Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 14

Differentiate the cycloid parametrization:The half-angle simplification is valid throughout $0<t<2\pi$, where $1-\cos t>0$. In particular the horizontal coordinate increases strictly along the arch.

The cycloid has limiting endpoints $(0,0)$ and $(2\pi a,0)$, with its highest point $(\pi a,2a)$ at $t=\pi$. The transformation $t\mapsto 2\pi -t$ sends $x$ to $2\pi a-x$ and preserves $y$, so the arch is symmetric about $x=\pi a$. Its slope decreases from $+\infty$ through zero to $-\infty$, giving vertical limiting tangent lines at the cusps and a horizontal tangent line at the top. More explicitly,so the arch is a concave function throughout its interior. The endpoint cusps are limiting points, excluded by the open parameter interval.

The original PDF specifies slope $\sqrt{3}$; the transcription's $\sqrt{-1}$ is erroneous. By part (a), $\cot (t/2)=\sqrt{3}$. Since $t/2\in (0,\pi )$ and cotangent is strictly decreasing there, the unique parameter value is $t=\pi /3$. ThereforeThe tangent line drawn in the preceding figure has the required positive slope.

Because $dx/dt>0$ along the arch, its area above the baseline is the definite integral $\int ydx$. Changing to the parameter givesAdding or removing the two endpoint points does not change the area. This is the area formula in area and length of a cycloid arch.

The arc length follows from the speed of the parametrization:Here $\sin (t/2)>0$ inside the given interval. ThereforeThe result scales as a length, while the preceding area scales as $a^2$, as required by dimensional analysis.