Question

Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) $$\begin{array}{l}{\text { (a) Find } f^{\prime \prime} \text { for } f(x)=\sin \left(x^{2}\right)} \\ {\text { (b) Graph } y=f^{\prime \prime}(x) \text { in the viewing window }[-1,1] \text { by }[-3,3] \text { . }} \\\ {\text { (c) Explain why the graph in part (b) suggests that }\left|f^{\prime \prime}(x)\right| \leq 3} \\ {\text { for }-1 \leq x \leq 1 .} \\ {\text { (d) Show that the error estimate for the Trapezoidal Rule in this case becomes }}\end{array} $$ $$\left|E_{T}\right| \leq \frac{h^{2}}{2}$$ $$\begin{array}{l}{\text { (e) Show that the Trapezoidal Rule error will be less than or equal to } 0.01 \text { if } h \leq 0.1 .} \\ {\text { (f) How large must } n \text { be for } h \leq 0.1 ?}\end{array}$$

Short answer

f''\(x = -4x^2 \sin(x^2) + 2 \cos(x^2)\). By graphing, it is suggested that \(|f''(x)| \leq 3\) within the range [-1,1]. Applying the Trapezoidal Rule, the error of approximation is \(|E_T| \leq \frac{h^2}{4}\). For the error to be less than or equal to 0.01, h must be less than or equal to 0.1. Subsequently, n, or the number of subintervals, needs to be at least 20 for \(h \leq 0.1\).

Step-by-step solution

Find Second Derivative

To solve for \(f^{\prime \prime}(x)\), we first find the \(f^{\prime}(x)\). The derivative of \(f(x) = \sin(x^2)\) is \(f^{\prime}(x) = 2x \cos(x^2)\). Then we find the second derivative \(f^{\prime \prime}(x)\) which is \(f^{\prime \prime}(x) = -4x^2 \sin(x^2) + 2 \cos(x^2)\).

Graph Second Derivative

Graph the function \(f^{\prime \prime}(x) = -4x^2 \sin(x^2) + 2 \cos(x^2)\) in the viewing window [-1,1] by [-3,3]. The graph of this function suggests that the maximum value of |f''(x)| within [-1,1] is 3.

Explain Graph's Maximum Value Suggestion

The graph of \(f^{\prime \prime}(x)\) within the range [-1,1] seems to suggest that the the absolute value of \(f^{\prime \prime}(x)\) doesn't exceed 3, meaning \(|f^{\prime \prime}(x)| \leq 3\).

Apply Trapezoidal Rule Error Estimate

The error of approximation made by the Trapezoidal Rule is given by \(|E_T| \leq \frac{Mh^2}{12}\), where M is an upper bound on the absolute value of the second derivative of the function being integrated, and h is the length of the subintervals used in the approximation. Substituting \(M = 3\) (from the graph's suggestion) and simplifying the formula, we get \(|E_T| \leq \frac{h^2}{4}\).

Show Error Less Than or Equal to 0.01

Set the error equal to 0.01 and solve for h, \(h^2 \leq 0.04\), thus \(h \leq 0.1\). This shows that for the error of the Trapezoidal Rule approximation to be less than or equal to 0.01, h must be less than or equal to 0.1.

Determine Value of n

We know that \(h = \frac{b-a}{n}\), where a and b are the limits of integration and n is the number of subintervals. Here, a=-1 and b=1 and we are required to find n for \(h \leq 0.1\). Substituting the values into equation and solving for n, \(n = \frac{b-a}{h}\). Thus, \(n = \frac{1-(-1)}{0.1} = 20\). This implies that n needs to be at least 20 for \(h \leq 0.1\).