3. Given the vector equation r(t) = (3+4 sint, -2 + 4 cos t), 0 < t < 2n, you should recognize that it represents a circle in the ry-plane. Identify the center, radius, initial and terminal points, and positive orientation (clockwise/counterclockwise) of the circle.
3. Given the vector equation r(t) = (3+4 sint, -2 + 4 cos t), 0 < t < 2n, you should recognize that it represents a circle in the ry-plane. Identify the center, radius, initial and terminal points, and positive orientation (clockwise/counterclockwise) of the circle.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**3. Given the vector equation**
\[
\vec{r}(t) = \langle 3 + 4 \sin t, \, -2 + 4 \cos t \rangle, \; 0 \leq t \leq 2\pi,
\]
**you should recognize that it represents a circle in the xy-plane. Identify the center, radius, initial and terminal points, and positive orientation (clockwise/counterclockwise) of the circle.**
**Explanation:**
- **Center of the Circle:** (3, -2)
- **Radius of the Circle:** 4
- **Initial Point:** When \( t = 0 \), \( \vec{r}(0) = \langle 3 + 4(0), \, -2 + 4(1) \rangle = (3, 2) \)
- **Terminal Point:** When \( t = 2\pi \), \( \vec{r}(2 \pi) = \langle 3 + 4(0), \, -2 + 4(1) \rangle = (3, 2) \)
- **Positive Orientation:** Counterclockwise
This vector equation describes the motion along a circle centered at (3, -2) with a radius of 4 units. The initial and terminal points coincide at (3, 2), and the movement follows a counterclockwise direction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7417eb4-c4f6-4b23-b299-1f887648c283%2F61a8c6fe-b8c3-4697-a2a8-c0d2c681013b%2Ff0rry_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**3. Given the vector equation**
\[
\vec{r}(t) = \langle 3 + 4 \sin t, \, -2 + 4 \cos t \rangle, \; 0 \leq t \leq 2\pi,
\]
**you should recognize that it represents a circle in the xy-plane. Identify the center, radius, initial and terminal points, and positive orientation (clockwise/counterclockwise) of the circle.**
**Explanation:**
- **Center of the Circle:** (3, -2)
- **Radius of the Circle:** 4
- **Initial Point:** When \( t = 0 \), \( \vec{r}(0) = \langle 3 + 4(0), \, -2 + 4(1) \rangle = (3, 2) \)
- **Terminal Point:** When \( t = 2\pi \), \( \vec{r}(2 \pi) = \langle 3 + 4(0), \, -2 + 4(1) \rangle = (3, 2) \)
- **Positive Orientation:** Counterclockwise
This vector equation describes the motion along a circle centered at (3, -2) with a radius of 4 units. The initial and terminal points coincide at (3, 2), and the movement follows a counterclockwise direction.
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