MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A cart moves along a straight track. Its velocity v(t), in cms−1, is defined for 0≤t≤10 seconds by the piecewise function
v(t)={3t−9,−t2+14t−36,0≤t<4,4≤t≤10.
Throughout [0,10] the velocity is continuous.
(1) Part 1: (a) Determine all times in [0,10] at which the cart is at rest.
(2) Part 2: (b) For each rest time, analyze the sign of v(t) immediately before and immediately after that time to decide whether the cart reverses direction or continues in the same direction after stopping. Clearly indicate all reasoning.
(3) Part 3: (c) Sketch a possible graph of v(t) that is consistent with the given information, labeling the rest times and indicating the sign of v(t) on each interval.