Problem 127 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) Use a graphing utility to complete the table for the function \(f(x)=(\ln x) / x.\) $$\begin{array}{|l|l|l|l|l|l|l|}\hline x & 1 & 5 & 10 & 10^{2} & 10^{4} & 10^{6} \\\\\hline f(x) & & & & & & \\\\\hline\end{array}$$ (b) Use the table in part (a) to determine what value \(f(x)\) approaches as \(x\) increases without bound. Use the graphing utility to confirm your result.

Step-by-Step Solution

Answer

The values for \(f(x)\) calculated using the function \(f(x) = \frac{{ln(x)}}{x}\) are: 0, 0.322, 0.230, 0.046, 0.009, and 0.0014 respectively for \(x=1, 5, 10, 10^2, 10^4, 10^6\). As \(x\) increases, \(f(x)\) approaches 0.

Step 1: Calculate Table Values

Use the function \(f(x) = \frac{{ln(x)}}{x}\) and fill in the blank spaces in the table. To do so, replace the value of \(x\) in the function with the associated \(x\) value in the table: \n\nAt \(x=1\), \(f(x) = \frac{{ln(1)}}{1} = 0\)\nAt \(x=5\), \(f(x) = \frac{{ln(5)}}{5} ≈ 0.322\)\nAt \(x=10\), \(f(x) = \frac{{ln(10)}}{10} ≈ 0.230\)\nAt \(x=10^2\), \(f(x) = \frac{{ln(10^2)}}{10^2} = \frac{2ln(10)}{100} ≈ 0.046\)\nAt \(x=10^4\), \(f(x) = \frac{{ln(10^4)}}{10^4} = \frac{4ln(10)}{10000} ≈ 0.009\)\nAt \(x=10^6\), \(f(x) = \frac{{ln(10^6)}}{10^6} = \frac{6ln(10)}{1000000} ≈ 0.0014

Step 2: Analyze the Pattern

As \(x\) increases, the value of \(f(x)\) is decreasing and getting closer to zero.

Step 3: Confirm with Graphing Tool

Look at the graph of the function \(f(x)=\frac{{ln(x)}}{x}\) using a graphing tool. As \(x\) goes into infinity, the graph of \(f(x)\) approaches y=0, which confirms the results found earlier.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool often used in mathematics to visualize the behavior of functions. In this particular exercise, the graphing utility helps us to understand how the function \( f(x) = \frac{\ln x}{x} \) behaves as \( x \) increases. When you input this function into the graphing tool, you will see how the graph behaves over different values: from small to very large.

Why is this important? Graphs provide a visual representation that shows not just the value of the function at specific points, but also how the values trend and change over a range. This visual can confirm what we calculate from a table.

• Visual confirmation of calculations
• Insight into the overall shape and direction of the function

By observing the graph, you'll notice \( f(x) \) decreases as \( x \) becomes very large, converging to 0. This helps us predict how the function behaves in the long run, which is a critical skill in calculus.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is used to transform multiplicative relationships into additive ones, which is useful in calculus to solve equations involving exponential growth or decay.

In this exercise, \( \ln x \) is used in the function \( f(x) = \frac{\ln x}{x} \). Understanding \( \ln x \) is vital because it influences the behavior of the function as \( x \) changes.
Some key properties of natural logarithms include:

• \( \ln 1 = 0 \): Any logarithm of 1 is 0 because any base raised to the power of 0 is 1.
• \( \ln(ab) = \ln a + \ln b \): This logarithmic identity shows how logarithms convert multiplication into addition.
• \( \ln(a^b) = b \cdot \ln a \): The power rule for logarithms, separating the exponent from the logarithm.

By applying these properties, we can see how the numerator \( \ln x \) grows at a slower rate compared to the denominator \( x \) in the function, leading the overall fraction to decrease as \( x \) increases.

Function Behavior Analysis

Function behavior analysis is about understanding how a function behaves as variables within the function change. In calculus, this often includes investigating limits, asymptotic behavior, and trends.

For the function \( f(x) = \frac{\ln x}{x} \), we are interested in what happens as \( x \) increases towards infinity. Here, we start with plugging specific values into our function and then observing the trend. What we notice is that although \( \ln x \) increases, it does so slowly compared to \( x \), making the output \( f(x) \) decrease.
Key elements of analyzing function behavior include:

• Identifying trends: Does the function increase or decrease?
• Determining limits: What value does the function approach as \( x \) gets very large?
• Using graphical tools for confirmation.

In this case, as \( x \) becomes extremely large, \( \ln x \) still increases but not fast enough to keep up with \( x \) itself, causing \( f(x) \) to approach 0. This shows us that understanding both the algebra and visualization of a function is crucial for complete function behavior analysis.