How large should n be to guarantee that the Simpson's Rule approximation to

Question

integral 0 to 1 e^(x^2) dx is accurate to within 0.00001?

I got f''''(x)=(16x^4+48x^2+12)e^(x^2) and now I have to show how to evaluate |(16x^4+48x^2+12)e^(x^2)|<=k

Please help me how to evaluate k "SPECIFICALLY"

I need to prove why k is something without just plugging numbers.

Answer 1

If you graph the 4th derivative you will see that it is increasing to the right of zero. You could also determine that the 5th derivative is positive to the right of zero (so 4th derivative is increasing). That means f⁽⁴⁾(1) has the largest value on [0,1]. So, k ≤ f⁽⁴⁾(1) ≤ 76e < 207.

So, 0.00001 ≤ 207(1-0)⁵/18n⁴, where 0.00001 is the error in Simpson's rule.