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: [tex]t \in [ -\infty, 1/6][/tex]
B) Counter-clockwise: [tex]t \in [ 1/6, \infty][/tex]
C) [tex]\theta \in [ 0, 2 \pi][/tex]
D) [tex]t= 0 \ or \ t=1/3[/tex]
For this exercise, the following equations were informed:
[tex]x= cos(3t^2-t)\\y= sin(3t^2-t)\\t \in [ -\infty, \infty][/tex]
taking the parameterization we have that:
[tex]\phi = 3t^2 - t= t(3t-1)[/tex]
As t increases from [tex][ -\infty, \infty][/tex] [tex]\phi[/tex] decreases, after 0 it becomes negative and after 1/3, goes on increasing. Also:
[tex]\frac{d\phi}{dt} = (6t-1)\\t= 1/6[/tex]
a) For clockwise begin [tex]\phi[/tex] must be decreasing, so:
[tex]t \in [ -\infty, 1/6][/tex]
b) For counter-clockwise [tex]\phi[/tex] must be increasing, so:
[tex]t \in [ 1/6, \infty][/tex]
c) Entise circle gets traced out. For we know:
[tex]x= cos\theta\\y= sin\theta[/tex]
Circle gets traced out once for:
[tex]\theta \in [ 0, 2 \pi][/tex]
d) When point (1, 0) so:
[tex]1= cos(3t^2-t)\\0= sin(3t^2-t)\\t= 0 \or \ t=1/3[/tex]
See more about parameterization at brainly.com/question/14770282
