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Magazine Chapter 5: 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
\(g(x) = \sqrt{x}\) grows faster than \(f(x) = \ln x\) as \(x\) approaches 30.
Step by step solution
01
Understand the Functions
We need to compare the rates of growth of two functions: \( f(x) = \ln x \) and \( g(x) = \sqrt{x} \). The function \( \ln x \) is the natural logarithm, which grows very slowly, whereas \( \sqrt{x} \) grows at a moderate pace.
02
Determine the Domain and Range
We will be graphing these functions over the interval \([-1, 30]\) on the x-axis and \([-1, 6]\) on the y-axis. Note that \( \ln x \) is undefined for \( x \leq 0 \), so its practical domain here will be \( (0, 30] \). \( \sqrt{x} \) is only defined for non-negative values of \( x\), so its domain is \([0, 30]\).
03
Plot \( f(x) = \ln x \)
Starting from just above \( x = 0 \) to \( x = 30 \), plot the natural logarithm function. At \( x = 1 \), \( \ln x = 0 \). As \( x \) increases, \( \ln x \) increases slowly. At \( x = e \approx 2.718 \), it's 1, and it approaches 3.4 as \( x \) reaches 30.
04
Plot \( g(x) = \sqrt{x} \)
Plot the square root function starting from \( x = 0 \). At \( x=0 \), \( g(x)=0 \). As \(x\) increases, \( g(x) \) increases at a slower rate compared to \( \ln x \). At \( x = 1 \), \( g(x) = 1 \), \( x = 4 \), \( g(x) = 2 \), and \( x = 30 \), \( g(x) \) reaches approximately 5.5.
05
Analyze and Compare the Graphs
From \( x=1 \) to \( x=30 \), observe both functions. \( \ln x \) starts lower and crosses \( \sqrt{x} \) between \( x=0 \) to \( x=1 \), indicating \( \ln x \) is initially less than \( \sqrt{x} \) but grows towards 3.4 slowly, whereas \( \sqrt{x} \) is steadier, finishing higher near 5.5 at \( x=30 \). \( g(x) = \sqrt{x} \) grows faster in the range.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
natural logarithm
The natural logarithm, denoted as \( \ln x \), is a logarithmic function with the base of the constant \( e \), which is approximately 2.718. This function is defined only for positive values of \( x \). Here are some important features of the natural logarithm:
- It grows very slowly compared to polynomial functions. As \( x \) increases, the increase in \( \ln x \) becomes less steep.
- When \( x = 1 \), \( \ln x = 0 \), because \( e^0 = 1 \).
- The curve of the natural logarithm is concave, meaning it bends upwards as \( x \) increases.
square root function
The square root function, expressed as \( g(x) = \sqrt{x} \), is another fundamental mathematical function. It is defined for non-negative values of \( x \) and is characterized by its moderate rate of growth. Here are some of its main properties:
- Starts at \( (0,0) \), implying \( g(0) = 0 \).
- The function is increasing, meaning as \( x \) becomes larger, \( g(x) \) also becomes larger.
- Has a moderate rate of growth鈥攊t grows faster than the natural logarithm but slower than linear or quadratic functions.
- The curve of \( \sqrt{x} \) is also concave, similar to \( \ln x \), but the rate of growth is more even across its domain.
domain and range
In mathematics, the domain and range of a function are critical elements that define where a function exists and the values it can take.
- Domain: This refers to all the possible input values (\( x \)) for which the function is defined. For the natural logarithm \( \ln x \), the domain is restricted to positive real numbers (\( 0, \infty \)), while for \( \sqrt{x} \), the domain is \( [0, \infty) \). Thus, any graphing on a real plane must respect these constraints.
- Range: This represents all the possible output values of the function. \( \ln x \) can take any real number as it slowly increases without bound. On the other hand, the range of \( \sqrt{x} \) starts at 0 and increases indefinitely as \( x \) increases.
