The velocity of a particle is given by $v(t) = t \cos(π t)$.

1. For what value(s) of $t$ is the particle at rest?

   Step (a):

   When $t = 0$ or $\cos(πt) = 0$.

2. What is the acceleration of the particle at $t = 2$?

   Step (b):

   $a\left(t\right)=\cos\left(\pi t\right)-t\left(\pi\right)\sin\left(\pi t\right)$

3. Write a function $x(t)$ for the position if $x(0) = 5$.

   Hint (c):

   $s(t)=\int{v(t)dt}$

   Step (c):

   Use integration by parts. Let $f = f$ and $dg = \cos(πt)dt$.

   Step 1:

   $\int t\cos(\pi t)dt=\frac{t}{\pi}\sin(\pi t)-\frac{1}{\pi}\int\sin(\pi t)dt$

   Step 2:

   $=\frac{t}{\pi}\sin(\pi t)+\frac{1}{\pi^2}\cos(\pi t)+C$

   Step 3:

   Solve for $C$ given $x(0) = 5$.