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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Attention: show sidework and proof using the board steps. The hint to follow is in the 2nd photo2- show how it's bounded
![### Mathematical Concepts on the Chalkboard
The image shows a section of a chalkboard with some mathematical expressions and concepts. Here's a detailed transcription and explanation:
1. **Expression Breakdown:**
- The main expression on the board is about evaluating the absolute value of the difference between two functions or terms.
- The expression is written as:
\[
|f(x) - f(u)| = | \sqrt{x} - \sqrt{u} |
\]
It simplifies to:
\[
= \left| \frac{(\sqrt{x} - \sqrt{u})(\sqrt{x} + \sqrt{u})}{\sqrt{x} + \sqrt{u}} \right|
\]
This can be further simplified to:
\[
= \frac{1}{\sqrt{x} + \sqrt{u}} |x - u|
\]
2. **Variable Definitions:**
- \( x \) and \( u \) are variables, potentially related to a limit or continuity problem.
- The expression involves simplifying radicals and exploring differences, relevant in calculus for examining limits.
3. **Highlighted Terms:**
- A portion of the expression is circled:
\[
\frac{1}{\sqrt{x} + \sqrt{u}}
\]
This could be a crucial factor in an inequality or an approximation.
4. **Inequality and Constants:**
- Indicates that an inequality or bounding step follows:
\[
\leq B |x-u|
\]
This suggests that \( B \) is a constant that serves as an upper bound.
- Terms \( \text{ABSC} \leq \epsilon \) are written, possibly highlighting a condition of absolute convergence or approximation.
The content appears to be focused on understanding and simplifying limits or continuity examples in calculus, especially dealing with the behavior of functions involving square roots. These steps are fundamental for calculus students to comprehend concepts such as limits, derivatives, or integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5203d42-0da6-4b1b-a1a9-5b37798d2cd6%2F2da0031a-6f71-4057-91a3-20e482fbe209%2Ftmgq9t.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Concepts on the Chalkboard
The image shows a section of a chalkboard with some mathematical expressions and concepts. Here's a detailed transcription and explanation:
1. **Expression Breakdown:**
- The main expression on the board is about evaluating the absolute value of the difference between two functions or terms.
- The expression is written as:
\[
|f(x) - f(u)| = | \sqrt{x} - \sqrt{u} |
\]
It simplifies to:
\[
= \left| \frac{(\sqrt{x} - \sqrt{u})(\sqrt{x} + \sqrt{u})}{\sqrt{x} + \sqrt{u}} \right|
\]
This can be further simplified to:
\[
= \frac{1}{\sqrt{x} + \sqrt{u}} |x - u|
\]
2. **Variable Definitions:**
- \( x \) and \( u \) are variables, potentially related to a limit or continuity problem.
- The expression involves simplifying radicals and exploring differences, relevant in calculus for examining limits.
3. **Highlighted Terms:**
- A portion of the expression is circled:
\[
\frac{1}{\sqrt{x} + \sqrt{u}}
\]
This could be a crucial factor in an inequality or an approximation.
4. **Inequality and Constants:**
- Indicates that an inequality or bounding step follows:
\[
\leq B |x-u|
\]
This suggests that \( B \) is a constant that serves as an upper bound.
- Terms \( \text{ABSC} \leq \epsilon \) are written, possibly highlighting a condition of absolute convergence or approximation.
The content appears to be focused on understanding and simplifying limits or continuity examples in calculus, especially dealing with the behavior of functions involving square roots. These steps are fundamental for calculus students to comprehend concepts such as limits, derivatives, or integrals.
Transcribed Image Text:The image contains handwritten mathematical notes. Here's a transcription suitable for an educational website:
---
**Uniform Continuity of \( f(x) = \sqrt{x} \)**
Consider the function \( f: [a, +\infty) \to \mathbb{R} \), where \( a > 0 \).
**Claim:** The function \( f \) is uniformly continuous on the interval \([a, +\infty)\).
**Proof Outline:**
- Define the function \( f(x) = \sqrt{x} \).
- Investigate the continuity of \( f \) on the specified domain.
- Demonstrate that \( f \) maintains uniform continuity by examining the behavior over the interval starting from \( a \).
This exploration helps in understanding how the square root function behaves over open intervals and asserts its continuity properties.
---
This outline summarizes the main ideas from the handwritten notes and explains the consideration of the function's uniform continuity.
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