Vector Functions — Question 7

Question 7

Analyze the planar vector function $\vec r(t)=\left\langle t-\sin t,1-\cos t\right\rangle$ for $0\le t\le 4\pi$. Find the points for $t=0,\pi,2\pi,3\pi,4\pi$, identify repeated heights, and describe the geometric motion.

Question 7 – Solution

Strategy Evaluate at multiples of $\pi$ and use the periodicity of sine and cosine.

Key points The points are $$(0,0),\ (\pi,2),\ (2\pi,0),\ (3\pi,2),\ (4\pi,0).$$ The height oscillates between 0 and 2 while $x=t-\sin t$ increases from 0 to $4\pi$.

Geometry The curve is two arches of a cycloid, moving left to right, with cusps at $(0,0),(2\pi,0),(4\pi,0)$.

Verification Since $1-\cos t\in[0,2]$ and $x'(t)=1-\cos t\ge 0$, the description is consistent.