For 0 ≤ t ≤ 2π, a particle is moving along the x-axis. The position of the particle is given by x(t) =
e
t
2 cos (
t
2
). The velocity of the particle is given by v(t) = 1
2
e
t
2
(cos (
t
2
) − sin (
t
2
)).
a) Find the time t at which the particle is farthest to the left. Justify your answer.
b) Is the speed of the particle increasing or decreasing at time t = π? Justify your answer.
c) The total distance traveled by the particle can be expressed as ∫ v(t) dt k
0
− ∫ v(t) dt 2π
k
. What is the
value of k? Justify your answer.