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....
I typed 0 for the time, since it was at (0,1) when it started, and slected yes for part c, I have part A and B wrong, I don't know how to write the intervals from part A and I didn't see any counter-clockwise motion... please explain...
To determine the direction of tracing of the parameterization on the unit circle, we need to examine the changes in the angle as t varies. Recall that in a Cartesian coordinate system, the positive x-axis is associated with an angle of 0 degrees or 0 radians, and the positive y-axis is associated with an angle of 90 degrees or π/2 radians.
(a) To find the intervals where the parameterization traces the unit circle in a clockwise direction, we need to identify the values of t for which the angle decreases. In this case, since the angle is determined by the arctangent of y/x, we need to find the intervals where the derivative of the angle (with respect to t) is negative.
First, let's find the derivative of the angle:
θ = arctan(y/x) = arctan(sint/cost)
Differentiating with respect to t using the chain rule:
d(θ)/dt = [1/(1+(sint/cost)^2)] * [(-cost*sint - sint*cos(t))/cost^2]
Simplifying and factoring out -sint:
d(θ)/dt = [-sint * (1 + sint^2/cost^2)] / (1 + (sint/cost)^2)
To determine when this derivative is negative, we need to consider when the numerator (-sint * (1 + sint^2/cost^2)) is negative, and when the denominator (1 + (sint/cost)^2) is positive, excluding the singular points where cost = 0.
However, this is quite a complex expression, and evaluating it analytically is cumbersome. Instead, let's use numerical methods to approximate the solution.
We will use software or a graphing calculator to plot the derivative of the angle as a function of t. By observing the intervals where the derivative is negative, we can determine the intervals where the parameterization traces the unit circle in a clockwise direction. Based on your initial observation, it seems that the parameterization traces the circle clockwise for some interval, but we need to determine the exact interval.
(b) To find the intervals where the parameterization traces the unit circle in a counterclockwise direction, we need to identify the intervals where the angle increases. Similarly, we need to find the intervals where the derivative of the angle (with respect to t) is positive.
Using the same derivative expression as above, we can plot the derivative and determine the intervals where the derivative is positive.
(c) To determine if the entire unit circle is traced by this parameterization, we need to verify if the parameterization covers the entire range of angles. Since t ranges from negative infinity to positive infinity, it seems plausible that the entire unit circle is traced. However, we need to examine if there are any values of t that lead to duplicate points.
We can verify this by analyzing the behavior of cos(4t^2 − t) and sin(4t^2 − t). By considering the range of t, check if the x and y values repeat or if there are any gaps on the unit circle.
(d) To find a time t at which the point being traced out on the circle is at (10), we need to solve for t when x = cos(4t^2 − t) and y = sin(4t^2 − t) satisfy the condition x = 10 and y = 0.
Solve the equation cos(4t^2 − t) = 10 and sin(4t^2 − t) = 0 numerically using software or a graphing calculator, which can provide an approximation of t that satisfies these equations.
Note: The specific intervals for clockwise and counterclockwise directions and whether the entire unit circle is traced can only be determined accurately by using numerical analysis software or a graphing calculator.