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

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Answer 1 · 2014-09-19T19:57:16

By taking the derivatives,

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

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 $3$ .

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