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Unformatted text preview: 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 , x 1 and x 2 , then Z x 2 x p ( x ) dx = h 3 ( f ( x ) + 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 and x 1 . Use Lagrange’s formula to prove that: Z x 1 x p ( x ) dx = h 2 ( f ( x ) + 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: Z 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....
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This note was uploaded on 10/28/2010 for the course MATH 135A taught by Professor Thomas during the Spring '10 term at UC Riverside.
- Spring '10