Question

Consider the parameterization of the unit circle given by x=cos(4t^2−t), y=sin(4t^2−t) for t in (−InF, INF). Describe in words and sketch how the circle is traced out, and use this to answer the following questions.

(a) When is the parameterization tracing the circle out in a clockwise direction?
_____________
(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals.

(b) When is the parameterization tracing the circle out in a counter-clockwise direction?
__________

(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals.

C)Does the entire unit circle get traced by this parameterization?

(d) Give a time t at which the point being traced out on the circle is at (10):
t=_________



Ok, I graphed it in my graphing calculator, the circle seemed to be tracing itself for a while, but I still don't get how to write the intervals. Since it appeared to be just tracing itself clockwise, I typed in the first one (my homework is online)
(0,1),(1,0),(0,-1),(-1,0)
A message was shown saying that the left endpoint must be less than the right endpoint, so I'm not sure how to solve this one. I believe the answer for B is none, and I don't know what to put for the time, please explain this to me....

Answer 1

Consider the parameterization of the unit circle given by x = cos(ln(2t)), y = sin(ln(2t)) for t in (0,∞). De- scribe in words and sketch how the circle is traced out, and use this to answer the following questions.
SOLUTION
Let h(t) = ln(2t). Then h′(t) = 1 . The particle is moving
t
counterclockwise when h′(t) > 0, that is, when t is in (0,∞).
Any other values of t, t ≤ 0, are not in the domain of the func- tion, so the particle is never moving clockwise.
The full circle is traced out if h(t) = ln(2t) produces values over one period of the functions sin(t) and cos(t). In this case h(t) produces all the needed values, so the full circle must be produced.
To find a point where the parameterization is tracing out the point (1,0), we are looking for a t such that h(t) = 0. Such a value is t = 0.5.
Correct Answers:
• None
• (0,infinity) •A
• 0.5