MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A particle moves along a straight line, and its position (in meters) at time t seconds is given by a differentiable function s(t). Selected values of the velocity v(t)=s′(t) are provided in the table below. Between the listed times, v(t) is either entirely positive or entirely negative as indicated.
t (s)
0
2
4
6
7
9
v(t) (m/s)
+
0
−
−
0
+
For t>9, the velocity remains positive until t=12.
(a) Determine all times in the interval 0<t<12 at which the particle changes direction. Justify your answer using the information in the table.
(b) For each time found in part (a), classify the corresponding position of the particle as a relative maximum or a relative minimum of s(t). Provide a clear explanation based on the sign of v(t).
(c) Suppose s(0)=5 m and s(12)=9 m. Without integrating, explain how the information in the table accounts for the net change in position from t=0 to t=12.