hw4 - tion is interpolated in each of the n subgroups of...

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Math 128a, fall 2011, Chorin, theory homework 4, due the week of Sept. 26. 1. Let I h ( f ) be the midpoint approximation with mesh h to the integral R b a f ( x ) dx . Show that R b a f ( x ) dx = I h ( f ) + Ch 2 + O ( h 4 ), where C is a constant independent of h . 2. Derive the coefficients in the integration rule in which the interval of in- tegration is divided into 3 n subintervals of equal lengths, and the func-
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Unformatted text preview: tion is interpolated in each of the n subgroups of three subintervals by a polynomial of degree 3 which coincides the function at the ends of the subintervals. Find a bound for the error, and check that the rule is exact for polynomials of degree 3 or less. 1...
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