1. 15x4- 13x²+ a is o(x*) . Hint: Apply theurem I.
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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Transcribed Image Text:**Big O Notation Examples**
1. **Example 1:**
- Expression: \(15x^4 - 13x^2 + 2\)
- Big O Notation: \(O(x^4)\)
- Hint: Apply theorem 1.
2. **Example 2:**
- Expression: \(\frac{x^4 + x^2 + 1}{x^3 + 1}\)
- Big O Notation: \(O(x)\)
- Hint: Use long division.
3. **Example 3:**
- Expression: \(1^4 + 2^4 + 3^4 + \ldots + n^4\)
- Big O Notation: \(O(n^5)\)
- Hint: For \(n > 1\), assess as \(1^4 \leq n^4\), \(2^4 < n^4\), etc.
4. **Example 4:**
- Expression: \(2^x + 17\)
- Big O Notation: \(O(3^x)\)
- Hint: \(2^x < 3^x\), for \(x > 1\) and \(17 < 3^x\), for \(x \geq 3\).
5. **Explanation Request:**
- Task: Carefully explain what is meant by \(O(1)\).
6. **Example 6:**
- Expression: \((x^2 + 2x + 1)^2\)
- Find \(n\) such that: \(O(x^n)\)
- Hint: Expand \((x^2 + 2x + 1)\) and apply theorem 1.
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