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sample_short - Final Exam, Math 151A/2, Winter 2001, UCLA,...

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Unformatted text preview: Final Exam, Math 151A/2, Winter 2001, UCLA, 03/21/2001, 8am-11am I. (a) Let x , x 1 , ..., x n be n +1 distinct points in [ a, b ], with x = a and x n = b , and f ∈ C n +1 [ a, b ]. Let P ( x ) = P , 1 ,...,n ( x ) be the Lagrange polynomial interpolating the points x , x 1 , ..., x n , such that P ( x i ) = f ( x i ), for all i = 0 , 1 , ..., n . Express f ( x ) in terms of P ( x ) and a remainder term (the error formula). (b) Consider the case n = 2 and the data x x = 0 x 1 = 1 x 2 = 2 f ( x ) = ln( x + 1) ln 2 ln 3 (i) Find polynomials L i ( x ), i = 0 , 1 , 2, such that L i ( x i ) = 1 and L i ( x j ) = 0 if i negationslash = j . (ii) Deduce the Lagrange polynomial P ( x ) = P , 1 , 2 ( x ) interpolating this data points. (iii) Write the error formula and find a bound for the error | f (0 . 5) − P (0 . 5) | . II. (a) Let i, j be two distinct integers in { , 1 , ..., n } . Express P , 1 ,...,n ( x ) in terms of P , 1 ,...,i − 1 ,i +1 ,...,n ( x ) and of P , 1 ,...,j − 1 ,j +1...
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This note was uploaded on 10/28/2010 for the course MATH 151a taught by Professor Staff during the Spring '08 term at UCLA.

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sample_short - Final Exam, Math 151A/2, Winter 2001, UCLA,...

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