Consider the vector-valued function
$\mathbf r(t)=
\begin{cases}
\langle t^{2},\,2t\rangle, & t\le 0,
\langle \sin t,\,e^{t}-1\rangle, & t>0.
\end{cases}$
Determine whether r′(0) exists. If it exists, give its value; if not, state that it does not exist.