Problem 79 - Precalculus Mathematics for Calculus

Question

Compare the rates of growth of the functions \(f(x)=\ln x\) and \(g(x)=\sqrt{x}\) by drawing their graphs on a common screen using the viewing rectangle \([-1,30]\) by \([-1,6]\).

Step-by-Step Solution

Answer

\(\sqrt{x}\) grows faster than \(\ln x\).

Step 1: Understand the Functions

The two functions we need to compare are \(f(x) = \ln x\) and \(g(x) = \sqrt{x}\). The logarithmic function \(\ln x\) grows at a slower rate than the square root function \(\sqrt{x}\) as \(x\) increases. First, let's understand their basic shapes: \(\ln x\) is a curve that increases but at a decreasing rate, and \(\sqrt{x}\) is a curve that increases at a decreasing rate as well but faster than \(\ln x\). We need to plot these functions to visually compare their rates of growth.

Step 2: Plot the Function \(f(x) = \ln x\)

On the graph spanning the range \([-1, 30]\) for \(x\) and \([-1, 6]\) for \(y\), plot the function \(\ln x\). The graph of \(\ln x\) will start at negative infinity as \(x\) approaches 0 from the right and will increase slowly, showing that it is defined only for positive \(x\) values.

Step 3: Plot the Function \(g(x) = \sqrt{x}\)

On the same graph, plot the function \(\sqrt{x}\). Unlike \(\ln x\), \(\sqrt{x}\) starts at \(0\) for \(x=0\), and increases more rapidly than \(\ln x\) for positive values of \(x\).

Step 4: Compare the Graphs

By observing the plots of \(f(x) = \ln x\) and \(g(x) = \sqrt{x\) on the same screen, compare how they increase. Notice how \(\sqrt{x}\) climbs higher initially and faster compared to \(\ln x\). This visualization confirms that \(g(x) = \sqrt{x}\) has a faster rate of growth than \(f(x) = \ln x\).

Step 5: Confirm with Derivatives

Calculate the derivatives to verify the rate of growth: \( \frac{d}{dx}\ln x = \frac{1}{x} \) and \( \frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}} \). As \(x\) increases, \(\frac{1}{x}\) decreases faster than \(\frac{1}{2\sqrt{x}}\), confirming that \(\sqrt{x}\) grows faster than \(\ln x\).

Key Concepts

Logarithmic Function

A logarithmic function is a mathematical function that grows very slowly as its input increases. In this article, we will focus on the natural logarithm, denoted by \( \ln x \). Its graph is defined only for positive values of \(x\) and exhibits a gentle upward curve.

• As \(x\) increases, \( \ln x \) grows, but the rate of growth diminishes.
• The value of \( \ln x \) becomes larger with larger \(x\) values, but the increase in value is less pronounced as \(x\) continues to grow.

This characteristic makes logarithmic functions very distinct in growth behavior compared to, for instance, polynomial or exponential functions. Despite its slow growth, the logarithmic function is significant in various scientific and engineering contexts due to its properties and behavior in calculations and real-world applications.

Square Root Function

The square root function, denoted by \( \sqrt{x} \), describes a different type of growth compared to the logarithmic function. Starting from zero, it increases as \(x\) grows, representing a curve that rises much like the \( \ln x \) curve but more quickly initially.

• The square root function begins at 0 when \(x = 0\) and grows as \(x\) increases.
• Its growth, however, is not uniform—the initial increase is rapid, but it slows down as \(x\) becomes larger.

The square root function is foundational in algebra and calculus, appearing frequently in geometry, physics, and various analysis problems where relationships involving squares and square roots are essential.

Graph Plotting

Graph plotting is a vital skill in comparing and analyzing mathematical functions like \( \ln x \) and \( \sqrt{x} \). To plot these functions, choose a range for your variables, ensuring that your graphs display the full behavior of each function in the scope you consider.

• Set an appropriate x-range, for instance, from -1 to 30 to avoid complex numbers for these real functions.
• Select a y-range that should capture the output values, like -1 to 6, based on expected function growth.

Using graphing tools, such as a calculator or software, visualize these on a single screen. Compare their paths and observe where one overtakes the other, indicating faster growth. Comparing graphical output gives a clear, visual means to understand otherwise abstract differences in growth rates.

Derivative Calculation

Derivative calculation is a powerful tool to understand how a function behaves and grows. For \(f(x) = \ln x\), the derivative \( \frac{d}{dx} \ln x = \frac{1}{x} \) highlights that as \(x\) becomes larger, the rate at which the function increases slows down significantly.

• The derivative of \( \ln x\), which is \( \frac{1}{x} \), demonstrates diminishing returns on incrementing \(x\).
• In comparison, the derivative of the square root function, \( \frac{d}{dx} \sqrt{x} = \frac{1}{2\sqrt{x}} \), suggests a still slowing but relatively faster growth for small \(x\) values than \( \ln x \).

By analyzing derivatives, we confirm that \( \sqrt{x} \) grows at a quicker pace initially compared to \( \ln x \), reinforcing insights gathered from graph plotting.