MHB Taylor Polynomials for $e^{-4x}$ at $x=0$

[karush]

$\tiny{206.11.1.16-T}$
$\textsf{Find the nth-order Taylor polynomials
centered at 0, for $n=0, 1, 2.$}\\$
$$\displaystyle f(x)=e^{-4x}$$
$\textsf{using}\\$
$$P_n\left(x\right)
\approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$
$\textsf{n=0}\\$
\begin{align}
f^0(x)&\approx e^{-4x}\therefore f^0(0)\approx1 \\
P_0\left(x\right)&\approx\frac{1}{0!}x^{0}\approx 1
\end{align}
$\textsf{n=1}\\$
\begin{align}
f^1(x)&\approx-4e^{-4x}\therefore f^1(0)\approx -4 \\
P_1 f(x)&\approx \frac{1}{0!}x^{0}
+\frac{-4}{1!}x^{1}
\approx 1-4x

\end{align}
$\textsf{n=2}\\$
\begin{align}
f^2 (x)&= 16e^{-4x}\therefore f^2 (0)=16\\
P_2 f(x)&\approx \frac{1}{0!} x^{0}
+\frac{-4}{1!}x^{1}+\frac{16}{2!}x^{2}
\approx 1- 4x+8x^{2}
\end{align}

hopefully
not too hard kinda subtle tho

 

[I like Serena]

Hi karush! ;)

It looks fine to me, except that every $\approx$ should be a $=$.
The approximation only applies to $P_i(x) \approx f(x)$ for $i=1, 2, ...$, but that's not in your list.

 

[karush]

was wondering about that... some examples were confusing..