A particle moves in the $xy$-plane so that its position at any time $t$, for $-\pi \leq t \leq \pi$, is given by $x(t) = \sin(3t)$ and $y(t) = 2t$. (a) Sketch the path of the particle in the $xy$-plane provided. Indicate the direction of motion along the path. (b) Find the range of $x(t)$ and the range of $y(t)$. (c) Find the smallest positive value of $t$ for which the $x$-coordinate of the particle is a local maximum. What is the speed of the particle at this time? (d) Is the distance traveled by the particle from $t = -\pi$ to $t = \pi$ greater than $5\pi$? Justify your answer.
A particle moves in the $xy$-plane so that its position at any time $t$, for $-\pi \leq t \leq \pi$, is given by $x(t) = \sin(3t)$ and $y(t) = 2t$.
(a) Sketch the path of the particle in the $xy$-plane provided. Indicate the direction of motion along the path.
(b) Find the range of $x(t)$ and the range of $y(t)$.
(c) Find the smallest positive value of $t$ for which the $x$-coordinate of the particle is a local maximum. What is the speed of the particle at this time?
(d) Is the distance traveled by the particle from $t = -\pi$ to $t = \pi$ greater than $5\pi$? Justify your answer.