EXERCISE 1.4 Given the parametric curve P(t) = (2cos(t) + cos(2t), 2 sin(t) – sin(2t)) where 0 ≤t≤ 2π, find out the t values that corresponds to the curve's cusps. (Hint: Try to find the solutions of equation P′(t) = (0, 0).)
EXERCISE 1.4 Given the parametric curve P(t) = (2cos(t) + cos(2t), 2 sin(t) – sin(2t)) where 0 ≤t≤ 2π, find out the t values that corresponds to the curve's cusps. (Hint: Try to find the solutions of equation P′(t) = (0, 0).)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Exercise 1.4
**Problem Statement:**
Given the parametric curve \(\mathbf{P}(t) = \left(2\cos(t) + \cos(2t), 2\sin(t) - \sin(2t)\right)\) where \(0 \le t \le 2\pi\), find the \(t\) values that correspond to the curve’s cusps.
**Hint:** Try to find the solutions of the equation \(\mathbf{P}'(t) = (0, 0)\).
**Steps to Solve:**
1. **Compute the first derivative of the parametric curve:**
\[
\mathbf{P}'(t) = \left(\frac{d}{dt}\left( 2\cos(t) + \cos(2t) \right), \frac{d}{dt}\left( 2\sin(t) - \sin(2t) \right)\right).
\]
2. **Set the first derivative equal to \( (0, 0) \) and solve for \( t \):**
\[
\mathbf{P}'(t) = (0, 0).
\]
3. **Analyze the results to determine the \( t \) values:**
Identify which of the \( t \) values in the given range \([0, 2\pi]\) create cusps on the curve.
This approach ensures that finding the solution to the derivatives set to zero will identify the cusps for the parametric curve in question.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2Fcf932e8f-084c-4c9f-9d4d-35670fe7a9fc%2Fr7dlvbtp_processed.png&w=3840&q=75)
Transcribed Image Text:### Exercise 1.4
**Problem Statement:**
Given the parametric curve \(\mathbf{P}(t) = \left(2\cos(t) + \cos(2t), 2\sin(t) - \sin(2t)\right)\) where \(0 \le t \le 2\pi\), find the \(t\) values that correspond to the curve’s cusps.
**Hint:** Try to find the solutions of the equation \(\mathbf{P}'(t) = (0, 0)\).
**Steps to Solve:**
1. **Compute the first derivative of the parametric curve:**
\[
\mathbf{P}'(t) = \left(\frac{d}{dt}\left( 2\cos(t) + \cos(2t) \right), \frac{d}{dt}\left( 2\sin(t) - \sin(2t) \right)\right).
\]
2. **Set the first derivative equal to \( (0, 0) \) and solve for \( t \):**
\[
\mathbf{P}'(t) = (0, 0).
\]
3. **Analyze the results to determine the \( t \) values:**
Identify which of the \( t \) values in the given range \([0, 2\pi]\) create cusps on the curve.
This approach ensures that finding the solution to the derivatives set to zero will identify the cusps for the parametric curve in question.
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