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 $\frac{dx}{dt} = \frac{\sqrt{t+2}}{e^{t}}$ and $\frac{dy}{dt} = \sin^{2} t$. (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 $\frac{dx}{dt} = \frac{\sqrt{t+2}}{e^{t}}$ and $\frac{dy}{dt} = \sin^{2} t$.
(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 $t = 4$.