Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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#### Part (a)
Let \( f \) be a function defined as the y-coordinate of the point on the long side of the pool that is nearest the swimmer at any given time during the swimmer's crossing of the pool. Determine the function \( f \) and sketch its graph.
Is \( f \) continuous? Explain.
##### Solution:
1. **Identification of Function \( f \):**
- The swimmer moves along the line from \( (0, 0) \) to \( (2b, b) \).
- The equation of the line representing the swimmer’s path can be given by:
\[
y = \frac{b}{2b} x = \frac{1}{2} x.
\]
- At any point \( (x, \frac{1}{2} x) \) along this path, \( f(x) \) must be the distance to the nearest long side of the pool.
- On the long side of the pool nearest to the swimmer, the y-coordinate is either 0 or \( b \).
2. **Defining the Function \( f \):**
- Since \( f \) is the y-coordinate of the point on the long side nearest to the swimmer:
- For \( 0 \leq x \leq b \):
\[
f(x](https://content.bartleby.com/qna-images/question/2dcd4d2b-e26a-484b-8c09-7130117e90c6/1062813f-5a54-4cd1-ab02-d3d495c95377/pipneqj.png)
Transcribed Image Text:### Problem 2
A swimmer crosses a pool of width \( b \) by swimming in a straight line from \( (0, 0) \) to \( (2b, b) \). (See figure below)
#### Diagram Explanation:
The diagram illustrates a rectangular swimming pool with coordinates system aligned such that one of its long sides lies along the x-axis, and its short side along the y-axis. The following points and labels are important:
- The starting point of the swimmer is \( (0, 0) \).
- The endpoint of the swimmer is \( (2b, b) \).
- The line representing the swimmer's path is a diagonal from \( (0, 0) \) to \( (2b, b) \).
- The vertical span (width) of the pool is denoted as \( b \).
- The corresponding axes are labeled with \( x \) and \( y \) coordinates, where the long side aligns with the x-axis and the short side aligns with the y-axis.
#### Part (a)
Let \( f \) be a function defined as the y-coordinate of the point on the long side of the pool that is nearest the swimmer at any given time during the swimmer's crossing of the pool. Determine the function \( f \) and sketch its graph.
Is \( f \) continuous? Explain.
##### Solution:
1. **Identification of Function \( f \):**
- The swimmer moves along the line from \( (0, 0) \) to \( (2b, b) \).
- The equation of the line representing the swimmer’s path can be given by:
\[
y = \frac{b}{2b} x = \frac{1}{2} x.
\]
- At any point \( (x, \frac{1}{2} x) \) along this path, \( f(x) \) must be the distance to the nearest long side of the pool.
- On the long side of the pool nearest to the swimmer, the y-coordinate is either 0 or \( b \).
2. **Defining the Function \( f \):**
- Since \( f \) is the y-coordinate of the point on the long side nearest to the swimmer:
- For \( 0 \leq x \leq b \):
\[
f(x
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