A particle moves along a straight line so that its velocity at time t seconds (0≤t≤6) is given by v(t)=e−0.2tsint meters per second. The particle is at the origin when t=0. A scientist records discrete velocity data every 1.5 seconds, shown in the table below.
Time t (s): 0 1.5 3.0 4.5 6.0
Velocity v(t) (m/s): 0 0.864 0.141\, \!-0.433 \!-0.220
(a) Explain why the displacement function s(t) satisfies s(t)=∫0tv(u)du.
(b) Using Simpson’s Rule with the subintervals defined by the data points, approximate the displacement of the particle at t=6 seconds.
(c) The scientist claims that the particle changes direction exactly once on the interval [0,6]. Justify or refute this claim based on the given velocity function.