User:Yongtao-Li/NM.HW4.2

=Problem 4.2 "One step in the proof of Simple Simpson's Error Theorem"=

Given
In order to prove Simple Simpson's Error Theorem, we define

Find
Find the first derivative of the first term of (4.2.1) since it is needed to compute the first derivative of (4.2.2).

Solution
Suppose there is a constant $$ k \in \left] { - t,t} \right[ $$ and then the first term can be written as

Then we can find out the first derivative of (4.2.3) is

Therefore the first derivative of the first term of (4.2.1) is