How do I show that $$\int_a^bf(x)dx - I_1 = - \frac{h^5}{90}f^{(4)} (\xi)$$ with $\xi \in [a,b]$ and $$I_1 := \frac{h}{3}(f(a) + 4f\left( \frac{a+b}{2}\right) + f(b))$$ and $h= \frac{b-a}{2}$. I was thinking about about the mean value theorem somewhere, but I am not sure how to apply it to get to the result.
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I think what you do is pick up a good Calculus textbook, and see how it's proved there. Also, you could look at https://math.stackexchange.com/questions/14306/prove-simpsons-rule-including-error-using-the-integral-remainder or https://math.stackexchange.com/questions/1732738/numerical-integration-error-for-simpsons-rule-through-taylor-series – Gerry Myerson Mar 15 '22 at 11:50
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You can go via the Taylor expansion in the midpoint, or using the mean value theorem like in https://math.stackexchange.com/questions/1768667. Adding sufficient detail, both amount to the same in the end, only the contributing theoretical parts are grouped differently. – Lutz Lehmann Mar 15 '22 at 12:46
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Süli and Mayers "An Introduction to Numerical Analysis" (Cambridge University Press, 2003) discuss this in chapter 4. No, it isn't a simple calculus exercise. – vonbrand Sep 03 '22 at 02:30