I have two simple questions and my ideas and want to check if they are correct.
i) Let $p(x)$ be the cubic polynomial of $f(x) = \sin(3\cdot x)+5\cdot \cos(x)$ at the nodes $(-3,0,2,3)$. Compute $p(0.5)$.
Here I calculate the Lagrange Polynomial and find $p(0.5)$ right? This is what is meant by a cubic polynomial?
ii) Given $(x_0,x_1,x_2,x_3)$ = $(-3,0,1,3)$ and $f(x) = \sin(3\cdot x)+5\cdot \cos(x)$, compute the coefficient of $(x-x_0)\cdot(x-x_1)\cdot(x-x_2)$ in the Newton interpolation polynomial.
This seems straight forward as well. Basicly I compute the Newton interpolation polynomial (not even necessary to compute the whole polynomial) and then look at the coefficient?
Especially if the first idea is right, I would just calculate it and then edit this post if somebody is interested. Thank you for your help in advance!