1. ## Quadratic spline interpolation

Hi, I have been working on this 3 hours, and I just can't figure it out. I have this article https://books.google.ee/books?id=5fxu15HQRJ0C&pg=PA60&lpg=PA60&dq=ruutspla in+pedas&source=bl&ots=8h47blCY7y&sig=ce8u4FB5U18A kvFDEyke4ChKnOw&hl=et&sa=X&ved=0ahUKEwiYjuL80dLZAh WFDCwKHf9JCPkQ6AEIJjAA#v=onepage&q=ruutsplain%20pe das&f=false and I need to derive the formula (10) from page 50. Can anyone show me how it's done?

I know that
$S'(x_i)=m_i, i=0,1,...,n-1$
and
$S'(x)=m_i+\frac{m_{i+1}-m_i}{h}(x-x_i), x\in [x_i,x_{i+1}]$
so integrate from $x_i$ to x
$S(x)-S(x_i)=m_i(x-x_i)+\frac{m_{i+1}-m_i}{h} \frac{(x-x_i)(x-x_i)}{2}$

2. ## Re: Quadratic spline interpolation

Also, if anyone thinks that the article is in Estonian, then it's not, just a little section. Everything else is in English.