Problem 2 - Calculus Early Transcendentals

Question

Sketch the graph of \(f(x)=\ln |x|\) and explain how the graph shows that \(f^{\prime}(x)=\frac{1}{x}\).

Step-by-Step Solution

Answer

Question: Describe why the derivative of the function \(f(x)=\ln|x|\) is equal to \(f'(x)=\frac{1}{x}\) for positive values of x, and \(f'(x)=-\frac{1}{x}\) for negative values of x. Answer: The derivative of the function \(f(x)=\ln|x|\) shows the rate of change of the function with respect to x. The graph of \(f(x)=\ln|x|\) shows that the slope increases as x moves away from the y-axis on either side (positive or negative). The positive values of the derivative \(f'(x)=\frac{1}{x}\) for x > 0 indicate that the function is increasing in the positive x-direction. Similarly, the negative values of the derivative \(f'(x)=-\frac{1}{x}\) for x < 0 show that the function is decreasing in the negative x-direction. These two separate expressions show complementary characteristics between the function and its derivative, reflecting the behavior of \(f(x)=\ln|x|\) in the respective x-directions.

Step 1: Sketch the function \(f(x)=\ln |x|\)

To sketch the graph, we will first analyze the properties of the natural logarithm function, \(\ln(x)\), and the absolute value function, \(|x|\). The natural logarithm function is only defined for positive numbers, and it increases monotonically (always increasing) as \(x\) increases. The absolute value function reflects the input, \(x\), over the y-axis if it is negative. So, \(\ln |x|\) will be the same as \(\ln(x)\) for \(x > 0\), and it will be the reflection of the graph of \(\ln(x)\) when we replace \(x\) by \(-x\), for \(x < 0\). 1. For \(x > 0\), \(f(x) = \ln x\). 2. For \(x < 0\), \(f(x) = \ln|-x|\) which will be the reflection of \(\ln x\) over the y-axis. Now, we can sketch the graph of \(f(x)=\ln |x|\) using these insights.

Step 2: Understanding the graph of \(f'(x) = \frac{1}{x}\)

Now that we have sketched the graph of \(f(x)=\ln |x|\), we can analyze its derivative, \(f'(x)\). We should first differentiate the function \(f(x)\) with respect to \(x\). Differentiating \(f(x) = \ln |x|\) with respect to \(x\) involves the use of the chain rule. Recall that the chain rule states that \((g(h(x)))' = g'(h(x)) \cdot h'(x)\). Here, let \(g(u) = \ln u\) and \(h(x) = |x|\). So, \(g'(u) = \frac{1}{u}\) and \(h'(x)\) depends on the value of \(x\). For \(x > 0\), \(h'(x) = 1\). For \(x < 0\), \(h'(x) = -1\). Now, applying the chain rule, we get: 1. For \(x > 0\), \(f'(x) = g'(h(x)) \cdot h'(x) = \frac{1}{|x|} \cdot 1 = \frac{1}{x}\). 2. For \(x < 0\), \(f'(x) = g'(h(x)) \cdot h'(x) = \frac{1}{|-x|} \cdot (-1) = -\frac{1}{x}\). Notice that the derivative is not defined at \(x=0\). Now, we can plot the graph of \(f'(x) = \frac{1}{x}\).

Step 3: Relating the graphs of \(f(x)=\ln |x|\) and \(f'(x)=\frac{1}{x}\)

From the graph of \(f(x) = \ln |x|\), we can observe that the slope of the function increases as \(x\) moves away from the y-axis on either side (positive or negative). This means that the rate of change of the function increases as the value of \(x\) increases (positively or negatively). When we look at the graph of \(f'(x) = \frac{1}{x}\), we see that the derivative has positive values for \(x > 0\) and negative values for \(x < 0\) (except undefined at \(x = 0\)). This indicates that the original function \(f(x) = \ln |x|\) is increasing in the positive x-direction and decreasing in the negative x-direction, which matches our observation from the graph of the function. In conclusion, the graph of \(f(x) = \ln |x|\) and its derivative \(f'(x) = \frac{1}{x}\) show complementary characteristics. The function \(f(x)\) increases as \(x\) moves away from the y-axis, and the derivative \(f'(x)\) reflects this behavior, confirming the relationship between the two graphs.

Key Concepts

Natural Logarithm Function

The natural logarithm function, denoted as \(\ln(x)\), is a special mathematical function that is only defined for positive values of \(x\). It is a way to express the logarithm to the base \(e\), where \(e\) is approximately equal to 2.718. The function has an important property: it increases monotonically, meaning it always gets larger as \(x\) increases.

This function is crucial in many areas of mathematics due to its simple differentiation and integration properties. For example, the derivative of \(\ln(x)\) is \(\frac{1}{x}\), which makes it straightforward to work with, especially when dealing with exponential growth or decay models. When we consider \(\ln |x|\), we are looking at how the natural logarithm behaves over the entire real line, with special handling through the absolute value function, which reflects part of the graph over the y-axis.

Absolute Value Function

The absolute value function, represented as \(|x|\), is a function that transforms any real number into a non-negative number. Essentially, it returns the magnitude of the number without considering its sign. This function can be formally expressed as:

* \(|x| = x\) if \(x \geq 0\)
* \(|x| = -x\) if \(x < 0\).

When combined with the natural logarithm function in \(\ln|x|\), the absolute value plays a crucial role in extending the domain of the logarithm to include negative numbers (except zero, where the logarithm is not defined). For \(x > 0\), \(\ln|x|\) behaves just like \(\ln(x)\). For \(x < 0\), the function essentially reads \(\ln(-x)\), giving us a reflection over the y-axis.

Chain Rule

Differentiating functions like \(f(x) = \ln|x|\) requires the use of the chain rule. The chain rule is a fundamental tool in calculus used to find the derivative of composite functions. It states that if we have a function \(g(h(x))\), its derivative is the product of the derivative of \(g\) with respect to its input, and the derivative of \(h(x)\) with respect to \(x\).

In our case, consider \(g(u) = \ln(u)\) and \(h(x) = |x|\). We differentiate as follows:

• For \(x > 0\), \(h'(x) = 1\); hence, \(f'(x) = \frac{1}{x}\).
• For \(x < 0\), \(h'(x) = -1\); hence, \(f'(x) = -\frac{1}{x}\).

Thus, the chain rule allows us to understand the changing behavior of \(\ln|x|\) across different regions of the x-axis. The derivative, \(f'(x)\), reflects these changes and shows how the slope of the function varies.

Graphical Analysis

Graphical analysis helps visualize the behavior of functions like \(f(x) = \ln|x|\) and their derivatives, such as \(f'(x) = \frac{1}{x}\). By sketching the graph of \(f(x)\), we observe symmetry about the y-axis. The natural logarithm function, when applied to absolute values, reflects the portion of the graph \(x < 0\) as its positive counterpart.

The derivative \(f'(x) = \frac{1}{x}\) has distinct characteristics:

• For \(x > 0\), the derivative is positive, indicating the function is increasing.
• For \(x < 0\), the derivative is negative, indicating the function is decreasing.
• The derivative is undefined at \(x = 0\), as division by zero is not possible.

This produces a graph where the function moves steeply upward on the positive side of the y-axis and downward on the negative side, establishing a clear relationship between the function and its rate of change. Graphical analysis confirms the analytic work, showing the function's symmetry and the derivative's asymptotic tendencies towards infinity.