Final_Fall_07-08 - AMERICAN UNIVERSITY OF BEIRUT Faculty of...

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Unformatted text preview: AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH 251 FINAL EXAM FALL 2007-2008 Closed Book, 2 HOURS WRITE YOUR ANSWERS ON THE QUESTION SHEET STUDENT NAME ID NUMBER Problem Out of Grade 1 20 2 10 3 20 4 15 5 25 6 10 TOTAL 100 1 The next 2 questions deal with the natural cubic spline S ( x ) on the following table of 5 data points for a function f ( x ): i 1 2 3 4 x i 1 2 3 4 5 f ( x i ) 1 4 11 17 23 1. (20 points) Let w i = S ( x i ) be the values of the moments of the cubic spline. (a) (10 points) Specify the system Aw = r . 2 (b) (10 points) Solve Aw = r using Naive Gauss Elimination followed by backward substitution. 3 2. (10 points) On the basis of w , give the formulae only of: (a) (5 points) The spline derivatives: z followed by z i = S ( x i ) , i = 1 ··· 4. (b) (5 points) The expression of S i ( x ), the spline S ( x ) restricted for x ∈ [ x i , x i +1 ]....
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Final_Fall_07-08 - AMERICAN UNIVERSITY OF BEIRUT Faculty of...

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