2. The velocity vector of a particle moving in the $x y$-plane has components given by

$$\frac { d x } { d t } = 14 \cos \left( t ^ { 2 } \right) \sin \left( e ^ { t } \right) \text { and } \frac { d y } { d t } = 1 + 2 \sin \left( t ^ { 2 } \right) , \text { for } 0 \leq t \leq 1.5 .$$

At time $t = 0$, the position of the particle is $( - 2,3 )$.\\
(a) For $0 < t < 1.5$, find all values of $t$ at which the line tangent to the path of the particle is vertical.\\
(b) Write an equation for the line tangent to the path of the particle at $t = 1$.\\
(c) Find the speed of the particle at $t = 1$.\\
(d) Find the acceleration vector of the particle at $t = 1$.

\begin{center}
\begin{tabular}{ | c | | c | c | c | c | c | c | c | }
\hline
$t$ & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\
\hline
$P ( t )$ & 0 & 46 & 53 & 57 & 60 & 62 & 63 \\
\hline
\end{tabular}
\end{center}

\includegraphics[max width=\textwidth, alt={}, center]{1e6e06d3-cac5-4bbc-b4de-43b4650eb6cb-3_283_850_880_949}\\