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Math 105/206 - Practice problems for Quiz 4 Problem 1 Compute Simpson’s approximation for the integral Z π/2 sin(2x) dx 0 with n = 4 subintervals, and estimate the absolute error. How big do you need to take n to have an error of at 1 ? most 1000 Problem 2 Find the general solution of the following separable first-order differential equations. dy dx = y(x2 + 1) where y > 0 √ • x2 dw = w(3x + 1). dx • Problem 3 Find the solution to the following first-order initial value problem. dz z2 = 2 dx x +1 z(0) = 16 Problem 4 You have a colony of bacteria that, if left undisturbed, grows proportionally to the population, with a factor 0.008 (and they never die......). Let’s say we use years as unit of time, and you harvest a quantity h of the bacteria every year, and say the initial population is P0 . • if P0 = 2000, what’s the value of h that gives you a constant population? • if h = 200, what’s the value of P0 that gives you a constant population? • if P0 = 100 and h = 50, find an expression for the population at a generic time t. Problem 5 Are the following functions CDFs for a continuous random variable? 1. F (x) = π1 (arctan(x) + π2 ) 2. F (x) = 1 + e−x 3. F (x) = 1 1+log(x2 +1) if x < 1 0 2x − 2 if 1 ≤ x < 32 4. F (x) = 1 if x ≥ 32 if x < 1 0 x − 1 if 1 ≤ x < 32 5. F (x) = 1 if x ≥ 23