微積分SECTION2.5MarginalAnalysis

1、1. Marginal Analysis 邊際分析邊際分析 SECTION 2.5 Marginal Analysis and Approximations Using IncrementsCalculus is an important tool in economics. In this section, we will use the derivative to explore rates of change involving economic quantities (量) such as cost , revenue, profit etc.In economics, the use

2、 of the derivative to approximate the change in a quantity that results from a 1-unit increase in production is called marginal analysis.Margin: n.C The empty space at the side of a printed page 頁(yè)邊空白 Marginal: adj. relating to a change in cost , value etc. when one more thing is produced 邊際的C(x)-the

3、 total cost of producing x units of a commodityC(x0)-the derivative (導(dǎo)數(shù)) of C(x) when x = x0 C(x0)的解釋：的解釋：0000()()1()limhC xhC xCxh（）平均變化率 0 xx當(dāng)時(shí)的成本的瞬時(shí)變化率0000()()()limhC xhC xC xh（2）000(1)()1CxxC xh當(dāng) 較大時(shí)（,取 =1）C(x0) is called the marginal cost of producing x0 units. (生產(chǎn) x0件的邊際邊際成本成本)00(1)()C xC x0-+

4、x生產(chǎn)第1件的實(shí)際成本NOTE 生產(chǎn)第x0+1件的實(shí)際成本生產(chǎn)x0件的邊際成本，即C(x0)-生產(chǎn)x0件的成本C(x0)-生產(chǎn)x0件的邊際邊際成本/生產(chǎn)第x0+1件的估計(jì)估計(jì)成本C(x0+1)-C(x0)-生產(chǎn)第x0+1件的實(shí)際成本000(1)()()C xC xC x區(qū)別區(qū)別-邊際分析近似公式邊際分析近似公式現(xiàn)有生產(chǎn)規(guī)模現(xiàn)有生產(chǎn)規(guī)模000(1)()()R xR xRxMarginal Revenue入邊際收 000(1)()()MarginalP xP xPxProfit潤(rùn)邊際利 邊際分析邊際分析就是利用邊際成本、收入和利潤(rùn)估計(jì)下一個(gè)產(chǎn)品的實(shí)際成本、收入和利潤(rùn)。Similarly,Examp

5、le 2.5.1 (P162) Studying Marginal Cost and Marginal Revenue A manufacturer estimates that when x units of a particular commodity are produced, the total cost will be 21( )3988C xxxdollars per unit.a. Find the marginal cost and the marginal revenue.b. Use marginal cost to estimate the cost of produci

6、ng the 37th unit. What is the actual cost of producing the 37th unit?c. Use marginal revenue to estimate the revenue derived from the sale of the 37th unit. What is the actual revenue derived from the sale of the 37th unit?dollars, and furthermore, that all x units will be sold when the price is 1(

7、)(75)3p xxExample 2.5.2 (P163) Using Marginal Analysis to Make a Business Decision A manager for a digital camera company estimates that when x hundred cameras are produced, the total profit will be P(x) = -0.0035x3 + 0.07x2 + 25x - 200thousand dollars.a. What is the marginal profit function?c. What

8、 decision should the manager make if the current level of production is x = 50 (5000 cameras)? What if x = 80 (8000 cameras are being produced)? b. The current level of production is x = 10 (1000 cameras). Based on the marginal profit at this level of production, should the manager recommend increas

9、ing or decreasing production to increase profit?2. Approximation by Increments 增量表示的近似增量表示的近似 If f (x) is differentiable (可導(dǎo)的) at x = x0 and x is a small change in x, then000()()()f xxf xfxx-一般近似公式一般近似公式Marginal analysis is an important example of a general approximation procedure.Incremental Approx

10、imation Formula：0000()()()limhf xhf xfxh: Analysis00()()()f xxf xxhxx 是的量，取很小一個(gè)+1.nxxxn 思考：當(dāng) 很小時(shí)，有近似公式1Suppose the total cost of manufacturing q hundred units of a certain commodity is C thousand dollars where C(q) = 3q2 + 5q + 10. If the current level of production is 4,000 units, estimate how the

11、total cost will change if 4050 units are produced.Example 2.5.3 (P164) Estimating Change in Cost :(40.5)(40)(40)(40)(0.5)CCCqC Hint3. Differentials 微分微分 If y = f (x) is a differentiable function of x, the approximation formula is ()( )( )f xxf xxxfxd記Approximation Formula： ydy 記dyy. .n Cdifferentialadj relating to differential calculus微分微分學(xué)的Example 2.5.7 (P167) Finding Differentials ( ).Find the differential of yf x:( )Hintdyfx dx32.(