University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Practice Exercises - Page 474: 45


$n \geq 16$

Work Step by Step

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 $n \geq 16$ .
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