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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-6

Vector Functions — Question 7

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Question 7

Analyze the planar vector function r→(t)=⟨t−sint,1−cost⟩\vec r(t)=\left\langle t-\sin t,1-\cos t\right\rangle for 0≤t≤4π0\le t\le 4\pi. Find the points for t=0,π,2π,3π,4πt=0,\pi,2\pi,3\pi,4\pi, identify repeated heights, and describe the geometric motion.

Original worksheet page 1: question and worked solution for 6-6-007
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Question 7 – Solution

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

See the diagram in the original worksheet below.

Key points The points are (0,0),(π,2),(2π,0),(3π,2),(4π,0).(0,0),\ (\pi,2),\ (2\pi,0),\ (3\pi,2),\ (4\pi,0). The height oscillates between 0 and 2 while x=t−sin⁡tx=t-\sin t increases from 0 to 4π4\pi.

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

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

Original worksheet page 2: question and worked solution for 6-6-007

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