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Match the following equation with the direction field. y'(t) = - !

Match the following equation with the direction field. y'(t) = - !

Calculus: Early Transcendentals
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.
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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