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-1- -1 -1- Which of the following equations would produce the direction field above? dy da dy dr dy dr dy 1 y dr

-1- -1 -1- Which of the following equations would produce the direction field above? dy da dy dr dy dr dy 1 y dr

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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**Direction Field Identification for Differential Equations** *Educational Content for Understanding Differential Equations and Direction Fields* --- **Description:** This lesson is designed to help students understand how to determine which differential equation corresponds to a given direction field. The image shows a direction field composed of short line segments, symbolizing the slopes of solutions to a differential equation at various points on the xy-plane. **Diagram Explanation:** The direction field illustrated in the diagram consists of small red lines representing the slope of the solution curve of a differential equation at the specific points on the plane. The field is plotted with respect to the coordinate axes marked from -2 to 2 on both the x-axis and y-axis. --- **Problem:** Identify which of the following differential equations would produce the direction field depicted in the above diagram: 1. \( \frac{dy}{dx} = x^2 \cdot y^2 \) 2. \( \frac{dy}{dx} = x \cdot y^2 \) 3. \( \frac{dy}{dx} = x \cdot y \) 4. \( \frac{dy}{dx} = x^2 \cdot y \) (A) \( \frac{dy}{dx} = x^2 \cdot y^2 \) (B) \( \frac{dy}{dx} = x \cdot y^2 \) (C) \( \frac{dy}{dx} = x \cdot y \) (D) \( \frac{dy}{dx} = x^2 \cdot y \) --- **Objective:** Students are expected to analyze the provided direction field and determine which proposed differential equation matches the given field. **Methodology:** 1. **Analyze the Given Direction Field:** Observe the pattern of the slopes across different regions of the plane. This includes recognizing how the slopes change as you move along the x-axis and y-axis. 2. **Compare to Differential Equations:** - For \( \frac{dy}{dx} = x^2 \cdot y^2 \): Check if the slope is positive and increasing in proportion to \( x^2 \) and \( y^2 \). - For \( \frac{dy}{dx} = x \cdot y^2 \): Observe if the dependency is more on y squared and linearly on x. - For \( \frac{dy}{dx} = x
Transcribed Image Text:**Direction Field Identification for Differential Equations** *Educational Content for Understanding Differential Equations and Direction Fields* --- **Description:** This lesson is designed to help students understand how to determine which differential equation corresponds to a given direction field. The image shows a direction field composed of short line segments, symbolizing the slopes of solutions to a differential equation at various points on the xy-plane. **Diagram Explanation:** The direction field illustrated in the diagram consists of small red lines representing the slope of the solution curve of a differential equation at the specific points on the plane. The field is plotted with respect to the coordinate axes marked from -2 to 2 on both the x-axis and y-axis. --- **Problem:** Identify which of the following differential equations would produce the direction field depicted in the above diagram: 1. \( \frac{dy}{dx} = x^2 \cdot y^2 \) 2. \( \frac{dy}{dx} = x \cdot y^2 \) 3. \( \frac{dy}{dx} = x \cdot y \) 4. \( \frac{dy}{dx} = x^2 \cdot y \) (A) \( \frac{dy}{dx} = x^2 \cdot y^2 \) (B) \( \frac{dy}{dx} = x \cdot y^2 \) (C) \( \frac{dy}{dx} = x \cdot y \) (D) \( \frac{dy}{dx} = x^2 \cdot y \) --- **Objective:** Students are expected to analyze the provided direction field and determine which proposed differential equation matches the given field. **Methodology:** 1. **Analyze the Given Direction Field:** Observe the pattern of the slopes across different regions of the plane. This includes recognizing how the slopes change as you move along the x-axis and y-axis. 2. **Compare to Differential Equations:** - For \( \frac{dy}{dx} = x^2 \cdot y^2 \): Check if the slope is positive and increasing in proportion to \( x^2 \) and \( y^2 \). - For \( \frac{dy}{dx} = x \cdot y^2 \): Observe if the dependency is more on y squared and linearly on x. - For \( \frac{dy}{dx} = x
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