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MATH 1100: QUANTITATIVE ANALYSIS TEST #4 (VERSION A) 1. Find the general solution to 4x dy = . dx y−3 Solution: y =3± p 4x2 + C 2. If consumption is $5 billion when disposable income is 0, and the marginal propensity to consume is dC 1 =1+ √ dy y find the national consumption function. Solution: √ C =5+y+2 y 3. Suppose that world population P (measured in billions of people) never exceeds 100 and has rate of grow dP = K(100 − P ). dt Suppose that at time t = 0 there are 1 billion people, and the rate of growth is dP = 90. dt Find P as a function of t. Solution: dP = K dt. 100 − P Now integrate. It is important to distinguish between the dP and the dt. The integral Z dt 100 − P is not the same as Z dP . 100 − P The final answer: P = 100 − 99e−(10/11)t . Date: November 2, 2001. 1 2 MATH 1100: QUANTITATIVE ANALYSIS TEST #4 (VERSION A) 4. Calculate the integral Z 100 x2 + 1 x dx Solution: 101 x2 + 1 + C. 202 There is a danger here: don’t imagine that Z f (x)g(x) dx will equal Z Z f (x) dx g(x) dx . Instead, try u = x2 + 1, so du = 2x dx. Make sure that your final answer is in terms of x, not u. 5. If the marginal revenue for a month is M R = x2 + 1, find the total revenue from sales of 10 units. Solution: 1030 R= . 3