For $0 \leq t \leq 12$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t ) = \cos \left( \frac { \pi } { 6 } t \right)$. The particle is at position $x = - 2$ at time $t = 0$. (a) For $0 \leq t \leq 12$, when is the particle moving to the left? (b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time $t = 0$ to time $t = 6$. (c) Find the acceleration of the particle at time $t$. Is the speed of the particle increasing, decreasing, or neither at time $t = 4$ ? Explain your reasoning. (d) Find the position of the particle at time $t = 4$.
For $0 \leq t \leq 12$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t ) = \cos \left( \frac { \pi } { 6 } t \right)$. The particle is at position $x = - 2$ at time $t = 0$.
(a) For $0 \leq t \leq 12$, when is the particle moving to the left?
(b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time $t = 0$ to time $t = 6$.
(c) Find the acceleration of the particle at time $t$. Is the speed of the particle increasing, decreasing, or neither at time $t = 4$ ? Explain your reasoning.
(d) Find the position of the particle at time $t = 4$.