4

4. [Maximum mark: 6]

A particle moves in a straight line such that its velocity, $v \text{ ms}^{-1}$, at time $t$ seconds is given by $v(t) = 1 + e^{-t} - e^{-\sin(2t)}$ for $0 \leq t \leq 2$.

(a) Find the velocity of the particle at $t = 2$. [1]

(b) Find the maximum velocity of the particle. [2]

(c) Find the acceleration of the particle at the instant it changes direction. [3]