## University Calculus: Early Transcendentals (3rd Edition)

$n \geq 16$
The error bound for Simpson's Rule is given as follows: $|E_S| \leq \dfrac{M(b-a)^5}{180n^4}$ Now, the maximum value of $|f^{4}(x)|$ on $[1,3]$ is: $|f^{4}(1)|=|\dfrac{24}{(1)^5}|=24$ $\implies M=24$ Further, $|E_S| \leq \dfrac{24 \times(3-1)^5}{180n^4}=\dfrac{32}{15n^4}$ we will choose $n$ such that $\dfrac{64}{15n^4} \leq 10^{-4}$ Therefore, $n \geq 14.4 \approx 15$ But as per Simpson's Rule, we need the even value of $n$ , thus $n \geq 16$ .