Good morning,

I want to learn more about cubic splines but unfortunately my class goes pretty quickly and we really only get the high level overview of why they're important and why they work.

To me it's clear why we dont use linear functions, we cant differentiate the spline at the points we have to interpolate. Due to this, we lose important information about the underlying function.

But now this is where it gets harder for me. I know we can't use hermite polynomials because we require the derivative and many times we dont have this information available to us.

So we *could* use quadratic polynomials between each point to approximate it so its smooth on the points and we can differentiate it. The book goes on to state

The difficulty arises because we generally need to specify conditions about the derivative of the interpolant at the endpoints $x_0$ and $x_n$. [In quadratic polynomials] there is not sufficient number of constants to ensure that condition will be satisfied.

How does the number of constant bear anything on the endpoint constraints? A quadratic polynomial is twice differentiable. Can anyone fill me in on gap I have here in my knowledge of why we need cubic splines?

thank you!