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**1. Homework Statement**

So this is a problem for my numerical methods class. We are doing integration and derivation. The part that I am stuck on is why does the midpoint method and trapezoidal method both have error of order ##h^2##, when the first one is one point integration method while the second one is a 2 points. I know how the error goes like, so I don't need to prove it mathematically, I just need to give a logical explanation.

**2. Homework Equations**

Trapezoidal method: ##\int_a^b f = (b-a)\frac{f(b)+f(a)}{2}##

Midpoint method: ##\int_a^b f = (b-a)f(\frac{a+b}{2})##

**3. The Attempt at a Solution**

In derivation, one point methods go like ##O(h)## while 2-point methods like ##O(h^2)##, as in the 2 point methods, you have more information. But the same would be here so I am not sure why the error is different. I was thinking that ##f(\frac{a+b}{2})## uses an actual value belonging to f, while ##\frac{f(b)+f(a)}{2}## is not a value of f, but I am not sure. Can someone help me? Thank you!