Let $f : (0, \infty)^2 \to \mathbb{R}$ with $f(x,y) := x^y$.
How can one find out an approximation to $1.05^{1.02}$ with Taylor's Theorem, i.e. $T_2f(a;(x,y))$ in the point $a = (1,1)$.
And how can one calculate, that the absolute error of the approximation above is smaller than $7 \cdot 10^{-5}$?
For the first question I thought about using the "Symmetry of second derivatives". Apparently we are allowed to use the following theorem:
Let $D \in \mathbb{R}^n$ be an open set and $f \in C^k (D;\mathbb{R})$.
Further, let $i_1,..,i_k \in \text{{1,...n}}$.
Then we get
$$\frac{\partial^k f}{\partial x_{ik} \cdot \cdot \cdot \partial x_{i1}}(x) = \frac{\partial^k f}{\partial x_{\sigma(ik)} \cdot \cdot \cdot \partial x_{\sigma (i1)}} (x) $$
for all $x \in D$ and every permutation $\sigma:$ {$i_1,...,i_k$} $\to$ {$i_1,...,i_k$}.
I don't know how to continue though..