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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![### Exploring Polar Coordinates: Visualizing Regions
**Objective:**
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions:
\[ r \geq 3, \quad 0 \leq \theta \leq \pi \]
**Detailed Graph Explanation:**
The image contains four separate plots, each illustrating a different region on the Cartesian plane. The focus is on the area determined by the polar coordinate conditions specified.
1. **First Plot:**
- A semicircle with radius 3, centered at the origin, and lying above the x-axis.
- The semicircle is shaded in blue, indicating that points satisfying the condition \( r \geq 3 \) and \( 0 \leq \theta \leq \pi \) are included.
- The radius extends from the origin to x = 3 on both the positive and negative sides.
2. **Second Plot:**
- The upper half of the graph is shaded in blue, extending from the x-axis upwards indefinitely, excluding the semicircle of radius 3.
- This represents the area outside the semicircle satisfying the condition \( r \geq 3 \), covering all points above the semicircle within \( 0 \leq \theta \leq \pi \).
3. **Third Plot:**
- Similar to the first plot but inverted, this plot highlights the semicircle lying below the x-axis.
- It remains unshaded, indicating exclusion of this region when considering the polar coordinate conditions given.
4. **Fourth Plot:**
- The semicircle is shaded in blue, like the first plot, but this visualization includes the region below the x-axis.
- This demonstrates the scenarios where the semicircle is fully included under discussed conditions as if the graph focused on other specified ranges of \(\theta\).
Overall, each diagram helps visualize how polar coordinates define specific regions based on the conditions set for \( r \) and \( \theta \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff42c59a5-908c-4bcd-9e36-90d243a82d72%2Fccda1eff-08e0-4a0f-b0e7-54d9f2dbd9dc%2Fjktmji_processed.png&w=3840&q=75)
Transcribed Image Text:### Exploring Polar Coordinates: Visualizing Regions
**Objective:**
Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions:
\[ r \geq 3, \quad 0 \leq \theta \leq \pi \]
**Detailed Graph Explanation:**
The image contains four separate plots, each illustrating a different region on the Cartesian plane. The focus is on the area determined by the polar coordinate conditions specified.
1. **First Plot:**
- A semicircle with radius 3, centered at the origin, and lying above the x-axis.
- The semicircle is shaded in blue, indicating that points satisfying the condition \( r \geq 3 \) and \( 0 \leq \theta \leq \pi \) are included.
- The radius extends from the origin to x = 3 on both the positive and negative sides.
2. **Second Plot:**
- The upper half of the graph is shaded in blue, extending from the x-axis upwards indefinitely, excluding the semicircle of radius 3.
- This represents the area outside the semicircle satisfying the condition \( r \geq 3 \), covering all points above the semicircle within \( 0 \leq \theta \leq \pi \).
3. **Third Plot:**
- Similar to the first plot but inverted, this plot highlights the semicircle lying below the x-axis.
- It remains unshaded, indicating exclusion of this region when considering the polar coordinate conditions given.
4. **Fourth Plot:**
- The semicircle is shaded in blue, like the first plot, but this visualization includes the region below the x-axis.
- This demonstrates the scenarios where the semicircle is fully included under discussed conditions as if the graph focused on other specified ranges of \(\theta\).
Overall, each diagram helps visualize how polar coordinates define specific regions based on the conditions set for \( r \) and \( \theta \).
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