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Which of the following graphs is the slope field for d 11 || \-|-~ \ || | | \-2+- \\ -37-11 -3 -2 -1 3 N W 2 1+~ -1 -2~// -3 = x³ ? = 1 1/2+ 17-1 -3-218 | | /-^|- / || 111-2-/ }} tri -31-201 || \-2|/ / 111-2+// -3

Which of the following graphs is the slope field for d 11 || \-|-~ \ || | | \-2+- \\ -37-11 -3 -2 -1 3 N W 2 1+~ -1 -2~// -3 = x³ ? = 1 1/2+ 17-1 -3-218 | | /-^|- / || 111-2-/ }} tri -31-201 || \-2|/ / 111-2+// -3

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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### Slope Fields and Differential Equations #### Question: Which of the following graphs is the slope field for \(\frac{dy}{dx} = x^3\)? #### Explanation: A slope field is a graphical representation of the possible slopes of a differential equation. Each small line segment in a slope field indicates the slope of the solution curve at that point. For the given differential equation \(\frac{dy}{dx} = x^3\), we want to determine which graph correctly represents this slope field. The four provided graphs each consist of a coordinate system with the x and y axes. They are as follows: 1. **Top Left Graph:** - At \((x = -3)\), the slopes are negative. - As x progresses towards \(0\) from \(-3\), the slopes become less steep but remain negative. - For \(x = 0\), the slopes are horizontal (indicating a slope of 0). - For \(x > 0\), the slopes are positive, becoming steeper as x increases. 2. **Top Right Graph:** - Slopes at the left and right of x-axis symmetrically appear similar to reversed direction. - For \(x = -3\) and \(x = 3\), they are positive with increasing steepness moving outward. 3. **Bottom Left Graph:** - The slopes gradually become steeper as we move away from the origin. - The pattern matches the curve where for \(x = -3\) slopes are large negative, and for \(x = 0\) slopes are 0. - As \(x\) increases to 3, slopes continuously increase in positive steepness. 4. **Bottom Right Graph:** - The slopes pattern does not change similarly across x-axis. - Central lines appear to be uniform, revealing quite different structure. ### Conclusion: The **Bottom Left Graph** is the correct slope field for the differential equation \(\frac{dy}{dx} = x^3\). This is because: - It shows the behavior where slopes become increasingly positive as \(x\) increases positively, and they become increasingly negative as \(x\) increases negatively. - The slopes are horizontal at \(x = 0\). ### Visualization Tips: - By observing the steepness and direction of short lines for each value of \(x\): - Horizontal lines at
Transcribed Image Text:### Slope Fields and Differential Equations #### Question: Which of the following graphs is the slope field for \(\frac{dy}{dx} = x^3\)? #### Explanation: A slope field is a graphical representation of the possible slopes of a differential equation. Each small line segment in a slope field indicates the slope of the solution curve at that point. For the given differential equation \(\frac{dy}{dx} = x^3\), we want to determine which graph correctly represents this slope field. The four provided graphs each consist of a coordinate system with the x and y axes. They are as follows: 1. **Top Left Graph:** - At \((x = -3)\), the slopes are negative. - As x progresses towards \(0\) from \(-3\), the slopes become less steep but remain negative. - For \(x = 0\), the slopes are horizontal (indicating a slope of 0). - For \(x > 0\), the slopes are positive, becoming steeper as x increases. 2. **Top Right Graph:** - Slopes at the left and right of x-axis symmetrically appear similar to reversed direction. - For \(x = -3\) and \(x = 3\), they are positive with increasing steepness moving outward. 3. **Bottom Left Graph:** - The slopes gradually become steeper as we move away from the origin. - The pattern matches the curve where for \(x = -3\) slopes are large negative, and for \(x = 0\) slopes are 0. - As \(x\) increases to 3, slopes continuously increase in positive steepness. 4. **Bottom Right Graph:** - The slopes pattern does not change similarly across x-axis. - Central lines appear to be uniform, revealing quite different structure. ### Conclusion: The **Bottom Left Graph** is the correct slope field for the differential equation \(\frac{dy}{dx} = x^3\). This is because: - It shows the behavior where slopes become increasingly positive as \(x\) increases positively, and they become increasingly negative as \(x\) increases negatively. - The slopes are horizontal at \(x = 0\). ### Visualization Tips: - By observing the steepness and direction of short lines for each value of \(x\): - Horizontal lines at
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