Problem 79 - Algebra and Trigonometry

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

\(g(x) = \sqrt{x}\) grows faster than \(f(x) = \ln x\) as \(x\) approaches 30.

Step 1: 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.

Step 2: 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 \([0, 30]\).

Step 3: 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.

Step 4: 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.

Step 5: 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.

Key Concepts

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.

It is important to note that near \( x = 0 \), the function approaches -infinity, and therefore is undefined for \( x \leq 0 \). This makes understanding its practical domain crucial when working with graphs. For example, in exercises like the one provided, the practical domain needs adjustment since it cannot include zero or negative values.

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—it 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.

For example, as \( x \) moves from 0 to 4, \( g(x) \) quickly climbs to 2, showcasing its faster initial growth compared to \( \ln x \). As \( x \) reaches larger values, such as 30, the function attains values around 5.5, illustrating its steady but moderate increase.

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.

When plotting these functions, understanding domain limitations ensures we graph only valid portions of each function. For example, in the context of the provided exercise, we're interested in the part of the graph visible within the specified viewing area \([-1,30] \) by \([-1,6] \). This requires careful attention to ensure the function respects its domain and the graph's view limits without straying into undefined regions.