3) Considering the similarity of the right triangles ACL and ALM in Figure 4, and recalling the geometric meaning of the derivative, verify that the value of the ordinate $d$ of the centre of the wheel remains constant during motion. Therefore, it seems to the cyclist that they are moving on a flat surface. [Figure] Figure 4 \footnotetext{${ } ^ { 1 }$ In general, the length of the arc of a curve with equation $y = \varphi ( x )$ between the abscissae $x _ { 1 }$ and $x _ { 2 }$ is given by $\int _ { x _ { 1 } } ^ { x _ { 2 } } \sqrt { 1 + \left( \varphi ^ { \prime } ( x ) \right) ^ { 2 } } d x$. }
Ministry of Education, University and Research
The graph of the function: $$f ( x ) = \frac { 2 } { \sqrt { 3 } } - \frac { e ^ { x } + e ^ { - x } } { 2 } , \quad \text { for } x \in \left[ - \frac { \ln ( 3 ) } { 2 } ; \frac { \ln ( 3 ) } { 2 } \right]$$ if replicated several times, can also represent the profile of a platform suitable for being traversed by a bicycle with very particular wheels, having the shape of a regular polygon.
3) Considering the similarity of the right triangles ACL and ALM in Figure 4, and recalling the geometric meaning of the derivative, verify that the value of the ordinate $d$ of the centre of the wheel remains constant during motion. Therefore, it seems to the cyclist that they are moving on a flat surface.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4247a904-6fcc-4fd2-a50f-9c2a64781d61-2_887_920_1674_520}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
\footnotetext{${ } ^ { 1 }$ In general, the length of the arc of a curve with equation $y = \varphi ( x )$ between the abscissae $x _ { 1 }$ and $x _ { 2 }$ is given by $\int _ { x _ { 1 } } ^ { x _ { 2 } } \sqrt { 1 + \left( \varphi ^ { \prime } ( x ) \right) ^ { 2 } } d x$.
}\section*{Ministry of Education, University and Research}
The graph of the function:
$$f ( x ) = \frac { 2 } { \sqrt { 3 } } - \frac { e ^ { x } + e ^ { - x } } { 2 } , \quad \text { for } x \in \left[ - \frac { \ln ( 3 ) } { 2 } ; \frac { \ln ( 3 ) } { 2 } \right]$$
if replicated several times, can also represent the profile of a platform suitable for being traversed by a bicycle with very particular wheels, having the shape of a regular polygon.