Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$
I'd like to know if the syntax is correct, how I solved this little task.
We have defined the Taylor-polynomial like that:
$$(T_{n, x_{0}}f)(x) = \sum_{k=0}^{n}\frac{1}{k!}f^{k}(x_{0})(x-x_{0})^{k}$$
$f^{0}(x) = \cos x$
$f^{1}(x) = -\sin x$
$f^{2}(x) = -\cos x$
$\Rightarrow$
$$(T_{2, 0}f)(x) = \frac{1}{0!}f^{0}(0)(x-0)^{0} + \frac{1}{1!}f^{1}(0)(x-0)^{1}+ \frac{1}{2!}f^{2}(0)(x-0)^{2}$$
$$(T_{2, 0}f)(x) = \frac{1}{1}1\cdot1 + \frac{1}{1}0x^{1} + \frac{1}{2}(-1)x^{2}$$
$$(T_{2, 0}f)(x) = 1 - \frac{1}{2}x^{2}$$
Please be as strict as possible, I don't want get minus for syntax in the exam because every single point is very important.
The syntax is mostly correct. The only improvement I see is to write the $k$-th derivative as $f^{(k)}$, not $f^k$. This also applies when $k$ is a known number.
There is an alternative notation with apostrophes. So you should write $f^{(0)}(x)$ or $f(x)$, $f^{(1)}(x)$ or $f'(x)$, $f^{(2)}(x)$ or $f''(x)$ and so forth. The second one is generally only used for the first few derivatives. I've seen up to three apostrophes.