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Suppose that g(x) = (integral symbol (x on top,  -2 on the bottom)) f(t)dt. is the rate of change in g increasing or decreasing at t=1? Explain your reasoning

Suppose that g(x) = (integral symbol (x on top,  -2 on the bottom)) f(t)dt. is the rate of change in g increasing or decreasing at t=1? Explain your reasoning

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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Suppose that g(x) = (integral symbol (x on top,  -2 on the bottom)) f(t)dt. is the rate of change in g increasing or decreasing at t=1? Explain your reasoning.

### Graph of Function \( f \) The provided image shows a graph labeled "Graph of \( f \)," which depicts the behavior of a mathematical function \( f \) over a range of \( x \)-values. #### Description and Key Points of the Graph: 1. **Axes and Scale:** - The horizontal axis (x-axis) and the vertical axis (y-axis) are depicted with a grid. - Each grid line along both axes represents a unit interval. 2. **Data Points:** - The function \( f \) passes through several specific points on the graph: - (-2, 0) - (-1, -1) - (0, 0) - (2, 3) - (5, 0) 3. **Shape and Behavior:** - The graph begins at the point (-2,0) on the x-axis. - It then decreases to the point (-1, -1). - Afterward, it returns to the origin (0,0). - The function \( f \) increases sharply to reach a peak at the point (2, 3). - Following the peak, the function gradually decreases and intersects the x-axis again at the point (5, 0). 4. **Key Features:** - **Origin:** The graph passes through the origin (0, 0). - **Maximum Point:** The highest value of the function is at (2, 3). - **Intercepts:** The x-intercepts of the graph are (-2, 0), (0, 0), and (5, 0). - **Negative and Positive Regions:** The function is negative between x = -2 and x = 0, positive between x = 0 and x = 5, and returns to zero at x = 5. This graph captures the intricate changes in the function \( f \) as it transitions through various values of \( x \), highlighting key points such as intercepts and the maximum value. Understanding the detailed behavior of functions through graphical representations is essential in mathematics and many scientific disciplines.
Transcribed Image Text:### Graph of Function \( f \) The provided image shows a graph labeled "Graph of \( f \)," which depicts the behavior of a mathematical function \( f \) over a range of \( x \)-values. #### Description and Key Points of the Graph: 1. **Axes and Scale:** - The horizontal axis (x-axis) and the vertical axis (y-axis) are depicted with a grid. - Each grid line along both axes represents a unit interval. 2. **Data Points:** - The function \( f \) passes through several specific points on the graph: - (-2, 0) - (-1, -1) - (0, 0) - (2, 3) - (5, 0) 3. **Shape and Behavior:** - The graph begins at the point (-2,0) on the x-axis. - It then decreases to the point (-1, -1). - Afterward, it returns to the origin (0,0). - The function \( f \) increases sharply to reach a peak at the point (2, 3). - Following the peak, the function gradually decreases and intersects the x-axis again at the point (5, 0). 4. **Key Features:** - **Origin:** The graph passes through the origin (0, 0). - **Maximum Point:** The highest value of the function is at (2, 3). - **Intercepts:** The x-intercepts of the graph are (-2, 0), (0, 0), and (5, 0). - **Negative and Positive Regions:** The function is negative between x = -2 and x = 0, positive between x = 0 and x = 5, and returns to zero at x = 5. This graph captures the intricate changes in the function \( f \) as it transitions through various values of \( x \), highlighting key points such as intercepts and the maximum value. Understanding the detailed behavior of functions through graphical representations is essential in mathematics and many scientific disciplines.
Below is a table that represents the velocity function \( v_P(t) \) for a particle P at different times \( t \). | \( t \) (hours) | 0 | 0.3 | 1.7 | 2.8 | 4 | |----------------------|-----|-----|-----|-----|-----| | \( v_P(t) \) (meters per hour) | 0 | 55 | -29 | 55 | 48 | **Explanation:** This table is used to present the velocity of a particle P at discrete points in time. The top row indicates the time \( t \) in hours, ranging from 0 to 4 hours. The bottom row indicates the velocity \( v_P(t) \) in meters per hour corresponding to each time value. - At \( t = 0 \) hours, the velocity \( v_P(t) \) is 0 meters per hour. - At \( t = 0.3 \) hours, the velocity \( v_P(t) \) is 55 meters per hour. - At \( t = 1.7 \) hours, the velocity \( v_P(t) \) is -29 meters per hour. - At \( t = 2.8 \) hours, the velocity \( v_P(t) \) is 55 meters per hour. - At \( t = 4 \) hours, the velocity \( v_P(t) \) is 48 meters per hour. This table can be used to analyze the motion of the particle P with respect to time, identifying changes in speed and direction based on the given velocities.
Transcribed Image Text:Below is a table that represents the velocity function \( v_P(t) \) for a particle P at different times \( t \). | \( t \) (hours) | 0 | 0.3 | 1.7 | 2.8 | 4 | |----------------------|-----|-----|-----|-----|-----| | \( v_P(t) \) (meters per hour) | 0 | 55 | -29 | 55 | 48 | **Explanation:** This table is used to present the velocity of a particle P at discrete points in time. The top row indicates the time \( t \) in hours, ranging from 0 to 4 hours. The bottom row indicates the velocity \( v_P(t) \) in meters per hour corresponding to each time value. - At \( t = 0 \) hours, the velocity \( v_P(t) \) is 0 meters per hour. - At \( t = 0.3 \) hours, the velocity \( v_P(t) \) is 55 meters per hour. - At \( t = 1.7 \) hours, the velocity \( v_P(t) \) is -29 meters per hour. - At \( t = 2.8 \) hours, the velocity \( v_P(t) \) is 55 meters per hour. - At \( t = 4 \) hours, the velocity \( v_P(t) \) is 48 meters per hour. This table can be used to analyze the motion of the particle P with respect to time, identifying changes in speed and direction based on the given velocities.
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