Problem 124 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(f(x)=\ln x\) and \(g(x)=x^{1 / n}\). (a) Use a graphing utility to graph \(g\) (for \(n=2\) ) and \(f\) in the same viewing window. (b) Determine which function is increasing at a greater rate as \(x\) approaches infinity. (c) Repeat parts (a) and (b) for \(n=3,4,\) and \(5 .\) What do you notice?

Step-by-Step Solution

Answer

The function \(f(x) = \ln x\) always increases at a slower rate than any of the \(g(x) = x^{1/n}\) functions as \(x \) approaches infinity, regardless of the value of \(n\). This is because the gradient of \(g(x)\) for \(x > 1\) is always greater than the gradient of \(f(x)\).

Step 1: Graphing for n=2

Using a graphing utility, graph the function \(f(x)=\ln x\) and \(g(x)=x^{1/2}\) on the same coordinate plane. Then, make observations about the rate at which each function increases as \(x\) tends towards infinity.

Step 2: Determining Rate of Increase for n=2

Determine which function is increasing at a greater rate as \(x\) approaches infinity by analyzing the gradients of the graphed functions.

Step 3: Graphing for n=3,4, and 5

Repeat step 1 for \(n=3,4,5\), separately graphing \(g(x)=x^{1/3}\), \(g(x)=x^{1/4}\), and \(g(x)=x^{1/5}\), each time alongside \(f(x)=\ln x\).

Step 4: Determining Rate of Increase for n=3,4, and 5

Determine which function is increasing at a greater rate as \(x\) approaches infinity for each of the new values of \(n\) by analyzing the gradients of the graphed functions.

Step 5: Observation

Analyze all the acquired data and summarize it by explaining your observation on how the rate of increase of \(f(x) = \ln x\) compares to the rate of increase of \(g(x) = x^{1/n}\) as \(n\) varies.

Key Concepts

Rate of Increase

When analyzing functions, understanding the rate at which they increase is often vital. The rate of increase of a function refers to how quickly its values grow as its input becomes larger. This is frequently examined using the gradient or slope of the function in graphical representations.

For example, in the functions from the original exercise, both the natural logarithm function, \(f(x) = \ln x\), and the root function, \(g(x) = x^{1/n}\), have different rates of increase. As \(x\) approaches infinity, their slopes change.

• For log functions like \(\ln x\), the rate of increase is constantly decreasing since the slope of \(\ln x\) becomes flatter as \(x\) grows larger.
• Conversely, for root functions \(x^{1/n}\), the rate of increase depends on the value of \(n\). As \(n\) becomes larger, the growth rate slows down more significantly.

This concept is crucial particularly in determining which function grows faster when \(x\) is large.

Natural Logarithm Function

The natural logarithm function, \(f(x) = \ln x\), is a fundamental mathematical function widely used in various fields like calculus and economics. Here's what you need to know:

• Definition: The natural logarithm is the logarithm to the base of \(e\), where \(e\) is an irrational constant approximately equal to 2.718281828. It is written as \(\ln(x)\).
• Behavior: As \(x\) increases, \(\ln x\) increases but at a decreasing rate, meaning that its growth becomes gradually slower.

In the context of our exercise, as \(x\) approaches infinity, \(\ln x\) continues to increase, but the rate is slower compared to many algebraic functions, including most root functions. This feature is characteristic of logarithmic functions, making them particularly interesting in comparative analyses of function growth rates.

Root Functions

Root functions take the form \(g(x)=x^{1/n}\), where \(n\) is a positive integer. These functions are mainly characterized by their growth behavior as \(x\) becomes large.

• Definition: A root function is essentially the inverse power function, meaning it increases more slowly than power functions.
• Rate of Increase: As \(n\) increases, the function \(x^{1/n}\) becomes flatter, implying slower growth. For instance, \(x^{1/2}\) (square root) grows faster than \(x^{1/3}\) and this pattern follows with increasing \(n\).

In the context of the exercise, when graphing both root functions and the logarithmic function \(\ln x\), an interesting observation is that for larger values of \(x\), root functions (for higher \(n\)) will eventually grow slower than \(\ln x\), showcasing the remarkable properties of logarithmic versus root functions over different scales.

Comparison of Functions

When comparing functions like \(f(x) = \ln x\) and \(g(x) = x^{1/n}\), the focus often lies in understanding how they behave as \(x\) grows. This involves determining which function increases more significantly, particularly as \(x\) approaches infinity.

Let's break this down:

• For smaller \(x\), root functions tend to increase faster than \(\ln x\) because the initial growth of root functions like \(x^{1/2}\) or \(x^{1/3}\) is steeper.
• However, as \(x\) becomes very large, \(\ln x\)'s rate of increase eventually surpasses that of root functions, especially for higher values of \(n\).
• This shift happens because the slopes of root functions flatten compared to the logarithmic function, whose slope though decreasing, does not flatten as much.

This explains the observations from the given exercise: for each increasing \(n\), \(x^{1/n}\) has a growth rate that diminishes more rapidly in comparison to \(\ln x\), making \(\ln x\) appear as the faster increasing function over greater \(x\). Understanding these distinctions helps in grasping the broader differences between these common mathematical functions.