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t 2. 3 4 A particle moves along the y-axis. The graph of the particle's position y(t) at time tis shown above for 0

t 2. 3 4 A particle moves along the y-axis. The graph of the particle's position y(t) at time tis shown above for 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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### Graph and Question Analysis The graph above represents the position \( y(t) \) of a particle moving along the \( y \)-axis over time \( t \) ranging from 0 to 5. The graph has five distinct segments: 1. **From \( t = 0 \) to \( t = 1 \):** The curve slopes upward, indicating positive velocity. 2. **From \( t = 1 \) to \( t = 2 \):** The segment is a flat line, suggesting zero velocity. 3. **From \( t = 2 \) to \( t = 3 \):** The curve slopes sharply upward, indicating increasing velocity. 4. **From \( t = 3 \) to \( t = 4 \):** The curve slopes downward, indicating negative velocity. 5. **From \( t = 4 \) to \( t = 5 \):** The downward slope becomes gentler, suggesting decreasing negative velocity with positive acceleration. ### Question: "A particle moves along the \( y \)-axis. The graph of the particle’s position \( y(t) \) at time \( t \) is shown above for \( 0 \leq t \leq 5 \). For what values of \( t \) is the velocity of the particle negative and the acceleration positive?" #### Choices: - **A:** \( 0 < t < 1 \) - **B:** \( 1 < t < 2 \) - **C:** \( 2 < t < 3 \) - **D:** \( 3 < t < 4 \) - **E:** \( 4 < t < 5 \) **Explanation:** To determine the intervals where the velocity is negative and acceleration is positive, focus on the sections of the graph where the slope is negative (indicating negative velocity) and where the rate of change of the slope is increasing (indicating positive acceleration). - **Correct Choice: E (\( 4 < t < 5 \))**: During this interval, the particle's velocity is negative (since the curve is sloping downward), but the slope is becoming less steep, indicating positive acceleration.
Transcribed Image Text:### Graph and Question Analysis The graph above represents the position \( y(t) \) of a particle moving along the \( y \)-axis over time \( t \) ranging from 0 to 5. The graph has five distinct segments: 1. **From \( t = 0 \) to \( t = 1 \):** The curve slopes upward, indicating positive velocity. 2. **From \( t = 1 \) to \( t = 2 \):** The segment is a flat line, suggesting zero velocity. 3. **From \( t = 2 \) to \( t = 3 \):** The curve slopes sharply upward, indicating increasing velocity. 4. **From \( t = 3 \) to \( t = 4 \):** The curve slopes downward, indicating negative velocity. 5. **From \( t = 4 \) to \( t = 5 \):** The downward slope becomes gentler, suggesting decreasing negative velocity with positive acceleration. ### Question: "A particle moves along the \( y \)-axis. The graph of the particle’s position \( y(t) \) at time \( t \) is shown above for \( 0 \leq t \leq 5 \). For what values of \( t \) is the velocity of the particle negative and the acceleration positive?" #### Choices: - **A:** \( 0 < t < 1 \) - **B:** \( 1 < t < 2 \) - **C:** \( 2 < t < 3 \) - **D:** \( 3 < t < 4 \) - **E:** \( 4 < t < 5 \) **Explanation:** To determine the intervals where the velocity is negative and acceleration is positive, focus on the sections of the graph where the slope is negative (indicating negative velocity) and where the rate of change of the slope is increasing (indicating positive acceleration). - **Correct Choice: E (\( 4 < t < 5 \))**: During this interval, the particle's velocity is negative (since the curve is sloping downward), but the slope is becoming less steep, indicating positive acceleration.
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