14.14 lagrange error bound\ncalculus - calculator allowed\nthe fifth - degree maclaurin polynomial for sin x…

Explanation:

Step1: Recall Lagrange error - bound formula

The Lagrange error - bound formula for the Maclaurin polynomial (P_n(x)) of a function (f(x)) is (|R_n(x)|\leq\frac{|f^{(n + 1)}(c)|}{(n+1)!}|x|^{n + 1}), where (c) is some number between (0) and (x), and (n) is the degree of the polynomial. Here, (n = 5), (f(x)=\sin x), and (x=\frac{\pi}{4}).

Step2: Find the ((n + 1)) - th derivative of (f(x))

The derivatives of (y = \sin x) follow a pattern: (f^{(1)}(x)=\cos x), (f^{(2)}(x)=-\sin x), (f^{(3)}(x)=-\cos x), (f^{(4)}(x)=\sin x), (f^{(5)}(x)=\cos x), (f^{(6)}(x)=-\sin x). The absolute - value of (f^{(6)}(c)=|-\sin c|\leq1) since (| \sin t|\leq1) for all real (t), and (c) is between (0) and (\frac{\pi}{4}).

Step3: Substitute values into the error - bound formula

We know that (n = 5), (x=\frac{\pi}{4}), and (|f^{(6)}(c)|\leq1). Then (|R_5(\frac{\pi}{4})|\leq\frac{|f^{(6)}(c)|}{(5 + 1)!}|\frac{\pi}{4}|^{6}). Substituting (|f^{(6)}(c)| = 1) and ((5+1)!=720) into the formula, we get (|R_5(\frac{\pi}{4})|\leq\frac{1}{720}(\frac{\pi}{4})^{6}).

Step4: Calculate the value

(\frac{1}{720}(\frac{\pi}{4})^{6}=\frac{\pi^{6}}{720\times4096}\approx\frac{961.389}{720\times4096}\approx\frac{961.389}{2949120}\approx0.000326).

Answer:

(0.000326)