Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch the graph of \(f(x)=\frac{1}{\sin x}\) on \([0,2 \pi]\). What is the period of \(f ?\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{1}{\sin x}\) has vertical asymptotes at \(x = 0, \pi, 2\pi\). The function is positive between \(0\) and \(\pi\), and negative between \(\pi\) and \(2\pi\). The period of the function is \(2\pi\).

Step 1: Identify the undefined points

For the function \(f(x)=\frac{1}{\sin x}\), \(x\) can be any real number except the points where \(\sin x = 0\) as it leads to division by zero, which is undefined. In the interval \([0,2\pi]\), \(\sin x = 0\) at \(x = 0, \pi, 2\pi\). So these are the points of discontinuity.

Step 2: Sketch the general shape of \(1/\sin x\)

Generally, \(\frac{1}{\sin x}\) is positive where \(\sin x > 0\) and is negative where \(\sin x < 0\). Also, the graph will trend towards positive or negative infinity (vertical asymptotes) at the points where \(\sin x = 0\). In the interval \([0, 2\pi]\), \(\sin x > 0\) for \(0 < x < \pi\) and \(\sin x < 0\) for \(\pi < x < 2\pi\). The graph will have vertical asymptotes at \(x = 0, \pi, 2\pi)\

Step 3: Calculation of the period

The period of \(f(x)\) is the length of the smallest interval over which the function repeats its values. In this case, \(f(x)=\frac{1}{\sin x}\) repeats its values over the same interval as \(\sin x\). Hence, the period of \(f(x)\) is \(2\pi\), same as the period of \(\sin x\).

Key Concepts

Discontinuity in functions

Discontinuity in functions occurs when there are places in the domain of the function where the function does not have a finite limit, or is not defined. For the function \(f(x) = \frac{1}{\sin x}\), it is undefined wherever \(\sin x = 0\) because division by zero is not possible. In our interval \([0, 2\pi]\), \(\sin x\) equals zero at the points \(x = 0, \pi,\) and \(2\pi\). This means there are vertical asymptotes at these points, making the function discontinuous there.
When sketching graphs of trigonometric functions that have discontinuities, it's crucial to identify these points first so you can avoid incorrect assumptions about where the graph might be continuous or defined. Discontinuities often appear in periodic functions, influencing the overall shape and features of the graph, such as asymptotes, cusps, or breaks.

Periodicity of functions

Periodicity in functions refers to the property that the function repeats its values over regular intervals. This is particularly common in trigonometric functions like sine, cosine, and their reciprocals. The function \(f(x) = \frac{1}{\sin x}\) inherits its periodicity from the \(\sin x\) function.
The period of the sine function, \(\sin x\), is \(2\pi\), meaning it repeats every \(2\pi\) units along the x-axis. As a result, \(f(x)\) also repeats every \(2\pi\), showing the same values and patterns in every cycle. Understanding periodicity helps in predicting the behavior of a function across its domain, especially when sketching or analyzing patterns.

• Look for repeated patterns.
• Identify key characteristic points like peaks, troughs, and zeros.
• Use the period to inform graphing over larger intervals.

Sketching graphs

Sketching graphs of trigonometric functions involves several steps to accurately represent the function's behavior. To sketch \(f(x) = \frac{1}{\sin x}\), start by identifying key points and features of the graph within the interval \([0, 2\pi]\).

First, mark the vertical asymptotes at the discontinuities \(x = 0, \pi, 2\pi\), resulting from \(\sin x = 0\). These asymptotes divide the interval into regions where the function is either positive or negative.

Note that \(f(x)\) is positive between each consecutive pair of asymptotes where \(\sin x > 0\) and negative where \(\sin x < 0\). This leads to pronounced upward or downward trending curves between these points.

• Identify positive and negative regions by referencing \(\sin x\).
• Draw vertical asymptotes where \(\sin x = 0\).
• Depict the direction of the curve as it approaches asymptotes (towards infinity).
• Verify the pattern repeats every \(2\pi\).

Capturing these behaviors visually can enrich your understanding and provide a reliable representation of the function across its domain.