# Science:Math Exam Resources/Courses/MATH101/April 2008/Question 05 (b)

MATH101 April 2008
Other MATH101 Exams

### Question 05 (b)

Let

${\displaystyle I=\int _{0}^{1}\cos(x^{2})\,dx}$

It can be shown that the 4th derivative of

${\displaystyle \displaystyle \cos(x^{2})}$

has absolute value at most 60 on the interval [0,1]. Using this bound, find the smallest positive integer n you can such that the Simpson's Rule approximation ${\displaystyle S_{n}}$ for ${\displaystyle I}$ has error less or equal to 0.001. You may use the fact that if

${\displaystyle |f^{(4)}(t)|\leq K}$

on the interval ${\displaystyle [a{\text{, }}b]}$, then the error in using ${\displaystyle S_{n}}$ to approximate

${\displaystyle \int _{a}^{b}f(x)\,dx}$

has absolute value less than or equal to

${\displaystyle {\frac {K(b-a)^{5}}{180n^{4}}}}$
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