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1. 15x4- 13x²+ a is o(x*) . Hint: Apply theurem I.

1. 15x4- 13x²+ a is o(x*) . Hint: Apply theurem I.

Advanced Engineering Mathematics
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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**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.
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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