The following functions have different growth rate (i.e., as n increases by 1, T(n) may increase to different extents depending or what T(n) is). List the following functions by growth rate in increasing order. • T1 = 20n • T2 = n² + 2n T. = n2 + 101ogn
The following functions have different growth rate (i.e., as n increases by 1, T(n) may increase to different extents depending or what T(n) is). List the following functions by growth rate in increasing order. • T1 = 20n • T2 = n² + 2n T. = n2 + 101ogn
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![**Understanding Function Growth Rates**
The following functions have different growth rates (i.e., as \( n \) increases by 1, \( T(n) \) may increase to different extents depending on what \( T(n) \) is). List the following functions by growth rate in increasing order.
- \( T_1 = 20n \)
- \( T_2 = n^2 + 2n \)
- \( T_3 = n^2 + 10\log_2 n \)
- \( T_4 = 2^n \)
### Explanation:
1. **\( T_1 = 20n \)**
- This function grows linearly with \( n \). As \( n \) increases, \( T_1 \) increases proportionally.
2. **\( T_2 = n^2 + 2n \)**
- This function is a quadratic function because the \( n^2 \) term will dominate as \( n \) becomes large.
3. **\( T_3 = n^2 + 10\log_2 n \)**
- Although this function has a logarithmic component, the \( n^2 \) term will dominate as \( n \) becomes large, making it essentially a quadratic function.
4. **\( T_4 = 2^n \)**
- This function is exponential, which grows much faster than polynomial functions for large \( n \).
### Ordered Growth Rates:
From the slowest growth rate to the fastest:
\[ T_1 < T_2 \approx T_3 < T_4 \]
#### Detailed Explanation of Growth:
- **Linear Growth (\( T_1 = 20n \))**:
The simplest growth pattern, where the function increases by a constant amount as \( n \) increases.
- **Quadratic Growth (\( T_2 \) and \( T_3 \))**:
These functions increase much more quickly than linear functions because their dominant term is \( n^2 \). Although \( T_2 \) and \( T_3 \) may seem different, the additional \( 2n \) and \( 10\log_2 n \) terms are insignificant relative to the \( n^2 \) term for large \( n \).
- **Exponential Growth (\( T_4 = 2^n \))**:
This](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1725b614-c61a-470a-b05f-008b5304bcc7%2F2053679c-f144-4a7d-821f-d149ac2b9fcf%2F7w83fx_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Function Growth Rates**
The following functions have different growth rates (i.e., as \( n \) increases by 1, \( T(n) \) may increase to different extents depending on what \( T(n) \) is). List the following functions by growth rate in increasing order.
- \( T_1 = 20n \)
- \( T_2 = n^2 + 2n \)
- \( T_3 = n^2 + 10\log_2 n \)
- \( T_4 = 2^n \)
### Explanation:
1. **\( T_1 = 20n \)**
- This function grows linearly with \( n \). As \( n \) increases, \( T_1 \) increases proportionally.
2. **\( T_2 = n^2 + 2n \)**
- This function is a quadratic function because the \( n^2 \) term will dominate as \( n \) becomes large.
3. **\( T_3 = n^2 + 10\log_2 n \)**
- Although this function has a logarithmic component, the \( n^2 \) term will dominate as \( n \) becomes large, making it essentially a quadratic function.
4. **\( T_4 = 2^n \)**
- This function is exponential, which grows much faster than polynomial functions for large \( n \).
### Ordered Growth Rates:
From the slowest growth rate to the fastest:
\[ T_1 < T_2 \approx T_3 < T_4 \]
#### Detailed Explanation of Growth:
- **Linear Growth (\( T_1 = 20n \))**:
The simplest growth pattern, where the function increases by a constant amount as \( n \) increases.
- **Quadratic Growth (\( T_2 \) and \( T_3 \))**:
These functions increase much more quickly than linear functions because their dominant term is \( n^2 \). Although \( T_2 \) and \( T_3 \) may seem different, the additional \( 2n \) and \( 10\log_2 n \) terms are insignificant relative to the \( n^2 \) term for large \( n \).
- **Exponential Growth (\( T_4 = 2^n \))**:
This
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