A particle moves along the $x$-axis so that its velocity $v$ at time $t \geq 0$ is given by $v ( t ) = \sin \left( t ^ { 2 } \right)$. The graph of $v$ is shown above for $0 \leq t \leq \sqrt { 5 \pi }$. The position of the particle at time $t$ is $x ( t )$ and its position at time $t = 0$ is $x ( 0 ) = 5$. (a) Find the acceleration of the particle at time $t = 3$. (b) Find the total distance traveled by the particle from time $t = 0$ to $t = 3$. (c) Find the position of the particle at time $t = 3$. (d) For $0 \leq t \leq \sqrt { 5 \pi }$, find the time $t$ at which the particle is farthest to the right. Explain your answer.
& 9 & 2 & 3 & 1
A particle moves along the $x$-axis so that its velocity $v$ at time $t \geq 0$ is given by $v ( t ) = \sin \left( t ^ { 2 } \right)$. The graph of $v$ is shown above for $0 \leq t \leq \sqrt { 5 \pi }$. The position of the particle at time $t$ is $x ( t )$ and its position at time $t = 0$ is $x ( 0 ) = 5$.
(a) Find the acceleration of the particle at time $t = 3$.
(b) Find the total distance traveled by the particle from time $t = 0$ to $t = 3$.
(c) Find the position of the particle at time $t = 3$.
(d) For $0 \leq t \leq \sqrt { 5 \pi }$, find the time $t$ at which the particle is farthest to the right. Explain your answer.