For $t \geq 0$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 2$, the particle is at position $( 1,5 )$. It is known that $\frac { d x } { d t } = \frac { \sqrt { t + 2 } } { e ^ { t } }$ and $\frac { d y } { d t } = \sin ^ { 2 } t$. (a) Is the horizontal movement of the particle to the left or to the right at time $t = 2$ ? Explain your answer. Find the slope of the path of the particle at time $t = 2$. (b) Find the $x$-coordinate of the particle's position at time $t = 4$. (c) Find the speed of the particle at time $t = 4$. Find the acceleration vector of the particle at time $t = 4$. (d) Find the distance traveled by the particle from time $t = 2$ to $t = 4$.
For $t \geq 0$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 2$, the particle is at position $( 1,5 )$. It is known that $\frac { d x } { d t } = \frac { \sqrt { t + 2 } } { e ^ { t } }$ and $\frac { d y } { d t } = \sin ^ { 2 } t$.
(a) Is the horizontal movement of the particle to the left or to the right at time $t = 2$ ? Explain your answer. Find the slope of the path of the particle at time $t = 2$.
(b) Find the $x$-coordinate of the particle's position at time $t = 4$.
(c) Find the speed of the particle at time $t = 4$. Find the acceleration vector of the particle at time $t = 4$.
(d) Find the distance traveled by the particle from time $t = 2$ to $t = 4$.