Problem 38 - Thomas Calculus

Question

Another way to avoid incorrect connections when using a graphing device is through the use of a “dot mode,” which plots only the points. If your graphing utility allows that mode, use it to plot the functions in Exercises 37–40. $$ y=\sin \frac{1}{x} $$

Step-by-Step Solution

Answer

Plot \( y = \sin \left( \frac{1}{x} \right) \) using dot mode to visualize discrete points.

Step 1: Understand the Function

The function given is \( y = \sin \left( \frac{1}{x} \right) \). This is a trigonometric function where the input to the sine function is the reciprocal of \( x \).

Step 2: Recognize Domain Restrictions

The function \( y = \sin \left( \frac{1}{x} \right) \) is defined for all real \( x \) except \( x = 0 \). At \( x = 0 \), the expression \( \frac{1}{x} \) is undefined, so \( x = 0 \) is not part of the domain.

Step 3: Configure Graphing Utility in Dot Mode

Use the graphing utility's settings to enable 'dot mode' which will plot only discrete points rather than connecting lines.

Step 4: Set Range for Graph

Choose a suitable range for \( x \) values to illustrate the behavior of the function. A range such as \([-5, 5]\) can be used, excluding \( x = 0 \).

Step 5: Plot the Function

In 'dot mode', input the function \( y = \sin \left( \frac{1}{x} \right) \) into the graphing utility and observe how the points are plotted across the chosen range, capturing the oscillatory nature of the sine of the reciprocal.

Key Concepts

Dot Mode

Dot mode is a helpful feature on many graphing utilities that simplifies the plotting process. Usually, graphs connect plotted points with lines, which can sometimes misrepresent the function, especially if it has discontinuities or sharp turns.
The essence of dot mode is that it only plots each point without connecting them. This ensures each point is accurate and there's no misleading connection between them.
Consider using dot mode when you have functions like trigonometric ones that can be misleading with straight line connections. By seeing only the plotted points, it's easier to visualize exactly where the function's values are and understand its behavior effectively.

Function Domain

The domain of a function is vital because it specifies the set of input values for which the function is defined. For the function \( y = \sin \left( \frac{1}{x} \right) \), the domain includes all real numbers, except for \( x = 0 \).
Why is \( x = 0 \) excluded? At \( x = 0 \), \( \frac{1}{x} \) becomes undefined, as division by zero is mathematically illogical. Therefore, any graph or analysis of \( y = \sin \left( \frac{1}{x} \right) \) should cater for this by excluding \( x = 0 \) from considerations.
Understanding a function's domain helps avoid errors when graphing or calculating, ensuring accurate mathematical modeling and interpretation.

Graphing Utility

Graphing utilities are powerful tools that help visualize mathematical functions. They transform equations into graphs, offering a clearer view of the function's behavior.
When graphing functionalities like \( y = \sin \left( \frac{1}{x} \right) \), adjusting the settings ensures an accurate graph. One such adjustment is using the dot mode, which prevents misleading connections between points.
Also, defining a suitable range for the \( x \) values, such as \([-5, 5]\), enhances the utility's effectiveness. This chosen range should accommodate enough detail to observe key aspects of the function, without overwhelming the viewer with extraneous data or occurring issues around the undefined points.

Sin Function

The sine function is one of the fundamental trigonometric functions fundamental in mathematics. Traditionally, it's known for its smooth oscillating wave when graphed over a range of values.
However, applying transformations, such as substituting the sine input with \( \frac{1}{x} \), changes its character significantly. Instead of a smooth curve, \( y = \sin \left( \frac{1}{x} \right) \) results in an intriguing pattern of extremely rapid oscillations, especially near the origin.
This rapid oscillation is due to the increasing values of \( \frac{1}{x} \) as \( x \) nears zero. As a student, understanding how these transformations affect the sine function will enhance your ability to analyze complex mathematical graphs and functions.