A particle moves along the $x$-axis so that its velocity at time $t$ is given by $$v(t) = -(t+1)\sin\left(\frac{t^2}{2}\right)$$ At time $t = 0$, the particle is at position $x = 1$. (a) Find the acceleration of the particle at time $t = 2$. Is the speed of the particle increasing at $t = 2$? Why or why not? (b) Find all times $t$ in the open interval $0 < t < 3$ when the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle from time $t = 0$ until time $t = 3$. (d) During the time interval $0 \leq t \leq 3$, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.
A particle moves along the $x$-axis so that its velocity at time $t$ is given by
$$v(t) = -(t+1)\sin\left(\frac{t^2}{2}\right)$$
At time $t = 0$, the particle is at position $x = 1$.\\
(a) Find the acceleration of the particle at time $t = 2$. Is the speed of the particle increasing at $t = 2$? Why or why not?\\
(b) Find all times $t$ in the open interval $0 < t < 3$ when the particle changes direction. Justify your answer.\\
(c) Find the total distance traveled by the particle from time $t = 0$ until time $t = 3$.\\
(d) During the time interval $0 \leq t \leq 3$, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.