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Could the function in the bottom plot be the derivative of the function in the top plot? Explain your answer. The plot of the FUNCTION (below) 5 4 3 2 1 -5 -4 -3 -1 -2 -3 -4 -5- The plot of its DERIVATIVE (below) 5 4 3 2 1 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 3 4 4

Could the function in the bottom plot be the derivative of the function in the top plot? Explain your answer. The plot of the FUNCTION (below) 5 4 3 2 1 -5 -4 -3 -1 -2 -3 -4 -5- The plot of its DERIVATIVE (below) 5 4 3 2 1 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 3 4 4

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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**Educational Content** **Question:** Could the function in the bottom plot be the derivative of the function in the top plot? Explain your answer. **Graphs and Description:** 1. **The Plot of the FUNCTION (above):** - The graph of the function has a wavy, sinusoidal shape. - Key points to note: - Peaks occur near \(x = -4\) and \(x = 2\). - Troughs occur near \(x = -1\) and \(x = 4\). - The function crosses the x-axis at points approximately \(x = -3\), \(x = 1\), and \(x = 3\). 2. **The Plot of its DERIVATIVE (below):** - The graph shows a dynamic curve that seems to reflect changes in slope of the function above. - The derivative is positive when the function itself is increasing, which is consistent with the behavior between the trough and the peak of the function. - The derivative is zero at points where the function has a peak or trough (local minimum and maximum). It crosses the x-axis at similar locations to where the function's slope changes direction. - The derivative features a notable increase close to \(x = -4\) and \(-1\), indicative of steep slopes on the function curve there. **Analysis:** When evaluating if the bottom plot could be the derivative of the function in the top plot, we look for consistent features: - **Zero Crossings:** The derivative is zero where the original function has local maxima or minima. These zero crossings should correspond with the peak and trough positions in the function graph. - **Sign of Derivative:** The sign of the derivative (positive or negative) reflects increasing or decreasing behavior in the function graph. For example, when the function rises, its derivative is positive; when it falls, the derivative is negative. - **Slope Behavior:** At points where the function sharply changes direction, the derivative will have steep increases or decreases. From the analysis, it appears that the function’s graph features and its derivative's graph features align in this manner. Therefore, based on these observations, it is plausible that the function in the bottom plot might be the derivative of the function in the top plot.
Transcribed Image Text:**Educational Content** **Question:** Could the function in the bottom plot be the derivative of the function in the top plot? Explain your answer. **Graphs and Description:** 1. **The Plot of the FUNCTION (above):** - The graph of the function has a wavy, sinusoidal shape. - Key points to note: - Peaks occur near \(x = -4\) and \(x = 2\). - Troughs occur near \(x = -1\) and \(x = 4\). - The function crosses the x-axis at points approximately \(x = -3\), \(x = 1\), and \(x = 3\). 2. **The Plot of its DERIVATIVE (below):** - The graph shows a dynamic curve that seems to reflect changes in slope of the function above. - The derivative is positive when the function itself is increasing, which is consistent with the behavior between the trough and the peak of the function. - The derivative is zero at points where the function has a peak or trough (local minimum and maximum). It crosses the x-axis at similar locations to where the function's slope changes direction. - The derivative features a notable increase close to \(x = -4\) and \(-1\), indicative of steep slopes on the function curve there. **Analysis:** When evaluating if the bottom plot could be the derivative of the function in the top plot, we look for consistent features: - **Zero Crossings:** The derivative is zero where the original function has local maxima or minima. These zero crossings should correspond with the peak and trough positions in the function graph. - **Sign of Derivative:** The sign of the derivative (positive or negative) reflects increasing or decreasing behavior in the function graph. For example, when the function rises, its derivative is positive; when it falls, the derivative is negative. - **Slope Behavior:** At points where the function sharply changes direction, the derivative will have steep increases or decreases. From the analysis, it appears that the function’s graph features and its derivative's graph features align in this manner. Therefore, based on these observations, it is plausible that the function in the bottom plot might be the derivative of the function in the top plot.
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