Question

question 10 of 14, step 1 of 1
find the maximum error for simpsons rule when it is used to approximate the integral of ( f ( x ) = 3 cos ( 4 x ) ) over the interval ( left \frac { pi } { 4 } , \frac { 3 pi } { 4 }
ight ). use ( n = 10 ). round
your answer to four decimal places if necessary.
answer
| e _ { s } | <
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Explanation:

Step1: Recall the error formula for Simpson's Rule

The error formula for Simpson's Rule is \(|E_S|\leq\frac{(b - a)^5}{180n^4}\max|f^{(4)}(x)|\), where \([a,b]\) is the interval of integration and \(n\) is the number of sub - intervals.
For \(y = f(x)=3\cos(4x)\), first find the fourth derivative.
\(y'=-12\sin(4x)\), \(y''=-48\cos(4x)\), \(y''' = 192\sin(4x)\), \(y^{(4)}=768\cos(4x)\).
The maximum value of \(|\cos(4x)|\) is \(1\), so \(\max|f^{(4)}(x)| = 768\).
The interval is \([a=\frac{\pi}{4},b = \frac{3\pi}{4}]\), so \(b - a=\frac{3\pi}{4}-\frac{\pi}{4}=\frac{\pi}{2}\), and \(n = 10\).

Step2: Substitute values into the error formula

Substitute \(a=\frac{\pi}{4}\), \(b=\frac{3\pi}{4}\), \(n = 10\), and \(\max|f^{(4)}(x)|=768\) into the formula \(|E_S|\leq\frac{(b - a)^5}{180n^4}\max|f^{(4)}(x)|\).
\(|E_S|\leq\frac{(\frac{\pi}{2})^5}{180\times10^4}\times768\).
First, calculate \((\frac{\pi}{2})^5=\frac{\pi^5}{32}\approx\frac{306.019684}{32}\approx9.563115\).
Then \(\frac{(\frac{\pi}{2})^5\times768}{180\times10^4}=\frac{9.563115\times768}{180\times10^4}\).
\(9.563115\times768 = 9.563115\times(700 + 60+8)=9.563115\times700+9.563115\times60 + 9.563115\times8=6694.1805+573.7869+76.50492=7344.47232\).
\(\frac{7344.47232}{180\times10^4}=\frac{7344.47232}{1800000}\approx0.00408\).

Answer:

\(0.0041\)