Problem 6 - Thomas Calculus in SI Units

Question

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? a. \(\log _{2}\left(x^{2}\right)\) b. \(\log _{10} 10 x\) c. \(1 / \sqrt{x}\) d. \(1 / x^{2}\) e. \(x-2 \ln x\) f. \(e^{-x}\) g. \(\ln (\ln x)\) h. \(\ln (2 x+5)\)

Step-by-Step Solution

Answer

a, b, h grow at the same rate; e grows faster; c, d, f, g grow slower.

Step 1: Identify the Type of Growth

We need to compare the growth rates of different functions with respect to the natural logarithm function \( \ln x \) as \( x \to \infty \). For each function, we'll evaluate it by substitution or direct comparison.

Step 2: Analyze \( a. \log _{2}(x^{2}) \)

\( \log _{2}(x^{2}) = 2 \log _{2}(x) = \frac{2 \ln x}{\ln 2} \). Since this expression is a multiple of \( \ln x \), it grows at the same rate as \( \ln x \).

Step 3: Analyze \( b. \log _{10}(10x) \)

\( \log _{10}(10x) = \log _{10}(10) + \log _{10}(x) = 1 + \log _{10}(x) \). \( \log _{10}(x) = \frac{\ln x}{\ln 10} \). This expression grows at the same rate as \( \ln x \).

Step 4: Analyze \( c. \frac{1}{\sqrt{x}} \)

As \( x \rightarrow \infty \), \( \frac{1}{\sqrt{x}} \rightarrow 0 \). Therefore, it grows slower than \( \ln x \).

Step 5: Analyze \( d. \frac{1}{x^{2}} \)

As \( x \rightarrow \infty \), \( \frac{1}{x^{2}} \rightarrow 0 \). Therefore, this function grows slower compared to \( \ln x \).

Step 6: Analyze \( e. x-2 \ln x \)

The dominant term as \( x \rightarrow \infty \) is \( x \) which grows faster than \( \ln x \). Thus, \( x-2 \ln x \) grows faster than \( \ln x \).

Step 7: Analyze \( f. e^{-x} \)

As \( x \rightarrow \infty \), \( e^{-x} \rightarrow 0 \). Therefore, this function grows slower than \( \ln x \).

Step 8: Analyze \( g. \ln(\ln x) \)

\( \ln(\ln x) \) is the logarithm of a logarithm and grows slower than \( \ln x \) as \( x \) increases.

Step 9: Analyze \( h. \ln(2x+5) \)

\( \ln(2x+5) \approx \ln 2 + \ln x \). As \( x \rightarrow \infty \), it behaves similarly to \( \ln x \), thus growing at the same rate as \( \ln x \).

Key Concepts

Growth Rates

Understanding how different functions grow is important, especially when analyzing their behavior as variables approach infinity. The term "growth rate" helps us determine how quickly the value of a function increases as its input becomes very large. This is particularly useful in mathematics and computer science where we need to compare and contrast functions.

• For instance, polynomial functions such as \(x^2\) have a higher growth rate than linear functions like \(x\).
• Exponentials, such as \(e^x\), grow even faster than polynomial functions.
• On the other hand, logarithmic functions like \(\ln x\) grow slower than linear functions but faster than functions that approach zero like \(\frac{1}{x}\).

This comparison gives us a tool to determine which functions dominate others in the long run.

Asymptotic Behavior

The concept of asymptotic behavior deals with the behavior of functions as they approach a limit, often infinity. This is especially relevant for comparing growth rates. Asymptotic behavior helps us make predictions about which functions will eventually surpass others.
Understanding asymptotic behavior involves:

• Identifying which terms in a function become negligible as \(x\) becomes very large.
• Recognizing dominant terms guiding the function's growth over time.
• Determining if two functions might "behave similarly" by checking if their dominant terms match.

By analyzing the asymptotic nature, we can simplify complex functions to their most critical components, focusing on what will continuously influence their growth.

Comparison of Functions

Comparing functions involves analyzing their growth rates and asymptotic behaviors directly against each other. This can be done by examining their dominant terms and calculating limits if needed.
When comparing functions:

• Functions like \(x - 2 \ln x\) might initially seem complex. Yet, when you identify \(x\) as the dominant term, it becomes clear this function grows faster than \(\ln x\).
• Contrastingly, functions like \(\ln(\ln x)\) show their slower-growing nature immediately, due to the additional logarithmic layer, which significantly diminishes growth.
• Through comparison, discovering such subtleties aids in identifying which functions hold their own as inputs grow ever larger.

By comparing multiple functions, students learn not only which functions are faster or slower but why this behavior occurs, reinforcing deeper understanding.