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Magazine Chapter 7: Problem 3
Which of the following functions grow faster than \(x^{2}\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(x^{2} ?\) Which grow slower? \begin{equation} \begin{array}{ll}{\text { a. } x^{2}+4 x} & {\text { b. } x^{5}-x^{2}} \\\ {\text { c. } \sqrt{x^{4}+x^{3}}} & {\text { d. }(x+3)^{2}} & \\ {\text { e. } x \ln x} & {\text { f. } 2^{x}} \\ {\text { g. } x^{3} e^{-x}} & {\text { h. } 8 x^{2}} \end{array} \end{equation}
Short Answer
Expert verified
Functions b and f grow faster; functions a, c, d, and h grow at the same rate; functions e and g grow slower.
Step by step solution
01
Understand Growth Rates
To determine the growth rate of functions, we compare each function's behavior as \(x \rightarrow \infty\) to the function \(x^2\). We categorize them as faster (\(f(x)/x^2 \rightarrow \infty\)), same rate (\(f(x)/x^2 \rightarrow C\), where \(Ceq 0, \infty\)), or slower (\(f(x)/x^2 \rightarrow 0\)).
02
Analyze Each Function
We will analyze each function one by one to categorize their growth.
03
Function a: \(x^2 + 4x\)
For \(f(x) = x^2 + 4x\), we have \(\frac{f(x)}{x^2} = \frac{x^2 + 4x}{x^2} = 1 + \frac{4}{x}\). As \(x \rightarrow \infty\), \(\frac{4}{x} \rightarrow 0\), thus, \(f(x)/x^2 \rightarrow 1\). This grows at the same rate as \(x^2\).
04
Function b: \(x^5 - x^2\)
For \(f(x) = x^5 - x^2\), \(\frac{f(x)}{x^2} = x^3 - 1\). As \(x \rightarrow \infty\), \(x^3 \rightarrow \infty\), so \(f(x)/x^2 \rightarrow \infty\). This grows faster than \(x^2\).
05
Function c: \(\sqrt{x^4 + x^3}\)
For \(f(x) = \sqrt{x^4 + x^3}\), \(\frac{f(x)}{x^2} = \frac{\sqrt{x^4(1 + x^{-1})}}{x^2} = \frac{x^2\sqrt{1 + x^{-1}}}{x^2} = \sqrt{1 + x^{-1}}\). As \(x \rightarrow \infty\), this approaches 1, so it grows at the same rate as \(x^2\).
06
Function d: \((x+3)^2\)
Expand \((x+3)^2\) to get \(x^2 + 6x + 9\), then \(\frac{f(x)}{x^2} = 1 + \frac{6}{x} + \frac{9}{x^2}\). As \(x \rightarrow \infty\), \(\frac{6}{x}\) and \(\frac{9}{x^2}\) go to 0, so \(f(x)/x^2 \rightarrow 1\), which grows at the same rate as \(x^2\).
07
Function e: \(x \ln x\)
For \(f(x) = x \ln x\), \(\frac{f(x)}{x^2} = \frac{\ln x}{x}\). As \(x \rightarrow \infty\), \(\ln x\) grows slower than \(x\), so \(\frac{\ln x}{x} \rightarrow 0\), meaning it grows slower than \(x^2\).
08
Function f: \(2^x\)
For \(f(x) = 2^x\), \(\frac{f(x)}{x^2} = \frac{2^x}{x^2}\). As \(x \rightarrow \infty\), \(2^x\) grows exponentially, much faster than any polynomial like \(x^2\), thus \(\frac{2^x}{x^2} \rightarrow \infty\). This grows significantly faster than \(x^2\).
09
Function g: \(x^3 e^{-x}\)
For \(f(x) = x^3 e^{-x}\), \(\frac{f(x)}{x^2} = x e^{-x}\). As \(x \rightarrow \infty\), \(e^{-x}\) decays much faster than any polynomial grows, so \(x e^{-x} \rightarrow 0\). This grows slower than \(x^2\).
10
Function h: \(8x^2\)
For \(f(x) = 8x^2\), \(\frac{f(x)}{x^2} = 8\). It grows at a constant multiple of \(x^2\), so it grows at the same rate as \(x^2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Comparison of Functions
When we compare functions, we are interested in how quickly they grow as they approach infinity. This is known as examining their growth rates. By comparing each function's behavior to a reference function鈥攍ike \( x^2 \) in our exercise鈥攚e categorize them into three groups:
- Faster Growth: If a function grows faster than \( x^2 \), this means that as \( x \) becomes very large, the function's value will increase more rapidly than \( x^2 \).
- Same Rate of Growth: If a function grows at the same rate as \( x^2 \), their growth speeds are the same, and neither one will outpace the other significantly over time.
- Slower Growth: If a function grows slower than \( x^2 \), this indicates that its increase is more gradual compared to the quadratic growth of \( x^2 \).
Asymptotic Analysis
Asymptotic analysis is the process of evaluating the long-term behavior of functions. It's crucial for understanding the growth rates of functions, especially as \( x \) approaches infinity. This type of analysis allows us to make simplified approximations about how functions behave when inputs are very large, which is essential in computation and algorithm analysis.
Key to asymptotic analysis is its focus on what's dominant as inputs grow to infinity. For example, in the function \( x^5 - x^2 \), when \( x \) is large, the \( x^5 \) term dominates the growth rate, making the entire expression grow faster than \( x^2 \). By stripping away smaller terms, we can see the core growth characteristics that matter most in a given context.
Key to asymptotic analysis is its focus on what's dominant as inputs grow to infinity. For example, in the function \( x^5 - x^2 \), when \( x \) is large, the \( x^5 \) term dominates the growth rate, making the entire expression grow faster than \( x^2 \). By stripping away smaller terms, we can see the core growth characteristics that matter most in a given context.
Big O Notation
Big O notation provides a formal way to express the upper bound of an algorithm's running time or the growth rate of a function. It's expressed as \( O(f(x)) \), meaning the function grows at most as fast as \( f(x) \) when \( x \) becomes very large.
This notation helps simplify complex evaluations by focusing only on the terms that grow the fastest, ignoring constants and low-order terms. For instance, for the function \( 8x^2 \), even though it equals exactly \( 8x^2 \), in Big O terms it is simply \( O(x^2) \), because the factor of 8 is a constant that doesn鈥檛 affect the form of the growth rate. Understanding Big O notation is essential in computer science for analyzing algorithm efficiency.
This notation helps simplify complex evaluations by focusing only on the terms that grow the fastest, ignoring constants and low-order terms. For instance, for the function \( 8x^2 \), even though it equals exactly \( 8x^2 \), in Big O terms it is simply \( O(x^2) \), because the factor of 8 is a constant that doesn鈥檛 affect the form of the growth rate. Understanding Big O notation is essential in computer science for analyzing algorithm efficiency.
Polynomial Growth
Polynomial growth refers to the growth rate of polynomial functions. These functions often look like \( x^n \) where \( n \) is a non-negative integer. The degree \( n \) indicates the highest power of \( x \) in the polynomial and determines the growth rate.
In our exercise, functions like \( x^5 - x^2 \) and \( 8x^2 \) exhibit polynomial growth, with the \( x^5 \) term growing significantly faster than \( x^2 \), whereas \( 8x^2 \) grows at the same rate but with a larger scale. Polynomial functions are foundational in mathematics and often appear in natural phenomena as well as engineering algorithms.
In our exercise, functions like \( x^5 - x^2 \) and \( 8x^2 \) exhibit polynomial growth, with the \( x^5 \) term growing significantly faster than \( x^2 \), whereas \( 8x^2 \) grows at the same rate but with a larger scale. Polynomial functions are foundational in mathematics and often appear in natural phenomena as well as engineering algorithms.
Exponential Growth
Exponential growth describes a situation in which a quantity grows at a rate proportional to its current value. This kind of growth is depicted in functions like \( 2^x \), where the base is constant, and the exponent \( x \) is the variable.
As \( x \) increases, the output of exponential functions grows much faster than any polynomial. In our analysis, \( 2^x \) grows vastly faster than \( x^2 \). Computer scientists and mathematicians often find exponential growth in algorithms that double in size, signals, or populations in ecology.
As \( x \) increases, the output of exponential functions grows much faster than any polynomial. In our analysis, \( 2^x \) grows vastly faster than \( x^2 \). Computer scientists and mathematicians often find exponential growth in algorithms that double in size, signals, or populations in ecology.
Logarithmic Functions
Logarithmic functions grow at a much slower rate compared to polynomial or exponential functions. They are expressed like \( \ln x \) or \( \log_{10} x \), with a typical real-world example being the time complexity in searching algorithms like binary search.
In our exercise, \( x \ln x \) demonstrates logarithmic-related growth, where the \( \ln x \) grows slowly, causing the whole expression to increase slower than a quadratic term like \( x^2 \). Even as \( x \) becomes very large, logarithmic functions tend to stabilize and grow less aggressively than their polynomial or exponential counterparts.
In our exercise, \( x \ln x \) demonstrates logarithmic-related growth, where the \( \ln x \) grows slowly, causing the whole expression to increase slower than a quadratic term like \( x^2 \). Even as \( x \) becomes very large, logarithmic functions tend to stabilize and grow less aggressively than their polynomial or exponential counterparts.
