User:Egm6341.s11.team4.kurth/hw3

=Problem 3.2: Investigating the Values of G(x) at All Nodes $$x_i$$= Refer to lecture slide [[media:Nm1.s11.mtg14.djvu|14-3]] for problem statement.

Given
Note: shorthand notation was used, however it is implied that these equations refer to the Lagrange interpolation of a continuous function $$\displaystyle f$$ which has $$(n+1)$$ continuous derivatives on $$\displaystyle I_t := \mathcal{H}(t,x_0,x_1,...,x_n)$$ as defined on lecture slides [[media:Nm1.s11.mtg11.djvu|11-2,11-3]].

Objective
Show that

Solution
First we will rearrange the second term in (2.1) as follows

From equation (2) on lecture slide [[media:Nm1.s11.mtg11.djvu|11-3]]:

where $$\displaystyle \xi \in I_t$$.

Rearranging (2.4) and substituting it back into (2.3) yields

From equation (3) on lecture slide [[media:Nm1.s11.mtg11.djvu|11-3]]

Upon inspection it can easily be seen that

From (2.5) and (2.7)

Solved by William Kurth.