Problem 5 - 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. \(\log _{3} x\) b. \(\ln 2 x\) c. \(\ln \sqrt{x}\) d. \(\sqrt{x}\) e. \(x\) f. \(5 \ln x\) g. \(1 / x\) h. \(e^{x}\)

Step-by-Step Solution

Answer

Functions growing faster are \(\sqrt{x}\), \(x\), and \(e^{x}\); those growing at the same rate are \(\log_3 x\), \(\ln 2x\), \(\ln \sqrt{x}\), \(5 \ln x\); slower is \(1/x\).

Step 1: Understand the Problem

We are tasked with determining the relative growth rates of given functions compared to the natural logarithm function, \(\ln x\), as \(x\) approaches infinity. The goal is to categorize each function as growing faster, at the same rate, or slower than \(\ln x\).

Step 2: Analyze Logarithmic Functions

Let's compare logarithmic functions first. - **a.** \(\log_3 x\) is another logarithmic function. By change of base formula, \( \log_3 x = \frac{\ln x}{\ln 3} \), which implies it grows at the same rate as \(\ln x\). - **b.** \(\ln 2x = \ln 2 + \ln x\), which grows at the same rate as \(\ln x\) because the constant term \(\ln 2\) does not affect growth rate as \( x \to \infty \).- **c.** \(\ln \sqrt{x} = \frac{1}{2} \ln x\) grows at the same rate as \(\ln x\).

Step 3: Analyze Polynomial and Exponential Functions

Now, let's look at non-logarithmic functions. - **d.** \(\sqrt{x}\) grows faster than \(\ln x\) because polynomial functions grow faster than logarithmic functions.- **e.** \(x\) grows faster than \(\ln x\); linear functions grow faster than logarithmic functions.- **f.** \(5 \ln x\) grows at the same rate as \(\ln x\) since it is a constant multiple of \(\ln x\).

Step 4: Analyze Inverses and Exponentials

Finally, consider the inverses and exponential functions.- **g.** \(\frac{1}{x}\) grows slower than \(\ln x\) because as \( x \to \infty \), \(\frac{1}{x}\) actually approaches zero.- **h.** \(e^x\) grows much faster than \(\ln x\); exponential functions grow much faster compared to logarithmic functions.

Key Concepts

Logarithmic Functions

Logarithmic functions are fundamental when it comes to understanding growth rates. When we talk about \(\ln x\), we're referring to the natural logarithm, which is the logarithm with base e (approximately 2.718).
Logarithmic functions grow very slowly compared to most other functions. As \(x\) becomes larger, the rate of increase of \(\ln x\) becomes very gradual. Still, there are subtle differences between different types of logarithmic functions, even if they are very closely related.

• For example, \(\log_3 x\) is essentially \(\ln x\) divided by a constant (\

Polynomial Growth

Polynomial growth represents functions that involve powers of \(x\), such as \(x^n\), where \(n\) is a positive real number. This kind of growth is generally much faster than logarithmic growth, especially as \(x\) becomes large.
In our list, \(\sqrt{x}\) and \(x\) are considered polynomial functions.

• \(\sqrt{x}\), or \(x^{1/2}\), grows faster than \(\ln x\). Despite being a root function, it outpaces logarithmic functions as \(x\) increases due to its polynomial nature.
• Similarly, \(x\) (or \(x^{1}\)) grows even quicker. This shows that even low-degree polynomials eventually surpass logarithmic growth.

Exponential Growth

Exponential functions, like \(e^x\), grow extraordinarily fast, much faster than any polynomial or logarithmic function. These functions involve raising a constant to the power of \(x\), which means that even small changes in \(x\) can lead to massive increases in the function's value.

• \(e^x\) is the quintessential example of exponential growth. As \(x\) increases, \(e^x\) keeps accelerating, surpassing all polynomial and logarithmic functions, making it one of the fastest-growing functions.

Exponential growth is often used to model real-world phenomena like population growth or radioactive decay due to its rapid increase.

Comparative Growth Analysis

Comparing the growth rates of different types of functions is a key exercise in understanding how they relate to each other. It helps in figuring out which functions dominate in the limit as \(x\) becomes very large.
We used the problem statement to categorize various functions by their growth rates compared to \(\ln x\):

• Functions like \(\log_3 x\), \(\ln 2x\), \(\ln \sqrt{x}\), and \(5 \ln x\) grow at the same rate as \(\ln x\), despite potential constant multipliers or additional constant terms.
• Polynomial functions \(\sqrt{x}\) and \(x\) outgrow \(\ln x\) due to their power-based growth.
• The exponential function \(e^x\) grows significantly faster than \(\ln x\).
• Meanwhile, functions like \(\frac{1}{x}\) showcase slower growth, often diminishing as \(x\) increases.

Understanding these differences is crucial for advanced study and applications in fields like mathematics, computer science, and beyond.