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Find the minimum and maximum of A over a particular interval [0,6]. A(x) = f(t) dt

Find the minimum and maximum of A over a particular interval [0,6]. A(x) = f(t) dt

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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Question
### Analysis of the Graph of \( y = f(x) \) The figure provided represents the graph of the function \( y = f(x) \). #### Detailed Description of the Graph: 1. **Axes and Scale:** - The horizontal axis (x-axis) ranges from -2 to 8. - The vertical axis (y-axis) ranges from -4 to 5. - The graph has a grid with a unit spacing of 1 along both axes. 2. **Behavior of the Graph:** - **From \( x = -2 \) to \( x = 0 \):** - The graph starts at the point (-2, -3) and ascends linearly to the origin (0, 0). - **From \( x = 0 \) to \( x = 2 \):** - The graph continues to rise linearly from the origin (0, 0) to the point (2, 4). - **From \( x = 2 \) to \( x = 4 \):** - The graph remains constant, forming a horizontal line at \( y = 4 \). - **From \( x = 4 \) to \( x = 5 \):** - The graph descends linearly from the point (4, 4) to the point (5, 0). - **From \( x = 5 \) to \( x = 6 \):** - The graph continues its descent linearly from the point (5, 0) to the point (6, -3). - **From \( x = 6 \) to \( x = 8 \):** - The graph remains constant again, forming a horizontal line at \( y = -3 \). These observations describe the key points and segments of the graph \( y = f(x) \), providing an understanding of its linear and constant segments within the specified range of \( x \).
Transcribed Image Text:### Analysis of the Graph of \( y = f(x) \) The figure provided represents the graph of the function \( y = f(x) \). #### Detailed Description of the Graph: 1. **Axes and Scale:** - The horizontal axis (x-axis) ranges from -2 to 8. - The vertical axis (y-axis) ranges from -4 to 5. - The graph has a grid with a unit spacing of 1 along both axes. 2. **Behavior of the Graph:** - **From \( x = -2 \) to \( x = 0 \):** - The graph starts at the point (-2, -3) and ascends linearly to the origin (0, 0). - **From \( x = 0 \) to \( x = 2 \):** - The graph continues to rise linearly from the origin (0, 0) to the point (2, 4). - **From \( x = 2 \) to \( x = 4 \):** - The graph remains constant, forming a horizontal line at \( y = 4 \). - **From \( x = 4 \) to \( x = 5 \):** - The graph descends linearly from the point (4, 4) to the point (5, 0). - **From \( x = 5 \) to \( x = 6 \):** - The graph continues its descent linearly from the point (5, 0) to the point (6, -3). - **From \( x = 6 \) to \( x = 8 \):** - The graph remains constant again, forming a horizontal line at \( y = -3 \). These observations describe the key points and segments of the graph \( y = f(x) \), providing an understanding of its linear and constant segments within the specified range of \( x \).
**Finding Extrema of a Function Defined by an Integral** **Problem Statement:** Find the minimum and maximum of \( A \) over a particular interval \([0,6]\). \[ A(x) = \int_{0}^{x} f(t) \, dt \] **Explanation:** The objective is to determine the minimum and maximum values of the function \( A(x) \), which is defined as the integral of the function \( f(t) \) from 0 to \( x \), within the interval \([0,6]\).
Transcribed Image Text:**Finding Extrema of a Function Defined by an Integral** **Problem Statement:** Find the minimum and maximum of \( A \) over a particular interval \([0,6]\). \[ A(x) = \int_{0}^{x} f(t) \, dt \] **Explanation:** The objective is to determine the minimum and maximum values of the function \( A(x) \), which is defined as the integral of the function \( f(t) \) from 0 to \( x \), within the interval \([0,6]\).
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