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hwk6 - Homework 6 Due Monday 02/25 1 In order to derive...

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Homework # 6 Due Monday, 02/25 1. In order to derive Simpson’s rule, we proved that, if p ( x ) is the unique polynomial of degree 2 which interpolates the function f ( x ) at x 0 , x 1 and x 2 , then x 2 x 0 p ( x ) dx = h 3 ( f ( x 0 ) + 4 f ( x 1 ) + f ( x 2 )) In this problem, we will do the same for the trapezoidal rule: let p ( x ) be the unique polynomial of degree 1 which interpolate the function f ( x ) at x 0 and x 1 . Use Lagrange’s formula to prove that: x 1 x 0 p ( x ) dx = h 2 ( f ( x 0 ) + f ( x 1 )) Remark: In class we derived this result by drawing a picture and calculating the area of a trapezoid. Here I am asking you to derive this result in a more rigourous way: First use Lagrange’s formula and then integrate the polynomial that you obtained. 2. While proving that the error in the trapezoidal rule is O ( h 2 ), we used the fact that: x 4 x 3 ( x - x 3 )( x - x 4 ) dx = - 1 6 ( x 3 4 - 3 x 2 4 x 3 + 3 x 4 x 2 3 - x 3 3 ) = - ( x 4 - x 3 ) 3 6 Prove it. 3. Theorem. Suppose f ( x ) is twice continously differentiable, then there exist ξ n [ a, b ] such that b a f ( x ) dx - I n ( f ) = - ( b - a ) f ( ξ n ) 12 h 2 (1) Corrolary.
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