1. Graphs of the position functions of two particles are shown, where t is measured in seconds.When is each particle speeding up? When is it slowing down? Explain. **Graphs of Position Functions for Two Particles**The graphs below display the position functions of two particles over time, with \( t \) expressed in seconds.The objective is to determine when each particle speeds up and when it slows down. ### Graph (a)- The graph demonstrates an initial increase in position, reaching a peak, followed by a decrease below zero, and an eventual rise again.- **Speeding Up:** - A particle is speeding up when its velocity and acceleration have the same sign. - In the initial rise and in the final rise, both velocity and acceleration are positive. - When the graph is heading downward sharply, both are negative; hence the particle is speeding up.- **Slowing Down:** - A particle slows down when velocity and acceleration have opposite signs. - Nearing the transition points from the peak to the low point and vice versa, the particle is slowing down.### Graph (b)- The graph starts with a decrease in position, then levels off briefly before decreasing sharply again and finally rising.- **Speeding Up:** - The particle speeds up during the initial and final drops when both velocity and acceleration are negative.- **Slowing Down:** - During transitions (as the graph changes direction), velocity and acceleration have opposite signs.These inflection points and behavior changes allow us to identify when the particles are accelerating or decelerating. Understanding these dynamics is crucial in physics for analyzing motion.

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Answered: 1. Graphs of the position functions of…

1. Graphs of the position functions of two particles are shown, where t is measured in seconds. When is cach particle speeding up? When is it slowing down? Explain. (a) SA (b) 1

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

1. Graphs of the position functions of two particles are shown, where t is measured in seconds.
When is each particle speeding up? When is it slowing down? Explain.

**Graphs of Position Functions for Two Particles**

The graphs below display the position functions of two particles over time, with \( t \) expressed in seconds.

The objective is to determine when each particle speeds up and when it slows down.

### Graph (a)

- The graph demonstrates an initial increase in position, reaching a peak, followed by a decrease below zero, and an eventual rise again.
- **Speeding Up:**
- A particle is speeding up when its velocity and acceleration have the same sign.
- In the initial rise and in the final rise, both velocity and acceleration are positive.
- When the graph is heading downward sharply, both are negative; hence the particle is speeding up.
- **Slowing Down:**
- A particle slows down when velocity and acceleration have opposite signs.
- Nearing the transition points from the peak to the low point and vice versa, the particle is slowing down.

### Graph (b)

- The graph starts with a decrease in position, then levels off briefly before decreasing sharply again and finally rising.
- **Speeding Up:**
- The particle speeds up during the initial and final drops when both velocity and acceleration are negative.
- **Slowing Down:**
- During transitions (as the graph changes direction), velocity and acceleration have opposite signs.

These inflection points and behavior changes allow us to identify when the particles are accelerating or decelerating. Understanding these dynamics is crucial in physics for analyzing motion.

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