Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Polar Function Analysis**
We consider the function \( r = \frac{4}{\theta} \) defined for every \( \theta > 0 \). The following polar plot of the graph was created using the Maple command:
```
polarplot(4/theta, theta = 1/4*Pi..7*Pi);
```
**Polar Plot Description:**
The graph is a spiral that starts at the origin and moves outward, with decreasing density between the arcs as \( \theta \) increases. The polar plot represents the function \( r = \frac{4}{\theta} \), and the curve makes several complete loops as it progresses outward with θ from 0 to approximately \( 7\pi \).
**Tasks:**
a) **Cartesian Coordinates:**
Find the Cartesian coordinates of the point on the plot furthest from the origin. The coordinates of this point are:
- \( x = \) [Input Box] (write the exact value)
- \( y = \) [Input Box] (write the exact value)
b) **Asymptotic Behavior:**
Given that the graph of \( r = \frac{4}{\theta} \) approaches a horizontal asymptote \( y = a \), find the value of \( a \) and write it in the box below.
- \( a = 4 \)
c) **Explanation:**
Explain briefly how you computed the value of \( a \).
[Input Box for Explanation]](https://content.bartleby.com/qna-images/question/35de4fc2-6ad6-4f3e-a928-435a7e0716e8/5404558f-25fb-48ad-8970-c3a42c45e3e4/u988qbe_processed.png)
Transcribed Image Text:**Polar Function Analysis**
We consider the function \( r = \frac{4}{\theta} \) defined for every \( \theta > 0 \). The following polar plot of the graph was created using the Maple command:
```
polarplot(4/theta, theta = 1/4*Pi..7*Pi);
```
**Polar Plot Description:**
The graph is a spiral that starts at the origin and moves outward, with decreasing density between the arcs as \( \theta \) increases. The polar plot represents the function \( r = \frac{4}{\theta} \), and the curve makes several complete loops as it progresses outward with θ from 0 to approximately \( 7\pi \).
**Tasks:**
a) **Cartesian Coordinates:**
Find the Cartesian coordinates of the point on the plot furthest from the origin. The coordinates of this point are:
- \( x = \) [Input Box] (write the exact value)
- \( y = \) [Input Box] (write the exact value)
b) **Asymptotic Behavior:**
Given that the graph of \( r = \frac{4}{\theta} \) approaches a horizontal asymptote \( y = a \), find the value of \( a \) and write it in the box below.
- \( a = 4 \)
c) **Explanation:**
Explain briefly how you computed the value of \( a \).
[Input Box for Explanation]
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