MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A small drone moves across a horizontal field. Its position in meters at time t seconds is given by the vector function r(t)=⟨3t2,t3⟩ for 0≤t≤4. (Both components are measured from the same origin at the southwest corner of the field.)
(a) Describe in words and with units the physical meaning of r(t), r′(t), and r′′(t) for this situation.
(b) Without performing any calculations, explain how the direction of the velocity vector changes as t increases from 0 to 4.
(c) At approximately what time does the drone’s path change from moving predominantly east–west to predominantly north–south? Justify your answer using appropriate derivatives or their interpretations.
(d) Interpret the sign of the x–component of r′′(t) on the interval 0<t<4 in terms of the drone’s horizontal motion.