Error
The error in approximating an integral by Simpson's rule is
where is some number between and .
The error is asymptotically proportional to . However, the above derivations suggest an error proportional to . Simpson's rule gains an extra order because the points at which the integrand is evaluated are distributed symmetrically in the interval .
Since the error term is proportional to the fourth derivative of f at, this shows that Simpson's rule provides exact results for any polynomial f of degree three or less, since the fourth derivative of such a polynomial is zero at all points.
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