Question

Consider the parameterization of the unit circle given by x=cos(3t^2-t), y=sin(3t^2-t) for t in (-infinity, infinity). 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)). Put 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)). Put the word None if there are no such intervals.
(c) Does the entire unit circle get traced by this parameterization?
A. yes
B. no
(d) Give a time t at which the point being traced out on the circle is at (10):
t= ___________?

Answer 1

Answer and Step-by-step explanation:

The answer is attached below

Answer 2

In this exercise we have to use the knowledge of parameterization and calculate the direction and direction of the equation, so we have to:

A) Clockwise: $t \in [ -\infty, 1/6]$

B) Counter-clockwise: $t \in [ 1/6, \infty]$

C) $\theta \in [ 0, 2 \pi]$

D) $t= 0 \ or \ t=1/3$

For this exercise, the following equations were informed:

$x= cos(3t^2-t)\\y= sin(3t^2-t)\\t \in [ -\infty, \infty]$

taking the parameterization we have that:

$\phi = 3t^2 - t= t(3t-1)$

As t increases from $[ -\infty, \infty]$  $\phi$ decreases, after 0 it becomes negative and after 1/3, goes on increasing. Also:

$\frac{d\phi}{dt} = (6t-1)\\t= 1/6$

a) For clockwise begin $\phi$ must be decreasing, so:

$t \in [ -\infty, 1/6]$

b) For counter-clockwise  $\phi$ must be increasing, so:

$t \in [ 1/6, \infty]$

c) Entise circle gets traced out. For we know:

$x= cos\theta\\y= sin\theta$

Circle gets traced out once for:

$\theta \in [ 0, 2 \pi]$

d) When point (1, 0) so:

$1= cos(3t^2-t)\\0= sin(3t^2-t)\\t= 0 \or \ t=1/3$

See more about parameterization at brainly.com/question/14770282