A particle moves along the $x$-axis so that its velocity $v$ at any time $t$, for $0 \leq t \leq 16$, is given by $v(t) = e^{2\sin t} - 1$. At time $t = 0$, the particle is at the origin. (a) On the axes provided, sketch the graph of $v(t)$ for $0 \leq t \leq 16$. (b) During what intervals of time is the particle moving to the left? Give a reason for your answer. (c) Find the total distance traveled by the particle from $t = 0$ to $t = 4$. (d) Is there any time $t$, $0 < t \leq 16$, at which the particle returns to the origin? Justify your answer.
A particle moves along the $x$-axis so that its velocity $v$ at any time $t$, for $0 \leq t \leq 16$, is given by $v(t) = e^{2\sin t} - 1$. At time $t = 0$, the particle is at the origin.\\
(a) On the axes provided, sketch the graph of $v(t)$ for $0 \leq t \leq 16$.\\
(b) During what intervals of time is the particle moving to the left? Give a reason for your answer.\\
(c) Find the total distance traveled by the particle from $t = 0$ to $t = 4$.\\
(d) Is there any time $t$, $0 < t \leq 16$, at which the particle returns to the origin? Justify your answer.