How do you find the Taylor remainder term R_n(x;3) for f(x)=e^(4x)?

Sep 19, 2014

By taking the derivatives,

$f \left(x\right) = {e}^{4 x}$
$f ' \left(x\right) = 4 {e}^{4 x}$
$f ' ' \left(x\right) = {4}^{2} {e}^{4 x}$
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${f}^{\left(n + 1\right)} \left(x\right) = {4}^{n + 1} {e}^{4 x}$

So,
R_n(x;3)={f^{(n+1)}(z)}/{(n+1)!}(x-3)^{n+1}={4^{n+1}e^{4z}}/{(n+1)!}(x-3)^{n+1},
where $z$ is between $x$ and $3$.