MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle moves along a straight track, and its velocity v (in meters per second) is a continuous, differentiable function of time t (in seconds) on the interval 0≤t≤10. The graph of v versus t has the following characteristics.
• v>0 and is decreasing on 0<t<4.
• v<0 and is decreasing on 4<t<6.
• v<0 and is increasing on 6<t<8.
• v>0 and is increasing on 8<t<10.
• v(4)=0 and v(8)=0.
Throughout the interval, the acceleration a is non-zero except at isolated points where v attains local extrema.
Answer the following questions about the motion of the particle.
(a) Describe the intervals on 0<t<10 during which the particle is moving to the right and the intervals during which it is moving to the left. Justify your answer in terms of the sign of the velocity.
(b) For each of the four subintervals listed in the description of v, determine whether the particle is speeding up or slowing down. Explain your reasoning by referencing the signs of both velocity and acceleration.
(c) The particle changes direction twice on the interval 0<t<10. Identify the times at which these changes occur, classify each as a left-to-right or right-to-left reversal, and provide justification.
(d) Sketch a possible graph of the position function s on the same time interval that is consistent with the information given. Label key features (extrema and inflection points) and explain how they correspond to features of v and a. (A precise scale is not required.)