Problem 129 - Calculus and its applications

Question

Let \(f(x)=\ln |x|\). a) Using a graphing utility, sketch the graph of \(f\) b) Find the slopes of the tangent lines at \(x=-3,\) \(x=-2\) and \(x=-1\) c) How do your answers for part (b) compare to the slopes of the tangent lines at \(x=3, x=2\) and \(x=1 ?\) d) In your own words, explain how the Chain Rule can be used to show that \(\frac{d}{d x} \ln (-x)=\frac{1}{x}\).

Step-by-Step Solution

Answer

The slopes at \(x = -3, -2, -1\) are the negatives of those at \(x = 3, 2, 1\). This is due to the symmetry of \(f(x) = \ln |x|\).

Step 1: Understanding the Function and Its Domain

The function given is \(f(x)=\ln |x|\). The domain of this function is all real numbers except zero, because the natural logarithm function requires positive arguments, and the absolute value ensures the input to \( \ln \) is positive. Thus, \(x\) can be any nonzero real number.

Step 2: Sketching the Graph using a Graphing Utility

Let's graph the function \(f(x)=\ln |x|\) using a graphing utility. It will show two parts: for \(x > 0\), the graph should look like the natural log function \(\ln x\), and for \(x < 0\), the function becomes \(\ln (-x)\) which mirrors about the y-axis the graph of \(\ln x\).

Step 3: Finding the Derivative to Determine Tangent Slopes

Find the derivative of \(f(x) = \ln |x|\). Using the chain rule, \(f'(x) = \frac{d}{dx} \ln |x| = \frac{1}{x}\cdot \frac{d}{dx}(|x|) = \frac{x}{|x|x} = \frac{1}{x}\) since \(\frac{d}{dx}(|x|) = \frac{x}{|x|}\). So, the derivative \(f'(x) = \frac{1}{x}\).

Step 4: Calculating Tangent Slopes at Negative x-values

For \(x = -3\), \(f'(-3) = \frac{1}{-3}\). For \(x = -2\), \(f'(-2) = \frac{1}{-2}\). For \(x = -1\), \(f'(-1) = \frac{1}{-1} = -1\).

Step 5: Calculating Tangent Slopes at Positive x-values

For \(x = 3\), \(f'(3) = \frac{1}{3}\). For \(x = 2\), \(f'(2) = \frac{1}{2}\). For \(x = 1\), \(f'(1) = \frac{1}{1} = 1\).

Step 6: Comparing Slopes at Corresponding Positive and Negative x-values

Notice the slopes at \(x = -3, -2, -1\) are the negatives of the slopes at \(x = 3, 2, 1\) respectively. This symmetry is due to the function \(\ln |x|\) having different inputs on the mirrored parts.

Step 7: Using the Chain Rule to Differentiate \(\ln (-x)\)

Let \(u = -x\). Then \(\ln(-x) = \ln(u)\), and \(\frac{d}{dx} = \frac{d}{du}\ln(u) \cdot \frac{du}{dx} = \frac{1}{u} \cdot (-1) = -\frac{1}{u} = \frac{1}{x}\). This shows how the derivative \(\frac{d}{dx}\ln(-x) = \frac{1}{x}\) using the chain rule.

Key Concepts

Derivative

In calculus, a derivative represents the rate at which a function is changing at any given point. For a function, the derivative is a new function that gives the slope of the original function's tangent line at any given point.

When working with the function \(f(x) = \ln |x|\), the derivative tells us how steep the graph of the function is at any particular value of \(x\). For this particular case, the derivative is \(f'(x) = \frac{1}{x}\), which is obtained from the natural logarithm property and the properties of absolute values.

Since \(\frac{1}{x}\) changes sign with x, the slope of \(f(x)\) is positive when \(x > 0\) and negative when \(x < 0\), demonstrating that the function graph has a symmetric behavior about the y-axis.

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of a composite function. This rule states that to differentiate a composite function, say \(f(g(x))\), you take the derivative of the outer function evaluated at the inner function and multiply it by the derivative of the inner function.

For the function \(f(x) = \ln|x|\), understanding the chain rule is pivotal in calculating its derivative. Breaking it down, we see \(f(x)\) as \(\ln(u)\), where \(u = |x|\). The derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\) and the derivative of \(|x|\) with respect to \(x\) is \(\frac{x}{|x|}\). Applying the chain rule, we get \(f'(x) = \frac{1}{|x|} \cdot \frac{x}{|x|} = \frac{1}{x}\). This calculation shows how effectively the chain rule can simplify complex derivative processes.

Natural Logarithm

The natural logarithm function, denoted as \(\ln(x)\), is the inverse of the exponential function \(e^x\). Its main feature is its connection with exponential growth and compound interest calculations.

The natural log of a number \(x\) gives the power to which the base \(e\) must be raised to produce \(x\). It is defined only for positive values of \(x\), which makes the concept of \(\ln|x|\) very interesting. Here, it allows us to extend the domain of the natural logarithm to include negative numbers by inserting an absolute value term, thereby modifying it to \(\ln|x|\).

The graph of \(\ln(x)\) is a curve that ascends slowly and has a vertical asymptote at x=0. This profound connection between exponentials and logarithms allows this natural logarithm to find applications in various fields such as physics, statistics, and finance.

Function Graphing

Graphing functions is an essential skill in calculus that helps visualize how a function behaves across its domain. It provides insights into key properties such as intercepts, asymptotes, and intervals of increase or decrease.

For the function \(f(x) = \ln |x|\), graphing involves understanding how the absolute value modifies the input, creating symmetry about the y-axis.

• For \(x > 0\), \(f(x)\) behaves like \(\ln(x)\).
• For \(x < 0\), \(f(x) = \ln(-x)\) is essentially the mirror image of \(\ln(x)\) about the y-axis.

Using a graphing utility, you will notice that as \(x\) approaches zero from both sides, the function tends toward negative infinity, highlighting that the y-axis is a vertical asymptote.

This helps to understand how the function \(\ln |x|\) behaves and allows students to predict values and behavior beyond simple calculations.