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1.0 0.5 g -2 -1 1 2 h -0.5

1.0 0.5 g -2 -1 1 2 h -0.5

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Explanation of Graphs This image presents the graphs of three functions labeled \( f \), \( g \), and \( h \). Each function is represented by a different colored curve: - **Function \( f \)**: Represented by a blue curve, which is monotonically increasing over the displayed range. - **Function \( g \)**: Represented by an orange curve, displaying a downward-opening parabolic shape that reaches a maximum around \( x = 0 \). - **Function \( h \)**: Represented by a green curve, showing an upward-opening parabolic shape with a minimum around \( x = 0 \). The graph is plotted with the x-axis ranging from -2 to 2, and the y-axis from -1 to 1. ### Analytical Observations The functions' behavior suggests the following relationships: - The curve \( g \) could represent the first derivative of \( f \), as it is consistent with \( f \) showing increasing behavior with a change in curvature. - The curve \( h \) might represent the second derivative of \( f \), given that it indicates curvature changes at its peak and trough. ### Matching Statements The goal is to determine which of the given statements best describes the relationships between the graphs of \( f \), \( g \), and \( h \): - \( f' = g \) and \( f'' = h \) - \( f' = h \) and \( f'' = g \) - \( g' = h \) and \( g'' = f \) - \( h' = g \) and \( h'' = f \) ### Correct Answer Based on the observed characteristics of the functions, the statement that appears consistent with the graphs is: - \( f' = g \) and \( f'' = h \) This choice assumes standard characteristics of derivatives, where \( f' \) describes the local slopes of \( f \), and \( f'' \) the concavity.
Transcribed Image Text:### Explanation of Graphs This image presents the graphs of three functions labeled \( f \), \( g \), and \( h \). Each function is represented by a different colored curve: - **Function \( f \)**: Represented by a blue curve, which is monotonically increasing over the displayed range. - **Function \( g \)**: Represented by an orange curve, displaying a downward-opening parabolic shape that reaches a maximum around \( x = 0 \). - **Function \( h \)**: Represented by a green curve, showing an upward-opening parabolic shape with a minimum around \( x = 0 \). The graph is plotted with the x-axis ranging from -2 to 2, and the y-axis from -1 to 1. ### Analytical Observations The functions' behavior suggests the following relationships: - The curve \( g \) could represent the first derivative of \( f \), as it is consistent with \( f \) showing increasing behavior with a change in curvature. - The curve \( h \) might represent the second derivative of \( f \), given that it indicates curvature changes at its peak and trough. ### Matching Statements The goal is to determine which of the given statements best describes the relationships between the graphs of \( f \), \( g \), and \( h \): - \( f' = g \) and \( f'' = h \) - \( f' = h \) and \( f'' = g \) - \( g' = h \) and \( g'' = f \) - \( h' = g \) and \( h'' = f \) ### Correct Answer Based on the observed characteristics of the functions, the statement that appears consistent with the graphs is: - \( f' = g \) and \( f'' = h \) This choice assumes standard characteristics of derivatives, where \( f' \) describes the local slopes of \( f \), and \( f'' \) the concavity.
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