Question

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

Answer 1

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`.