An interpolating polynomial of degree 20 is to be used to approximate e−x on the interval [0, 2]. How accurate will it be? (Use 21 uniform nodes, including the endpoints of the interval. Compare results, using Theorems 1 and 2.)
The solutions from the textbook would be $4.105 × 10^−14 (Thm 1), 1.1905 × 10^−23 (Thm 2)$
First pic is thm 2 second pic is thm 1. Theorem 2 is not too hard to compute since e^-x to the 20th derivative is still e^-x so M the upper bound for that would be 1 for [0,2] interval as can be seen if plotted. Then i plug in n=20 to get the answer. How do i use the first theorem? I am confused about how to find ξ?
Continuing from my comments. I actually made a mistake and should have said $|\prod_{i=0}^n (x-x_i)| < 2^{n+1}$. We want to find the maximum error for $n=20$, that is,
$$|f(x)-p(x)| <\frac{1}{(20+1)!}|e^{-x}|.2^{21}= \frac{2^{21}}{21!} = 4.105 \times 10^{-14}$$