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MATH 1100: QUANTITATIVE ANALYSIS TEST #5 (VERSION A) 1. Suppose that a business has continuous income stream with an annual rate of flow at time t given by f (t) = 10000. The interest rate is r = 10%, compounded continuously. Find the capital value of the income stream. Solution: Z ∞ CV = 10000e−0.1t dt = $100000. 0 2. Suppose that ( ce−kx f (x) = 0 if x > 0 otherwise where k > 0. What must the value of c be (in terms of k) so that f (x) is a probability density function? Solution: c = k 3. Suppose that the supply function for a product is p = x4 + 1, and the equilibrium price is $2. Find the producer’s surplus. Solution: At p = $2 = x4 + 1, we have x = 1. Z P S = x1 p1 − p(x) dx Z 1 = (1)(2) − x4 + 1 dx 0 = 2 − (1/5 + 1) = 4/5 = .80 4. Suppose that ( f (x) = 4 9π √ 9 − x2 0 if 0 < x < 3 otherwise. Find the mean value of a random variable with f (x) as probability density function. Solution: Z 3 p 4 µ= x 9 − x2 dx 9π 0 4 = 9 9π 4 = . π Date: November 16, 2001. 1 2 MATH 1100: QUANTITATIVE ANALYSIS TEST #5 (VERSION A) To do the integral, try u = 9 − x2 , so du = −2x dx. 5. Using n rectangles of width 1/n whose left shoulder touches the graph of f (x) = 2x, approximate the area under the graph of f (x) between x = 0 and x = 1. You might need to know that n X 1=n k=1 n X k = n(n + 1)/2. k=1 Solution: The area of all of the rectangles is 1 1− . n