# Approximation of second derivative of a smooth function

1. Oct 16, 2015

Hi,

I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is there any other reason? Also, what exactly is $$\mathcal{O}(h^2)$$ and why is it included? Isn't the first whole term (i.e. the fraction) the only necessary term to describe the second derivative of $\varphi$?

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2. Oct 16, 2015

### mathman

The second derivative is the limit to the fraction as h>0. $O(h^2)$ means the remainder for small h is roughly a constant times $h^2$.