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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![## Matching Differential Equations with Direction Fields
### Task
**Equation:**
\[ y'(t) = -\frac{y}{t} \]
**Question:**
Which of the following direction fields matches the given equation?
### Options:
Four graphs labeled A, B, C, and D are provided to illustrate potential direction fields. Each graph is a direction field represented on a coordinate system with pink arrows indicating the slope at different points.
### Graph Analysis
- **Graph A:**
- Displays a pattern of arrows that change direction rapidly around a central point.
- Arrows point inward in a spiral-like fashion towards the origin, suggesting a varied rate of change in both the x and y directions.
- **Graph B:**
- Features arrows arranged in a radial pattern, pointing inwards towards the center (origin).
- This pattern indicates a uniform, central approach, matching the structure of many standard differential equations which have symmetrical solutions around the origin.
- **Graph C:**
- Arrows in this graph seem to spiral outward from the center.
- This indicates a set of solutions moving away from the origin, unlike what would be expected from the given differential equation.
- **Graph D:**
- Shows arrows directing outwards in a uniform manner.
- This pattern suggests a steadily increasing or decreasing function, which does not directly match the given equation's divisive structure.
### Conclusion
Based on the visual analysis, **Graph B** is the most likely candidate that matches the differential equation \( y'(t) = -\frac{y}{t} \). The inward radial pattern of the arrows suggests solutions that proportionally decrease with respect to the t coordinate, fulfilling the equation's condition.
**Answer: Graph B**
---
This educational content guides students through the process of comparing differential equations with their corresponding direction fields, encouraging visual and analytical skills essential in understanding and solving differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F918808b3-3bab-4c24-a49a-fe439aaeecc0%2F16f1094c-b7e9-47e8-880c-554b6e139e3d%2Fp5xrf1_processed.png&w=3840&q=75)
Transcribed Image Text:## Matching Differential Equations with Direction Fields
### Task
**Equation:**
\[ y'(t) = -\frac{y}{t} \]
**Question:**
Which of the following direction fields matches the given equation?
### Options:
Four graphs labeled A, B, C, and D are provided to illustrate potential direction fields. Each graph is a direction field represented on a coordinate system with pink arrows indicating the slope at different points.
### Graph Analysis
- **Graph A:**
- Displays a pattern of arrows that change direction rapidly around a central point.
- Arrows point inward in a spiral-like fashion towards the origin, suggesting a varied rate of change in both the x and y directions.
- **Graph B:**
- Features arrows arranged in a radial pattern, pointing inwards towards the center (origin).
- This pattern indicates a uniform, central approach, matching the structure of many standard differential equations which have symmetrical solutions around the origin.
- **Graph C:**
- Arrows in this graph seem to spiral outward from the center.
- This indicates a set of solutions moving away from the origin, unlike what would be expected from the given differential equation.
- **Graph D:**
- Shows arrows directing outwards in a uniform manner.
- This pattern suggests a steadily increasing or decreasing function, which does not directly match the given equation's divisive structure.
### Conclusion
Based on the visual analysis, **Graph B** is the most likely candidate that matches the differential equation \( y'(t) = -\frac{y}{t} \). The inward radial pattern of the arrows suggests solutions that proportionally decrease with respect to the t coordinate, fulfilling the equation's condition.
**Answer: Graph B**
---
This educational content guides students through the process of comparing differential equations with their corresponding direction fields, encouraging visual and analytical skills essential in understanding and solving differential equations.
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