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pset_VI-06-2 - MMAE 350 D Rempfer Problem Set VI Page 1 of...

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MMAE 350 D. Rempfer Problem Set VI Page 1 of 3 Due Date: November 06, 2006 1. You are given the following set of ( x, y ) pairs of values, describing (error-prone) measurements of some unknown function f ( x ): x i -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 y i 1.0 1.8 2.5 3.0 2.8 3.1 3.1 2.5 2.4 2.0 0.8 Find a least-squares approximation of the function f ( x ) that is of the form f ( x ) a 0 log ( 1 + x 2 ) + a 1 exp ( - x 2 ) + a 2 x 2 . (1) 2. Calculate the coe ffi cients of the Fourier expansion ˜ f ( t ) = a 0 + 5 j = 1 a j cos(2 πj t ) + 5 i = j b j sin(2 πj t ) for the following two functions: a. f ( t ) = 1 2 - 1 2 t, x [0 , 1) b. f ( t ) = 1 2 t - 1 2 t 2 , x [0 , 1] For each of the above, plot both the periodic extension of f and ˜ f in the interval [0 , 2]. Note that the integrals that give these coe ffi cients can be solved in closed form, so you do not have to integrate numerically. 3. You are given the task to derive an integration formula of Newton-Cotes type that integrates a function that is given at the three non-equidistant points x 0 , x 1 = x 0 + 2 h , and x 2 = x 1 + 3 h .
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