The position (in meters) of a particle moving along a horizontal line is given by the differentiable function
s(t)=tsint,0<t<4π,
where t is measured in seconds.
(a) Derive explicit expressions for the velocity v(t) and the acceleration a(t).
(b) Determine every value of t in (0,4π) at which the particle changes direction. Explain how your answer follows from the velocity function you obtained in part (a).
(c) Using the signs of v(t) and a(t), identify all open intervals on (0,4π) where the particle’s speed is increasing and all open intervals where the speed is decreasing. Provide clear justification for each interval cited.
(d) Compute the absolute maximum value of the speed on the closed interval [0,4π] and the time(s) at which it occurs. Justify that your answer indeed gives the absolute maximum.