yearly demand for their hamburgers is given by P = 800 000 - X 200 000 (p in dollars)