How do you find the remainder term $R_3(x;1)$ for $f(x)=sin(2x)$ ?

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Answer 1 · 2014-11-02T11:42:41

Remainder Term of Taylor Series

$R_n(x;c)={f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}$ ,

where $z$ is a number between $x$ and $c$ .

Let us find $R_3(x;1)$ for $f(x)=sin2x$ .

By taking derivatives,

$f'(x)=2cos2x$
$f''(x)=-4sin2x$
$f'''(x)=-8cos2x$
$f^{(4)}(x)=16sin2x$

So, we have

$R_3(x;1)={16sin2z}/{4!}(x-1)^4$ ,

where $z$ is a number between $x$ and $1$ .

I hope that this was helpful.

How do you find the remainder term R_3(x;1)R_3(x;1) for f(x)=sin(2x)f(x)=sin(2x)?