Past exam of the mathematical methods II course of the University of Oxford / 2022 / 4

Put $h=x-1/\sqrt{2}$ and $k=y-1/(2\sqrt{2})$. At the expansion point the exponent is one and the gradient is $(\sqrt{2}e,\sqrt{2}e)$. The first-order multivariable Taylor expansion isThe directional error of a first-order Taylor approximation is determined by the omitted terms, not just the equal first partial derivatives. The Hessian matrix at the expansion point is $e\left(\begin{array}{cc}4& 2\\ 2& 2\end{array}\right)$, soFor equal small steps along the coordinate axes, the leading error is $2e{h}^2$ along $x$ and $e{k}^2$ along $y$. The error grows faster along the $x$ axis, although the first-order change in the function is equal along the two axes.