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eqe - awn 0:201 W WNW!“ SW30 AMSC/CMSC 460 Final Exam...

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Unformatted text preview: awn 0:201 W WNW!“ SW30? AMSC/CMSC 460 Final Exam Show all your work. Use Gaussian elimination (not your calculator) to solve linear systems. Draw a box around your answers. 1. (20 pts) We are given the following data points (t,-, 31,): (—2, 2), (—1,1), (0,2), (1,2). Find the best least squares ﬁt g(t) of the form g(t) = c1 ltl + c2t2. 2. (20 pts) We want to find \$1,272,103 which satisfy the nonlinear system 4\$1+\$§‘-1 = 0 4\$2+\$§—1 = 0 4:33—2:3—1 :2 0 Perform one step of the Newton method starting with the initial guess x(°) = (1,1, 1)T. 3. (40 pts) Consider the initial value problem y” + yy’ = t, y(1) = 1, y’(1)= 2 We want to ﬁnd y(T) for T = 2. (a) (10 pts) Perform two steps of the Euler method with h = % Give the resulting approximation for y(T). - (b) (10 pts) Perform one step of the RK2 method with h = 1. Give the resulting approximation for y(T). . (c) (10 pts) Consider now the initial values y(1) = 1, y'(1) = '0. Write an m—ﬁle solT.m so that z = solT(v) returns an approximation for y(T). (d) (10 pts) We try solT(-2) and get a negative value, then we try solT(—1) and get a positive value. Write an m—ﬁle which ﬁnds a value 'u so that y(T) = O. 4. (20 pts) We want to approximate I := ff ﬁdx using the composite trapezoid rule with N subinter— vals of equal size. (a) (10 pts) Find an approximation Q using N = 2 subintervals. (b) (10 pts) Use the error formula to determine a number N such that IQ —- 1| 3 10‘15. ...
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