Past exam of the mathematical methods II course of the University of Oxford / 2013 / 4 / b

Let $s\ge 0$ be distance along the unit direction from part (a). The point is $\mathbf{r}(s)=(1,2)+s(1,2)/\sqrt{5}$, whose distance from the origin is $\sqrt{5}+s$. For finite increments of a logarithmic radial function, the exact change isThus an increase of $0.2$ requiresIf the intended calculation is the local directional derivative estimate, its first-order value is $s\simeq \Delta T/|\nabla T|=\boxed{\sqrt{5}/10}$. The latter approximates the exact finite distance rather than equaling it. Numerically they are about $0.235$ and $0.224$, respectively.