Question

simpson's rule integration help

Answer 1

find the least possible value of N for which Error(SN)≤1×10−9 in approximating
\[\int\limits_{0}^{1}4e ^{x ^{2}}dx\] using the result that \[Error(S _{N})\le \frac{ K _{4}(b-a)^4 }{ 180N ^{4} }\] where K_4 is the least upper bound for all absolute values of the fourth derivatives of the function 4e^(x^2) on [a,b].

Answer 2

do i have to find the fourth derivative first?

Answer 3

i found the fourth derivative:
= e^(x^2) (64x^4 + 192x^2 + 48)
then i plugged in 1 for x and got 826.3577
then i plugged that into \[1\times10^{-9 }\le \frac{ K(b-a)^5 }{180N^4}\]
\[1\times10^{-9}\le \frac{ 826.3577(1)^5 }{ 180N^4 }\]
\[(180N^4)(1\times10^-9) \le 826.3577\]