Problem 19 - Calculus

Question

Sketch the graph of \(f(x),\) and use this graph to sketch the graph of \(f^{\prime}(x)\). $$f(x)=\ln x$$

Step-by-Step Solution

Answer

Graph \( f(x)=\ln x \) and its derivative \( f'(x)=\frac{1}{x}\) for \( x>0 \), showing increasing "flattening out" behavior.

Step 1: Understand the Function

The given function is \( f(x) = \ln x \), which is the natural logarithm function. The natural logarithm is defined for \( x > 0 \) and it has a vertical asymptote at \( x = 0 \). The graph of \( f(x) \) increases as \( x \) increases, but at a decreasing rate.

Step 2: Sketch the Graph of f(x)

The graph of \( f(x) = \ln x \) is a curve that passes through the point \( (1,0) \) because \( \ln 1 = 0 \). As \( x \) approaches 0 from the right, \( f(x) \rightarrow -\infty \). As \( x \) increases, the graph continues to rise but flattens out. The curve lies exactly above the x-axis but does not touch it and extends to infinity as it moves towards the right.

Step 3: Determine f'(x)

To find the derivative of \( f(x) = \ln x \), we use the derivative rule for natural logarithms: \( f'(x) = \frac{1}{x} \) for \( x > 0 \). This derivative describes the slope of the tangent lines to the curve of \( f(x) \).

Step 4: Analyze f'(x)

The derivative \( f'(x) = \frac{1}{x} \) indicates that the slope of \( f(x) \) is positive for \( x > 0 \). As \( x \) approaches 0, \( f'(x) \) approaches infinity, meaning the curve \( f(x) \) is very steep near the y-axis. As \( x \) increases, the slope decreases, meaning that the curve becomes less steep.

Step 5: Sketch the Graph of f'(x)

The graph of \( f'(x) = \frac{1}{x} \) is a curve that is always above the x-axis for \( x > 0 \), starts at a very large value near \( x = 0^+ \), and approaches zero as \( x \to \infty \).

Key Concepts

Graph sketching

Sketching the graph of a function like the natural logarithm, represented by \( f(x) = \ln x \), is a great way to understand its behavior visually. Start by noting that \( \ln x \) is defined only for \( x > 0 \). It makes a dramatic dive towards negative infinity as \( x \) approaches zero from the right. Imagine standing at \( x = 0^+ \) and looking at the graph shooting down steeply. As you move to the right along the x-axis, the graph begins to climb but at a slower rate.To sketch this, remember a few key points:

• It passes through the point \( (1,0) \) because \( \ln 1 = 0 \).
• There's a vertical asymptote along the y-axis at \( x = 0 \), where the graph heads towards negative infinity.
• The graph continues upward and to the right, but it never crosses the x-axis, showing that it stays positive for \( x > 1 \).

This sketch not only helps in visualizing \( f(x) = \ln x \) but also prepares for sketching its derivative.

Natural logarithm

The natural logarithm function \( f(x) = \ln x \) is a vital part of high school and college-level mathematics. It is a logarithm to the base \( e \), where \( e \approx 2.718 \), an irrational and transcendental number. Understanding its properties can greatly enhance your problem-solving capabilities.Key characteristics include:

• **Domain:** It is only defined for positive \( x \) values, or \( x > 0 \).
• **Range:** The output can be any real number, as the function ranges from negative infinity to infinity.
• **Asymptotic Behavior:** As \( x \) approaches zero, \( \ln x \rightarrow -\infty \).
• **Growth Rate:** While it continuously increases, it does so at a decreasing rate, becoming less steep as \( x \) grows.

These properties make the natural logarithm useful in calculus for differentiation and integration. For instance, the derivative of \( \ln x \) is \( \frac{1}{x} \), a simple yet significant result that helps in various mathematical analyses.

Function analysis

Analyzing the function \( f(x) = \ln x \) and its derivative provides insights into their behavior across different domains. Calculating the derivative as \( f'(x) = \frac{1}{x} \) showcases the relationship between the function and its slope.When conducting a function analysis, consider the following points:

• **Derivative Meaning:** The derivative \( f'(x) = \frac{1}{x} \) indicates the slope of the original function \( f(x) = \ln x \) at any point \( x > 0 \). A steep slope near the y-axis becomes more gentle as \( x \) increases.
• **Behavior Near Zero:** As you approach \( x = 0 \), \( f'(x) \) becomes very large, reflecting the near-vertical rise of \( \ln x \) close to the y-axis.
• **Decreasing Slope:** As \( x \) increases, \( f'(x) \) decreases, slowly approaching zero but never becoming negative or zero.

By studying \( f'(x) \), one can sketch its graph, starting high near \( x = 0^+ \) and steadily lowering towards zero as \( x \to \infty \). This aids in understanding how functions behave, helping students predict and describe changes in graphical representations.