Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 3A / Solution

$s(t)={\int }^t|\dot{x}(u)|du$, $t=dx/ds$, $\kappa =|dt/ds|$, and $n={\kappa }^{-1}dt/ds$. Differentiating $t\cdot t=1$ proves $t\perp n$. Here $|\dot{x}|=2e^t$ and

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