Example+4

4. A particle moves along the x-axis so that its velocity at time t, for 0 􀂅 t 􀂅 6, is given by a differentiable function v whose graph is shown above. The velocity is 0 at t 􀀠 0, t 􀀠 3, and t 􀀠 5, and the graph has horizontal tangents at t 􀀠 1 and t 􀀠 4. The areas of the regions bounded by the t-axis and the graph of v on the intervals 􀀾0, 3􀁀, 􀀾3, 5􀁀, and 􀀾5, 6􀁀 are 8, 3, and 2, respectively. At time t 􀀠 0, the particle is at x 􀀠 􀀐2. (a) For 0 􀂅 t 􀂅 6, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer. (b) For how many values of t, where 0 􀂅 t 􀂅 6, is the particle at x 􀀠 􀀐8 ? Explain your reasoning. (c) On the interval 2 􀀟 t 􀀟 3, is the speed of the particle increasing or decreasing? Give a reason for your answer. (d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.