Question

Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) $$\begin{array}{l}{\text { (a) Find } f^{(4)} \text { for } f(x)=\sin \left(x^{2}\right). \text { (You may want to check your work with a CAS if you have one available.) }}\end{array} $$ (b) Graph \(y=f^{(4)}(x)\) in the viewing window \([-1,1]\) by \([-30,10] .\) (c) Explain why the graph in part (b) suggests that \(\left|f^{(4)}(x)\right| \leq 30\) for \(-1 \leq x \leq 1\) (d) Show that the error estimate for Simpson's Rule in this case becomes $$\left|E_{S}\right| \leq \frac{h^{4}}{3}$$ (e) Show that the Simpson's Rule error will be less than or equal to 0.01 if \(h \leq 0.4 .\) (f) How large must \(n\) be for \(h \leq 0.4 ?\)

Short answer

The fourth derivative of \(f(x)=\sin(x^2)\) is \(f^{(4)}(x) = 4[3x^2\sin(x^2) + 4\cos(x^2) - 8x^4\sin(x^2)]\). From the graph of \(f^{(4)}(x)\), it is observed that \(\left|f^{(4)}(x)\right|\) does not exceed 30 within [-1,1]. The error estimate \( \left|E_{S}\right|\) for Simpson's Rule in this case is \( \frac{h^{4}}{3}\). The error will be less than or equal to 0.01 if \(h ≤ 0.4\). Hence the number of subintervals 'n' must be at least 5.

Step-by-step solution

Calculate the fourth derivative of \(f(x)=\sin \left(x^{2}\right)\)

Using the chain rule, we can begin by finding the first derivative. Differentiating \(f(x)=\sin(x^2)\) once gives \(f'(x) = 2x \cos(x^2)\). Differentiating a few times we find \(f^{(4)}(x) = 4[3x^2\sin(x^2) + 4\cos(x^2) - 8x^4\sin(x^2)]\).

Graph \(y=f^{(4)}(x)\) using the defined viewing window

You should sketch the graph of the fourth derivative \(f^{(4)}(x)\) in the given closed interval [-1, 1] with y-values in the range [-30,10], the pattern of the graph indicates a maximum and minimum value.

Analyze the graph

From the graph, it is clear that \(\left|f^{(4)}(x)\right|\) varies between -30 and 10. This suggests that the magnitude of \(f^{(4)}(x)\) is less than or equal to 30 for \(-1 \leq x \leq 1\).

Error Estimation for Simpson's Rule

According to Simpson's Rule, the absolute error \( \left|E_{S}\right|\) is given by \( \frac{K}{180}(\frac{b-a}{n})^4 \) where \(K\) is the maximum of \( |f^{(4)}(x)| \) on \([a,b]\), and \(n\) is the number of sub intervals. In this case, \(K ≤ 30\), \(a = -1\) and \(b = 1\), therefore \( \left|E_{S}\right|\) is less than or equal to \( \frac{30}{180}(\frac{1+1}{n})^4 = \frac{h^{4}}{3}\) where \( h=2/n \).

Find Condition for the Error to be Less Than or Equal to 0.01

From step 4, you know that the error is less than or equal to 0.01 if \( \frac{h^{4}}{3} ≤ 0.01\). Solution to this equation gives \(h ≤ 0.4\).

Calculate 'n' Ensuring \(h ≤ 0.4\)

We know that \(h = \frac{2}{n}\), substituting \(h ≤ 0.4\) gives \(n ≥ 5\). This means the number of sub intervals 'n' should be at least 5 to ensure that the error is less or equal to 0.01.