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Approximation of second derivative of a smooth function

  1. Oct 16, 2015 #1
    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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  3. Oct 16, 2015 #2

    mathman

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