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

Normalize the specified direction to $\hat{\mathbf{u}}=(1,1)/\sqrt{2}$. At distance $s\ge 0$ the squared radius is $(1+s/\sqrt{2}{)}^2+(2+s/\sqrt{2}{)}^2=5+3\sqrt{2}s+s^2$. The exact change $\Delta T=0.12$ therefore requires $5+3\sqrt{2}s+s^2=5e^{0.12}$. The positive root isFor comparison, the initial directional derivative is $(2/5,4/5)\cdot (1,1)/\sqrt{2}=3\sqrt{2}/5$. Its first-order distance estimate isThese give approximately $0.145$ and $0.141$. The exact expression accounts for the changing gradient along a finite straight displacement.