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 (1,0):
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 believed the answer for B was none, but that's not it, I typed Yes for the third part and 0 for the last one since it started at that point...
Can someone please explain to me how to do the first and 2nd part?
To determine whether the parameterization is tracing the circle in a clockwise or counterclockwise direction, we need to analyze the changes in the angle as t varies. Let's break it down step by step:
Step 1: Solving for t
In order to specify the intervals in terms of t, we need to solve the given equation for t. However, since this equation involves trigonometric functions, we can't solve it analytically. Instead, we can analyze the behavior of the parameterization.
Step 2: Analyzing the behavior
Consider the function 4t^2 − t. As t approaches positive infinity, this function increases without bound. Similarly, as t approaches negative infinity, the function decreases without bound. This indicates that the parameterization traces out an infinite loop of the unit circle.
Step 3: Determining the direction of tracing
To determine the direction of tracing, we can look at the derivative of the angle with respect to t. In this case, the angle can be represented as arctan(y/x). Differentiating this angle with respect to t will give us the derivative.
By using the chain rule to differentiate arctan(y/x) with respect to t, we get:
d(arctan(y/x))/dt = (1/(1 + (y/x)^2))*((x dy/dt - y dx/dt)/(x^2))
Step 4: Analyzing the derivative
To analyze the values of the derivative, we can substitute the expressions for x and y into the derivative expression derived above. Simplifying the equation will give us the expression for the derivative in terms of t.
After simplification, we observe that the denominator of the derivative expression is always positive. This means that the sign of the derivative depends solely on the numerator of the derivative expression.
Step 5: Determining the direction intervals
To determine the intervals where the parameterization traces the circle in each direction, we need to analyze the sign of the numerator expression mentioned in step 4.
Since the numerator is dependent on specific values of t, we need to evaluate it at certain critical points. These critical points occur where the numerator changes sign or is equal to zero.
For this particular parameterization, one critical point occurs when x = 0. By substituting x = 0 into the equation for y, we can find the corresponding value of t.
Once we have the critical points, we can determine the intervals where the parameterization traces the circle clockwise or counterclockwise. In these intervals, the numerator expression must be negative or positive, respectively.
Step 6: Answering the questions
(a) When is the parameterization tracing the circle out in a clockwise direction?
To answer this question, we need to find the intervals where the numerator expression is negative. These intervals represent the clockwise direction of tracing.
(b) When is the parameterization tracing the circle out in a counterclockwise direction?
To answer this question, we need to find the intervals where the numerator expression is positive. These intervals represent the counterclockwise direction of tracing.
(c) Does the entire unit circle get traced by this parameterization?
Since the parameterization traces an infinite loop, it effectively traces the entire unit circle.
(d) Give a time t at which the point being traced out on the circle is at (1,0):
To find this time, we need to solve the equation x = 1 for t. Evaluating the given equation x = cos(4t^2 − t) at t = 0, we can find the initial point. By comparing the x-coordinate to 1, we can determine a specific time t.
It is important to note that the intervals and specific times mentioned above cannot be determined without directly evaluating the equation by numerical methods or using advanced mathematical techniques like inverse functions or transcendental equations solvers.