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For the given graph of the function f, on the left, sketch the graph of of f' on the right. 4 2 -4 -2 4 -4 -2 2 4 -2 -4

For the given graph of the function f, on the left, sketch the graph of of f' on the right. 4 2 -4 -2 4 -4 -2 2 4 -2 -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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**Task Description** For the given graph of the function \( f \), on the left, sketch the graph of \( f' \) on the right. **Analysis of the Graph on the Left** The graph on the left represents the function \( f \). It is characterized by two vertical asymptotes at \( x = -2 \) and \( x = 2 \). As \( x \) approaches these values from either side, the function \( f(x) \) tends towards infinity positively or negatively. The function is symmetric around the y-axis and takes the form of an "M" with peaks at \( x = -2 \) and \( x = 2 \). **Graph Description** - The graph has asymptotes at \( x = -2 \) and \( x = 2 \). - The function approaches negative infinity as \( x \to -2^+ \) and \( x \to 2^- \). - The function approaches positive infinity as \( x \to -2^- \) and \( x \to 2^+ \). - The graph is relatively flat as \( x \) tends towards negative or positive infinity. **Instructions for Sketching \( f' \)** Since \( f' \) represents the derivative of \( f \), it will reflect the slope of \( f \). Observe the following trends: - **Slope at Asymptotes:** The derivative \( f' \) will have undefined values at \( x = -2 \) and \( x = 2 \) due to the vertical asymptotes. - **Slope Trend:** - As \( x \) approaches each asymptote from the left, the slope becomes steeper positive, indicating a high positive or negative value for \( f' \). - As \( x \) moves away from each asymptote, the slope becomes less steep, trending towards zero. **Drawing \( f' \)** The graph of \( f' \) should: - Show high values near the asymptotes, corresponding to the steep slopes of \( f \). - Be undefined exactly at \( x = -2 \) and \( x = 2 \). - Approach zero as \( x \) moves away from the asymptotes. This depiction helps visualize how the rate of change in \( f \) is distributed along the x-axis.
Transcribed Image Text:**Task Description** For the given graph of the function \( f \), on the left, sketch the graph of \( f' \) on the right. **Analysis of the Graph on the Left** The graph on the left represents the function \( f \). It is characterized by two vertical asymptotes at \( x = -2 \) and \( x = 2 \). As \( x \) approaches these values from either side, the function \( f(x) \) tends towards infinity positively or negatively. The function is symmetric around the y-axis and takes the form of an "M" with peaks at \( x = -2 \) and \( x = 2 \). **Graph Description** - The graph has asymptotes at \( x = -2 \) and \( x = 2 \). - The function approaches negative infinity as \( x \to -2^+ \) and \( x \to 2^- \). - The function approaches positive infinity as \( x \to -2^- \) and \( x \to 2^+ \). - The graph is relatively flat as \( x \) tends towards negative or positive infinity. **Instructions for Sketching \( f' \)** Since \( f' \) represents the derivative of \( f \), it will reflect the slope of \( f \). Observe the following trends: - **Slope at Asymptotes:** The derivative \( f' \) will have undefined values at \( x = -2 \) and \( x = 2 \) due to the vertical asymptotes. - **Slope Trend:** - As \( x \) approaches each asymptote from the left, the slope becomes steeper positive, indicating a high positive or negative value for \( f' \). - As \( x \) moves away from each asymptote, the slope becomes less steep, trending towards zero. **Drawing \( f' \)** The graph of \( f' \) should: - Show high values near the asymptotes, corresponding to the steep slopes of \( f \). - Be undefined exactly at \( x = -2 \) and \( x = 2 \). - Approach zero as \( x \) moves away from the asymptotes. This depiction helps visualize how the rate of change in \( f \) is distributed along the x-axis.
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