Use the given graph of f to find a number & such that if |x-1| < 8 then [f(x) - 1| < 0.2. 8 = 1.2 1 0.8 0.7 1 1.1
Use the given graph of f to find a number & such that if |x-1| < 8 then [f(x) - 1| < 0.2. 8 = 1.2 1 0.8 0.7 1 1.1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q1. Please answer the question
![**Title: Understanding Limits with Graphical Analysis**
**Objective:**
Use the given graph of the function \( f \) to determine a number \( \delta \) such that if \( |x - 1| < \delta \), then \( |f(x) - 1| < 0.2 \).
**Graph Explanation:**
The graph displays a curve representing the function \( f \). The x-axis is labeled \( x \) and the y-axis is labeled \( y \).
Key points on the graph:
- The curve \( f \) passes through key y-values at 1.2 and 0.8.
- Horizontal lines span these y-values to indicate the possible values for \( f(x) \) that are within 0.2 units of 1.
- Vertical lines highlight the corresponding x-values on the graph centered around \( x = 1 \).
- Specifically, the vertical lines demarcate the interval \( x = 0.7 \) and \( x = 1.1 \).
**Task:**
Identify \( \delta \) by observing where the lines parallel to \( y = 1 \) intersect the curve \( f \). You should determine \( \delta \) such that \( |x - 1| < \delta \) implies \( 0.8 < f(x) < 1.2 \).
**Solution:**
From the graph, we can observe that for the interval \( 0.7 < x < 1.1 \), the function \( f(x) \) satisfies the condition \( |f(x) - 1| < 0.2 \).
Thus, \( \delta \) is calculated as:
\[ \delta = 0.3 \]
**Conclusion:**
By analyzing the graph, \( \delta \) is determined to be 0.3. This ensures that within this interval, changes in \( x \) result in corresponding \( f(x) \) values that are close to 1 within a margin of 0.2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b1f7975-4b0c-45ff-85b5-d245674b95f7%2F06a215be-4bc3-4522-b8b1-7a6be2d052c1%2F3591r1_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Limits with Graphical Analysis**
**Objective:**
Use the given graph of the function \( f \) to determine a number \( \delta \) such that if \( |x - 1| < \delta \), then \( |f(x) - 1| < 0.2 \).
**Graph Explanation:**
The graph displays a curve representing the function \( f \). The x-axis is labeled \( x \) and the y-axis is labeled \( y \).
Key points on the graph:
- The curve \( f \) passes through key y-values at 1.2 and 0.8.
- Horizontal lines span these y-values to indicate the possible values for \( f(x) \) that are within 0.2 units of 1.
- Vertical lines highlight the corresponding x-values on the graph centered around \( x = 1 \).
- Specifically, the vertical lines demarcate the interval \( x = 0.7 \) and \( x = 1.1 \).
**Task:**
Identify \( \delta \) by observing where the lines parallel to \( y = 1 \) intersect the curve \( f \). You should determine \( \delta \) such that \( |x - 1| < \delta \) implies \( 0.8 < f(x) < 1.2 \).
**Solution:**
From the graph, we can observe that for the interval \( 0.7 < x < 1.1 \), the function \( f(x) \) satisfies the condition \( |f(x) - 1| < 0.2 \).
Thus, \( \delta \) is calculated as:
\[ \delta = 0.3 \]
**Conclusion:**
By analyzing the graph, \( \delta \) is determined to be 0.3. This ensures that within this interval, changes in \( x \) result in corresponding \( f(x) \) values that are close to 1 within a margin of 0.2.
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