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Sketch the graph of f' given the graph of f below. YA W. 0

Sketch the graph of f' given the graph of f below. YA W. 0

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:** Sketch the graph of \( f' \) given the graph of \( f \) below. **Graph Explanation:** The graph displayed represents a function \( f \) on a coordinate plane with the x-axis and y-axis intersecting at the origin, labeled as \( 0 \). **Analysis of Graph of \( f \):** 1. **Shape and Behavior:** - The graph displays a continuous curve, resembling two concave parabolic shapes connected at a point. - Starting from the left, the curve decreases sharply, reaches a minimum point, then increases, hits a maximum point, decreases again to another minimum point, and finally, rises sharply as it exits. 2. **Key Features:** - **Local Minima and Maxima:** The function has two local minima and one local maximum. - The slope of the function \( f \) (derivative \( f' \)) changes sign at these extrema: - Negative to positive slope indicates a local minimum. - Positive to negative slope indicates a local maximum. **Sketching \( f' \):** - As you sketch the derivative \( f' \): - The graph of \( f' \) should intersect the x-axis at points where \( f \) has its local maxima and minima. - \( f' \) will be zero at these points because the tangent is horizontal. - The sign of \( f' \) should change around these points: - Positive when \( f \) is increasing. - Negative when \( f \) is decreasing. This conceptual guide will aid in creating a sketch of the graph of the derivative, focusing on critical points where the slope is zero and the intervals of increasing/decreasing behavior of the function.
Transcribed Image Text:**Task:** Sketch the graph of \( f' \) given the graph of \( f \) below. **Graph Explanation:** The graph displayed represents a function \( f \) on a coordinate plane with the x-axis and y-axis intersecting at the origin, labeled as \( 0 \). **Analysis of Graph of \( f \):** 1. **Shape and Behavior:** - The graph displays a continuous curve, resembling two concave parabolic shapes connected at a point. - Starting from the left, the curve decreases sharply, reaches a minimum point, then increases, hits a maximum point, decreases again to another minimum point, and finally, rises sharply as it exits. 2. **Key Features:** - **Local Minima and Maxima:** The function has two local minima and one local maximum. - The slope of the function \( f \) (derivative \( f' \)) changes sign at these extrema: - Negative to positive slope indicates a local minimum. - Positive to negative slope indicates a local maximum. **Sketching \( f' \):** - As you sketch the derivative \( f' \): - The graph of \( f' \) should intersect the x-axis at points where \( f \) has its local maxima and minima. - \( f' \) will be zero at these points because the tangent is horizontal. - The sign of \( f' \) should change around these points: - Positive when \( f \) is increasing. - Negative when \( f \) is decreasing. This conceptual guide will aid in creating a sketch of the graph of the derivative, focusing on critical points where the slope is zero and the intervals of increasing/decreasing behavior of the function.
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