Problem 3: Give the asymptotic values of the following functions, using the e-notation: (a) 7n² + 2n¹ +3n+1 5 log: (n²) log₂ n (b) + n +n=& (c) 2n(n logn + n log³ n) + 3n²√√n (d) 25n¹2+n³ log¹ n + 1.25" (e) n4" +5n6.3n Justify your answer. (Here, you don't need to give a complete rigorous proof. For problems (b) - (e), give only an informal explanation using asymptotic relations between the functions nº, logn, and c".)
Problem 3: Give the asymptotic values of the following functions, using the e-notation: (a) 7n² + 2n¹ +3n+1 5 log: (n²) log₂ n (b) + n +n=& (c) 2n(n logn + n log³ n) + 3n²√√n (d) 25n¹2+n³ log¹ n + 1.25" (e) n4" +5n6.3n Justify your answer. (Here, you don't need to give a complete rigorous proof. For problems (b) - (e), give only an informal explanation using asymptotic relations between the functions nº, logn, and c".)
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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![**Problem 3: Give the asymptotic values of the following functions, using the Θ-notation:**
(a) \( 7n^2 + 2n^4 + 3n + 1 \)
(b)
\[ \frac{5}{n} + \frac{\log_3 (n^2)}{\log_2 n} + n^{-\frac{1}{2}} \]
(c) \( 2n(n \log n + n \log^3 n) + 3n^2 \sqrt{n} \)
(d) \( 25n^{12} + n^3 \log^4 n + 1.25^n \)
(e) \( n4^n + 5n^6 \cdot 3^n \)
*Justify your answer. (Here, you don’t need to give a complete rigorous proof. For problems (b) - (e), give only an informal explanation using asymptotic relations between the functions \(n^c\), \(\log n\), and \(c^n\).)*
---
**Explanation:**
This problem involves finding the asymptotic values of given functions in terms of Θ-notation, which is used in computer science to describe the limiting behavior of a function, especially in terms of input size \(n\). Specifically, Θ-notation describes the growth rate of a function, often used to analyze algorithms.
---
**Graphs and Diagrams:**
No graphs or diagrams are provided or required. The focus is strictly on the provided mathematical expressions.
**Steps for Solution:**
1. Identify the term in each expression that grows the fastest as \(n\) increases.
2. Use the identified term to express the asymptotic behavior of the entire function.
3. For each of the given functions in (b) to (e), compare the different components (polynomial, logarithmic, exponential) to determine their respective growth rates.
Refer to asymptotic notations and simplifications to justify the choice of the fastest growing term, showcasing a fundamental understanding of analyzing algorithm efficiency.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7451af8-c677-4bc1-8ff0-c69e4b49e989%2F0b347fc8-a597-4592-bdf8-6b9936691622%2Fw7s2u0r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Give the asymptotic values of the following functions, using the Θ-notation:**
(a) \( 7n^2 + 2n^4 + 3n + 1 \)
(b)
\[ \frac{5}{n} + \frac{\log_3 (n^2)}{\log_2 n} + n^{-\frac{1}{2}} \]
(c) \( 2n(n \log n + n \log^3 n) + 3n^2 \sqrt{n} \)
(d) \( 25n^{12} + n^3 \log^4 n + 1.25^n \)
(e) \( n4^n + 5n^6 \cdot 3^n \)
*Justify your answer. (Here, you don’t need to give a complete rigorous proof. For problems (b) - (e), give only an informal explanation using asymptotic relations between the functions \(n^c\), \(\log n\), and \(c^n\).)*
---
**Explanation:**
This problem involves finding the asymptotic values of given functions in terms of Θ-notation, which is used in computer science to describe the limiting behavior of a function, especially in terms of input size \(n\). Specifically, Θ-notation describes the growth rate of a function, often used to analyze algorithms.
---
**Graphs and Diagrams:**
No graphs or diagrams are provided or required. The focus is strictly on the provided mathematical expressions.
**Steps for Solution:**
1. Identify the term in each expression that grows the fastest as \(n\) increases.
2. Use the identified term to express the asymptotic behavior of the entire function.
3. For each of the given functions in (b) to (e), compare the different components (polynomial, logarithmic, exponential) to determine their respective growth rates.
Refer to asymptotic notations and simplifications to justify the choice of the fastest growing term, showcasing a fundamental understanding of analyzing algorithm efficiency.
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