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Unformatted text preview: UCLA MATH 151A, WINTER 2000, FINAL EXAM, MONDAY, MARCH 19 NAME STUDENT ID # This is a closedbook and closednote examination. No calculators are allowed. Please show all your work. Partial credit will be given to partial answers. There are 9 problems of total 100 points. PROBLEM 1 2 3 4 5 6 7 8 9 TOTAL SCORE Problem 1 : Let [ a, b ] be an interval of R , x , x 1 , ..., x n be n + 1 points in [ a, b ] and f ∈ C n +1 ( R ). 1a Give the form of the Lagrange polynomial of degree at most n such that f ( x k ) = P ( x k ) , for each k = 0 , 1 , ..., n. In the sequel, we will note this polynomial P x ,x 1 ,...,x n . 1b Does another polynomial Q , of degree at most n , exist such that f ( x k ) = Q ( x k ) , for each k = 0 , 1 , ..., n. 1c Deduce from the preceding question that any polynomial of degree at most n is exactly interpolated by a Lagrange polynomial. 1d State the formula which, for x ∈ [ a, b ], expresses f ( x ) in terms of P x ,x 1 ,...,x n ( x ) (the Lagrange polyno mial defined in question 1a) and a remainder term. 1e Deduce from the preceding question that any polynomial of degree at most n is exactly interpolated by a Lagrange polynomial. 1f Let ( i, j ) ∈ { , 1 , ..., n } 2 be two distinct integers. Taking the notation defined in question 1a, express P x ,x 1 ,...,x n in terms of P x ,x 1 ,...,x i 1 ,x i +1 ,...,x n and P x ,x 1 ,...,x j 1 ,x j +1 ,...,x n ....
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 Math, Numerical Analysis, Composite Simpson

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