Divided Differences - Silly Mistake to Avoid

I went and completed an entire table of divided differences to use for an estimate of sin(0.63) (Oct 09).

While this is enjoyable in a twisted kind of way, if you look at that question, you'll see it's not necessary. In fact the x column of that table could be restated as ...

x
0.6
0.7
0.8
red
herring
kipper
toby

This is because your point needs to be centred inside the steps of the table. Here 0.63 only fits between 0.6 and 0.7. This means the polynomial can only go up to first function difference (even spacing means no need to divide).

Had it been 0.73 you could've brought in point 0.8 and 0.9, even...

That's good news isn't it? No need to calculate the whole lot unless explicitly required to do so.

So the answer would be f₀ + s[delta f₀] + error.

And the error is from the next term rule. Do one more row of differences and get the second divided difference.