Problem 34 - Calculus: Early Transcendentals

Question

Locate the discontinuities of the function and illustrate by graphing. \( y = \ln (\tan^2 x) \)

Step-by-Step Solution

Answer

Discontinuities occur at \( x = \frac{\pi}{2} + k\pi \), where the function has vertical asymptotes.

Step 1: Identify the domain of tangent function

The function within the natural logarithm is \( \tan^2 x \). The tangent function, \( \tan x \), is undefined where \( x = \frac{\pi}{2} + k\pi \) for integers \( k \). Thus, \( \tan^2 x \) is also undefined at these points.

Step 2: Determine where the argument of ln is undefined or non-positive

The natural logarithm function, \( \ln(x) \), is undefined for \( x \leq 0 \). Since \( \tan^2 x \geq 0 \) for any \( x \), and \( \tan^2 x \) is undefined when \( \tan x \) is undefined, the original function \( \ln(\tan^2 x) \) is undefined at \( x = \frac{\pi}{2} + k\pi \).

Step 3: Observe continuity between discontinuities

Between each interval \( x = \frac{\pi}{2} + k\pi \) and \( x = \frac{\pi}{2} + (k+1)\pi \), the function \( \ln(\tan^2 x) \) is defined and smooth, meaning it is continuous in these intervals.

Step 4: Graph the function considering discontinuities

Plot the function keeping the domains of continuity in mind. At each \( x = \frac{\pi}{2} + k\pi \), there will be a vertical asymptote. Between these asymptotes, \( \ln(\tan^2 x) \) will show the graph rising and falling repeatedly, resembling a series of 'U' shapes between asymptotes.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln(x) \), is a fundamental function in mathematics that can only be applied to positive numbers. The natural logarithm is the inverse function of the exponential function \( e^x \).

It is defined such that \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \) for any positive \( x \). This means that you cannot calculate the natural logarithm of zero or a negative number.

In this exercise, the natural logarithm is applied to \( \tan^2 x \). Since \( \tan^2 x \) is always non-negative, \( \ln(\tan^2 x) \) is undefined when \( \tan^2 x = 0 \), which means when \( \tan x = 0 \). It's important because \( \ln(x) \) is discontinuous when its argument is zero or less.

Tangent Function

The tangent function, denoted as \( \tan(x) \), is one of the basic trigonometric functions. It is periodic, meaning its values repeat in a regular pattern. The tangent function is undefined when \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer, due to the sine component in its ratio \( \frac{\sin(x)}{\cos(x)} \) becoming zero.

These points of undefined values cause vertical asymptotes in the graph, which are lines the curve gets infinitely close to but never touches.

Because \( \tan(x) \) is crucial to our problem, understanding where it is undefined directly informs us about the discontinuities of \( \ln(\tan^2 x) \). By squaring the tangent function, \( \tan^2 x \), it retains the location of these asymptotes, but the function itself is always non-negative.

Intervals of Continuity

Intervals of continuity are the regions in the domain where a function has no breaks, jumps, or undefined points. For the function \( \ln(\tan^2 x) \), the intervals of continuity are between the points where the tangent function is undefined.

Specifically, these intervals are between each point \( x = \frac{\pi}{2} + k\pi \) and \( x = \frac{\pi}{2} + (k+1)\pi \). Within these intervals, \( \tan(x) \) is defined, positive, and \( \ln(\tan^2 x) \) is smooth, leading to a continuous curve in the graph.

Understanding where a function is continuous is crucial for graphing it accurately. It helps in predicting the behavior of the function between its discontinuities.

Vertical Asymptote

A vertical asymptote is a characteristic of a graph where the function approaches but never quite reaches or crosses a vertical line. For the function \( \ln(\tan^2 x) \), vertical asymptotes appear at points \( x = \frac{\pi}{2} + k\pi \), where the tangent function is undefined.

At these points, \( \tan(x) \) goes to infinity, making \( \tan^2 x \) go to infinity as well, causing \( \ln(\tan^2 x) \) to be undefined. When graphing, you will see the curve rising or falling sharply, hinting at the presence of an asymptote.

These asymptotes delineate the boundaries of the intervals of continuity, creating a repeated pattern when the graph is plotted. Vertical asymptotes are vital in understanding the broader behavior of a function, especially in predicting points of discontinuity.