A particle moves along a straight track so that its position at time (in seconds) is…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle moves along a straight track so that its position at time t (in seconds) is modeled by a twice–differentiable function s=f(t) (in meters). Selected values of f′(t) and f′′(t) are summarized verbally below.
• For 0<t<2, f′(t)>0 and f′′(t)<0.
• At t=2, f′(2)=0.
• For 2<t<4, f′(t)<0 while f′′(t)<0.
• At t=4, f′(4)=−3 and f′′(4)=0.
• For 4<t<6, f′(t)<0 and f′′(t)>0.
• At t=6, f′(6)=0.
• For t>6, f′(t)>0 and f′′(t)>0.
Answer the following, providing justification that refers explicitly to the given information.
(a) Describe the direction of motion of the particle on each of the intervals (0,2), (2,6), and (6,∞). Explain your reasoning based on f′(t).
(b) Determine the time intervals on which the particle is speeding up and the intervals on which it is slowing down. Explain how you use the signs of f′(t) and f′′(t) in your answer.
(c) At which of the given times, if any, does the particle attain a local minimum position? Justify your conclusion.