At time $t$, the position of a particle moving in the $xy$-plane is given by the parametric functions $( x ( t ) , y ( t ) )$, where $\frac { d x } { d t } = t ^ { 2 } + \sin \left( 3 t ^ { 2 } \right)$. The graph of $y$, consisting of three line segments, is shown in the figure above. At $t = 0$, the particle is at position $( 5,1 )$. (a) Find the position of the particle at $t = 3$. (b) Find the slope of the line tangent to the path of the particle at $t = 3$. (c) Find the speed of the particle at $t = 3$. (d) Find the total distance traveled by the particle from $t = 0$ to $t = 2$.
At time $t$, the position of a particle moving in the $xy$-plane is given by the parametric functions $( x ( t ) , y ( t ) )$, where $\frac { d x } { d t } = t ^ { 2 } + \sin \left( 3 t ^ { 2 } \right)$. The graph of $y$, consisting of three line segments, is shown in the figure above. At $t = 0$, the particle is at position $( 5,1 )$.\\
(a) Find the position of the particle at $t = 3$.\\
(b) Find the slope of the line tangent to the path of the particle at $t = 3$.\\
(c) Find the speed of the particle at $t = 3$.\\
(d) Find the total distance traveled by the particle from $t = 0$ to $t = 2$.