MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A particle moves along a straight line. Its position at time t seconds is given by the function s whose graph is described below.
• For 0≤t≤2, the graph of s is a straight line segment with slope 4 joining (0,0) to (2,8).
• For 2<t≤5, the graph is a quarter of a circle centered at (5,8) with radius 3 and lies below the center.
• For t>5, the graph is a smooth curve that decreases, has a single point of inflection at t=6, and approaches the horizontal line s=5 as t→∞.
Velocity is defined by v(t)=s′(t) and acceleration by a(t)=s′′(t).
(a) Construct a piecewise description or sketch of the velocity function v(t) for all t>0. Your answer must indicate exact numerical values of v on intervals where it is constant and accurately reflect any discontinuities or non‐existence points.
(b) Identify all open time intervals on which the particle is speeding up. Justify your answer using both v(t) and a(t).
(c) Determine the exact time(s) at which the acceleration changes sign. Explain how each sign change relates to the geometric properties of the position graph described above.