A particle moves along a straight line so that its position at time (measured in…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A particle moves along a straight line so that its position at time t (measured in seconds) is given by s(t) meters. Its velocity is described by the piecewise function
v(t)=s′(t)=⎩⎨⎧t2−6t+8,−21(t−3)2+1,0≤t<3,3≤t≤6.
Assume v is differentiable on (0,6) and s(0)=0.
(a) Determine the acceleration a(t)=s′′(t) for 0<t<3 and for 3<t<6. Express your answers in simplest form.
(b) Identify all times in the interval 0<t<6 at which the velocity attains a local maximum or minimum. Classify each as a maximum or minimum, supplying evidence from calculus.
(c) Using your results from part (b), state every time t at which the position function s may have a point of inflection. Fully justify each candidate by discussing the concavity of s immediately before and after the time in question.
(d) For each inflection point found in part (c), compute the corresponding position s(t) of the particle. Interpret physically what an inflection point of the position function means in the context of the particle’s motion.