(d) How large should we take n in order to guarantee that the Simpson's Rule approximation for 1¹6 g(x) dx would be accurate to within 10-6 ?
(d) How large should we take n in order to guarantee that the Simpson's Rule approximation for 1¹6 g(x) dx would be accurate to within 10-6 ?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PART D PLEASE
![**Problem 4**
(a) A table of values of a function \( g \) is given. Use Simpson’s Rule to estimate \( \int_{0}^{1.6} g(x) \, dx \).
\[
\begin{array}{|c|c|c|c|}
\hline
x & g(x) & x & g(x) \\
\hline
0.0 & 12.1 & 1.0 & 11.2 \\
0.2 & 11.6 & 1.2 & 11.6 \\
0.4 & 11.3 & 1.4 & 12.0 \\
0.6 & 11.1 & 1.6 & 12.2 \\
0.8 & 11.7 & & \\
\hline
\end{array}
\]
(b) If \(-6 \leq g^{(4)}(x) \leq 2\) for \(0 \leq x \leq 1.6\), estimate the error involved in the approximation in part (a).
(c) What is an average value of \( g(x) \) on \(0 \leq x \leq 1.6\)?
(d) How large should we take \(n\) in order to guarantee that the Simpson’s Rule approximation for \(\int_{0}^{1.6} g(x) \, dx\) would be accurate to within \(10^{-6}\)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b08cbe3-b052-4a50-ae7b-da07c66e348c%2Fad5ffb5a-3656-4963-9186-276a3d66b741%2Fibx4w8s_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4**
(a) A table of values of a function \( g \) is given. Use Simpson’s Rule to estimate \( \int_{0}^{1.6} g(x) \, dx \).
\[
\begin{array}{|c|c|c|c|}
\hline
x & g(x) & x & g(x) \\
\hline
0.0 & 12.1 & 1.0 & 11.2 \\
0.2 & 11.6 & 1.2 & 11.6 \\
0.4 & 11.3 & 1.4 & 12.0 \\
0.6 & 11.1 & 1.6 & 12.2 \\
0.8 & 11.7 & & \\
\hline
\end{array}
\]
(b) If \(-6 \leq g^{(4)}(x) \leq 2\) for \(0 \leq x \leq 1.6\), estimate the error involved in the approximation in part (a).
(c) What is an average value of \( g(x) \) on \(0 \leq x \leq 1.6\)?
(d) How large should we take \(n\) in order to guarantee that the Simpson’s Rule approximation for \(\int_{0}^{1.6} g(x) \, dx\) would be accurate to within \(10^{-6}\)?
Expert Solution
Step 1
The error bound in composite Simpson's 1/3rd rule for approximating is given by
where .
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