Problem 51 - Calculus: Early Transcendentals

Question

Graph the function \( f(x) = \sin (\pi/x) \) of Example 4 in the viewing rectangle \( [-1, 1] \) by \( [-1, 1] \). Then zoom in toward the origin several times. Comment on the behavior of this function.

Step-by-Step Solution

Answer

The function oscillates increasingly as \( x \) approaches zero, becoming infinitely dense near the origin.

Step 1: Understanding the Function

The function given is \( f(x) = \sin (\pi/x) \). This means for each input value \( x \), the output is the sine of \( \pi \) divided by \( x \). The sine function will oscillate, producing values between -1 and 1, and the behavior will be particularly interesting as \( x \) approaches zero, since the argument of the sine function, \( \pi/x \), will approach infinity.

Step 2: Setting Up the Viewing Rectangle

The problem requires graphing the function in the rectangle \( [-1, 1] \) by \( [-1, 1] \). This means the x-axis will range from -1 to 1, and the y-axis will also range from -1 to 1. Since \( x \) can't be 0 (division by zero is undefined), there will be a gap at \( x = 0 \) in the graph.

Step 3: Graphing the Function

Plot the function \( f(x) = \sin(\pi/x) \) between x-values -1 and 1. Due to the nature of \( \sin \), you will observe oscillations that become faster and have higher frequency as \( x \) approaches zero from both the positive and negative sides. Near zero, the curve will show increasing, rapid oscillations.

Step 4: Zooming In Toward the Origin

Zooming in towards the origin will enhance the observation of oscillations. As you zoom closer, the frequency of oscillations near \( x = 0 \) appear infinite, with the curve zig-zagging rapidly between the bounds of the y-axis, showing denser oscillation due to the term \( \pi/x \) elevating infinitely.

Step 5: Analyzing the Behavior

As \( x \) gets closer to zero, the function \( \sin(\pi/x) \) oscillates with increasing, possibly infinite, frequency. This makes the function appear dense with oscillations near the origin, and it is undefined at \( x = 0 \). This behavior is integral to understanding how the sine of reciprocals operates.

Key Concepts

Oscillations

When you hear the word "oscillations," you might think of a pendulum swinging back and forth. In math, particularly with trigonometric functions like sine, oscillations refer to the repetitive up-and-down behavior of the function. For the function \( f(x) = \sin(\pi/x) \), this oscillating behavior is evident because the sine function naturally alternates between values of -1 and 1.
However, in this specific function, the oscillations become even more intriguing, especially as \( x \) approaches zero. Imagine the peaks and valleys of a wave getting closer together—this is what happens with \( f(x) = \sin(\pi/x) \) near the origin. As \( x \) gets very close to zero, you will see rapid oscillations that appear dense on the graph, signaling very high-frequency changes in value.

Sine Function

The sine function is one of the most common trigonometric functions and is often easy to recognize thanks to its wave-like, oscillating graph. It calculates the y-coordinate of a point on the unit circle as it moves around a circle, which is why its values range between -1 and 1.
In the function \( f(x) = \sin(\pi/x) \), we're dealing with the sine of reciprocal values of \( x \). This alters the regular rhythm of the usual sine wave into one where oscillations vary dramatically depending on \( x \). Instead of a gentle, even wave, the function contours shift rapidly as \( x \) nears zero, producing more intense fluctuations than the typical sine curve.

Undefined at Zero

Variables that make mathematical expressions undefined are always key points of interest. For the function \( f(x) = \sin(\pi/x) \), the term \( \pi/x \) becomes problematic at \( x = 0 \) because division by zero is undefined.
This creates a "gap" or a "hole" at \( x = 0 \) on the graph, indicating that the function does not exist at that point. Any attempt to calculate \( \sin(\pi/0) \) doesn't work because it would involve dividing by zero, which mathematics does not allow. Hence, it is crucial to understand that the function \( f(x) = \sin(\pi/x) \) is not continuous across all real numbers; it is specifically undefined at zero.

Frequency of Oscillations

The frequency of oscillations in a function denotes how often the value cycles within a particular range. For \( f(x) = \sin(\pi/x) \), frequency is not constant as it is in a regular sine wave function, but it varies dramatically with \( x \).
As \( x \) gets smaller, the term \( \pi/x \) increases, leading to a rapid succession of oscillations, especially as you approach zero. This means that the closer you zoom in towards \( x = 0 \), the more frequently the oscillations occur, appearing infinite and densely-packed.
This characteristic makes the function an excellent example of how changes in the denominator of a fraction inside a sine function can drastically affect the frequency of oscillations, allowing for engaging visualizations when graphed.