Choose interpolation nodes

Question

How to determine power of interpolation polynomial and interpolation nodes so this polynomial approximates function $y = 2 * sin(3x)$ at $[0,1]$ with estimate error less or equal to $0.2$ ? How should I solve this problem ?

Answer 1

**hint $$

if $f (x)=2\sin (3x) $ then

$f^{n+1}(x)=2. (3^{n+1})\sin (3x+(n+1)\frac {\pi}{2}) $ and $$|f^{n+1}(x)|\leq 2. (3^{n+1}) $$

the error $E $ is given by

$$E(x)=\frac {\prod_{i=0}^n (x-x_i)}{(n+1)!}f^{n+1}(\xi) $$

and you should have $$|E (x)|\leq 0.2.$$ or $$\frac {3^{n+1}}{(n+1)!}\leq 0.1$$ using the fact that $|x-x_i|\leq 1$.