Hi, I'm trying to derive the error estimate in the trapezium rule but can't find any derivations on the internet and don't understand my lecturers notes after one point.

Using one strip for the trapezium rule, between a and (a+h). So the approximate integral is 0.5h[f(a) + f(a+h)]

so the error on this approximation is:

E(h) = (between a and a+h)int(f(x))dx - 0.5h[f(a) + f(a+h)]

Differentiating both sides twice wrt h gives:

E''(h) = -0.5h*f''(a+h) (this is in my notes and I understand it up to here, but it's the next part I don't understand)

let

M = max[abs(f''(x))] between a and (a+h)

Then

-0.5hM < E''(h) < 0.5hM (note I use < for less than or equal to, as I couldn't find the proper symbol)

Where have the last two steps come from? After these two steps we integrate twice to get

abs(E) < (1/12)M*h^3 , but I understand that part. It's just I don't get why -f''(a+h) < max[abs(f''(x))]

Thanks!