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

Nov 2, 2014

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 $f \left(x\right) = \sin 2 x$.

By taking derivatives,

$f ' \left(x\right) = 2 \cos 2 x$
$f ' ' \left(x\right) = - 4 \sin 2 x$
$f ' ' ' \left(x\right) = - 8 \cos 2 x$
${f}^{\left(4\right)} \left(x\right) = 16 \sin 2 x$

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.