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Magazine Chapter 4: Problem 79
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 \([-1,6]\).
Short Answer
Expert verified
\(\sqrt{x}\) grows faster than \(\ln x\).
Step by step solution
01
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.
02
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.
03
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 \(x\).
04
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 \(f(x) = \ln x\).
05
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 \(\ln x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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鈥攖he initial increase is rapid, but it slows down as \(x\) becomes larger.
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.
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 \( \ln x \).
