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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![**Problem 4: Calculating the Total Distance Traveled by a Particle**
**Objective:**
To find the total distance traveled by a particle that moves with a velocity function \( V(t) = 2t - 4 \) over the time interval \([0,5]\).
**Approach:**
To determine the total distance traveled, we need to compute the integral of the absolute value of the velocity function over the given interval. This involves finding where the velocity changes sign to break the interval into sections where the velocity is consistently positive or negative.
**Steps:**
1. **Identify the Critical Points:**
- Find when \( V(t) = 0 \):
\[
2t - 4 = 0 \implies t = 2
\]
2. **Evaluate the Integral of \(|V(t)|\):**
- Split the interval \([0,5]\) into two parts: \([0,2]\) and \([2,5]\).
- Calculate the integral of the velocity function in these intervals accounting for change in sign:
- From \([0,2]\), \( V(t) = 2t - 4 \leq 0 \).
- From \([2,5]\), \( V(t) = 2t - 4 \geq 0 \).
3. **Calculate the Total Distance:**
- Total distance is the sum of the absolute integrals over each interval.
By following these steps, the total traveled distance can be computed accurately, considering all changes in the direction of movement.
**Note:** For a more detailed computation, follow the full solution process with integrals and absolute values to ensure understanding and accuracy.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6401e82b-1884-43b6-8725-0773860dca12%2Fa75b3dda-0b12-4163-823a-7c013a483002%2F6zaxzlw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4: Calculating the Total Distance Traveled by a Particle**
**Objective:**
To find the total distance traveled by a particle that moves with a velocity function \( V(t) = 2t - 4 \) over the time interval \([0,5]\).
**Approach:**
To determine the total distance traveled, we need to compute the integral of the absolute value of the velocity function over the given interval. This involves finding where the velocity changes sign to break the interval into sections where the velocity is consistently positive or negative.
**Steps:**
1. **Identify the Critical Points:**
- Find when \( V(t) = 0 \):
\[
2t - 4 = 0 \implies t = 2
\]
2. **Evaluate the Integral of \(|V(t)|\):**
- Split the interval \([0,5]\) into two parts: \([0,2]\) and \([2,5]\).
- Calculate the integral of the velocity function in these intervals accounting for change in sign:
- From \([0,2]\), \( V(t) = 2t - 4 \leq 0 \).
- From \([2,5]\), \( V(t) = 2t - 4 \geq 0 \).
3. **Calculate the Total Distance:**
- Total distance is the sum of the absolute integrals over each interval.
By following these steps, the total traveled distance can be computed accurately, considering all changes in the direction of movement.
**Note:** For a more detailed computation, follow the full solution process with integrals and absolute values to ensure understanding and accuracy.
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