009C Sample Final 2, Problem 8

Find $n$ such that the Maclaurin polynomial of degree $n$ of $f(x)=\cos(x)$ approximates $\cos {\frac {\pi }{3}}$ within 0.0001 of the actual value.

Foundations:
Taylor's Theorem
Let $f$ be a function whose $n+1$ th derivative exists on an interval $I$ and let $c$ be in $I.$
Then, for each $x$ in $I,$ there exists $z_{x}$ between $x$ and $c$ such that
$f(x)=f(c)+f'(c)(x-c)+{\frac {f''(c)}{2!}}(x-c)^{2}+\cdots +{\frac {f^{(n)}(c)}{n!}}(x-c)^{n}+R_{n}(x),$
where $R_{n}(x)={\frac {f^{n+1}(z_{x})}{(n+1)!}}(x-c)^{n+1}.$
Also, $\|R_{n}(x)\|\leq {\frac {\max \|f^{n+1}(z)\|}{(n+1)!}}\|(x-c)^{n+1}\|.$

Solution:

Step 1:
Using Taylor's Theorem, we have that the error in approximating $\cos {\frac {\pi }{3}}$ with
the Maclaurin polynomial of degree $n$ is $R_{n}{\bigg (}{\frac {\pi }{3}}{\bigg )}$ where
${\bigg \|}R_{n}{\bigg (}{\frac {\pi }{3}}{\bigg )}{\bigg \|}\leq {\frac {\max \|f^{n+1}(z)\|}{(n+1)!}}{\bigg \|}{\bigg (}{\frac {\pi }{3}}-0{\bigg )}^{n+1}{\bigg \|}.$

Step 2:

We note that

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |f^{n+1}(z)|=|\cos(z)|\leq 1} or

|f^{n+1}(z)|=|\sin(z)|\leq 1.

Therefore, we have

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\bigg |}R_{n}{\bigg (}{\frac {\pi }{3}}{\bigg )}{\bigg |}\leq {\frac {1}{(n+1)!}}{\bigg (}{\frac {\pi }{3}}{\bigg )}^{n+1}.}

Now, we have the following table.

$n$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \approx {\frac {1}{(n+1)!}}{\bigg (}{\frac {\pi }{3}}{\bigg )}^{n+1}}
$1$	$0.548311$
$2$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0.191396}
$3$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0.050107}
$4$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0.01049}
$5$	$0.00183$
$6$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0.000274}
$7$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0.0000358}