A particle moves along a line so that its position (in meters) at time seconds is…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A particle moves along a line so that its position s (in meters) at time t seconds is given by the piecewise function
s(t)=⎩⎨⎧t3−6t2+9t,4,4+t−2,0≤t<2,t=2,2<t≤5.
Analyze the particle’s motion on the interval 0≤t≤5.
(1) Part 1: (a) Determine whether s is continuous on [0,5]. Be sure to address the behavior at t=2.
(2) Part 2: (b) For 0<t<2, obtain an expression for the average rate of change of s over the interval [t,2] and compute its limit as t→2−.
(3) Part 3: (c) For 2<t≤5, obtain an expression for the average rate of change of s over the interval [2,t] and compute its limit as t→2+.
(4) Part 4: (d) Using the results from parts (b) and (c), determine whether the particle has a well-defined instantaneous velocity at t=2. Justify your answer thoroughly.
(5) Part 5: (e) Identify every t in (0,5) at which the instantaneous velocity fails to exist. Provide justification for each point identified.