Call your values
and the corresponding function values
then the entries in the first difference column will be
.
If you were to divide each of these by you would get a discrete version of the derivative of
If this is to be/represent what does that suggest that should be ? Think calculus.
Make that (obvious) choice for and construct the first difference column, (a little algebraic simplification is required).
You should find that that doesn't quite work, but after tweaking slightly it does.
Hi,
I got the answer to the problem by referring to the solved answers to the similar finite differences problems. Here is the answer. We have $ e^x $ as first difference
$ e^x =e^x\frac {1-e}{1-e} $
$ e^x =\frac {e^x}{1-e}-\frac {e^{x+1}}{1-e} $
Hence,
the function is$\frac {e^{x+1}}{1-e} $