Maximum Rate of Change of X Coordinate of Cycloid

Theorem
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian coordinate plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.

Consider a point $P$ on the circumference of this circle.

Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.

The maximum rate of change of $x$ is $2 \mathbf v_0$, which happens when $P$ is at the top of the circle $C$.

Proof
From Rate of Change of Cartesian Coordinates of Cycloid, the rate of change of $x$ is given by:
 * $\dfrac {\d x} {\d t} = \mathbf v_0 \paren {1 - \cos \theta}$

This is a maximum when $1 - \cos \theta$ is a maximum.

That happens when $\cos \theta$ is at a minimum.

That happens when $\cos \theta = -1$.

That happens when $\theta = \pi, 3 \pi, \ldots$

That is, when $\theta = \paren {2 n + 1} \pi$ where $n \in \Z$.

That is, when $P$ is at the top of the circle $C$.

When $\cos \theta = -1$ we have:

Hence the result.