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Chapter 11: Problem 14

14) For each of the following expressions state whether or not it is periodic, and, if it is periodic, give the period. (a) \(\frac{\sin x}{x}, \quad x \neq 0\), (b) \(|\sin x|\).

Short Answer

Expert verified
a) Not periodic, b) Periodic with a period of 2蟺.

Step by step solution

01

Analyze the Expression for Periodicity

Determine if the given expression is periodic. For an expression to be periodic, there must be a positive number T such that the expression is the same when shifted by T. That is, the function f(x) must satisfy f(x + T) = f(x).
02

Evaluate Periodicity for \( \frac{\sin x}{x} \)

To determine if \( \frac{\sin x}{x} \) is periodic, check if there is any positive T such that \( \frac{\sin(x + T)}{x + T} = \frac{\sin x}{x} \) for all x. Given that \( \sin x \) is periodic with period 2蟺, it would be challenging for \( \frac{\sin x}{x} \) to have the same period because the denominator x is not a periodic function. Thus, \( \frac{\sin x}{x} \) is not periodic.
03

Evaluate Periodicity for \( |\sin x| \)

Assess the periodicity of \( |\sin x| \). The function \( \sin x \) is periodic with a period of 2蟺. Since taking the absolute value does not change the period, \( |\sin x| \) retains this periodicity. Therefore, \( |\sin x| \)is periodic with a period of 2蟺.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Periodicity
Periodicity in mathematics refers to a function repeating its values at regular intervals. If there's a number T such that a function f(x) satisfies f(x + T) = f(x) for all x in its domain, then T is called the period, and the function is periodic. A simple example is the sine function, where \(\text{sin}(x + 2\pi) = \text{sin}(x)\). To check if a function is periodic, you need to determine whether such a T exists.
Sine Function
The sine function \(\text{sin}(x)\) is a trigonometric function that describes a smooth periodic oscillation. It repeats its values every \2\pi\, so it has a period of \2\pi\. This means \(\text{sin}(x + 2\pi) = \text{sin}(x)\). The sine curve starts at 0, goes up to 1, back down through 0 to -1, and returns to 0 again, covering one period. Because of its symmetry and periodic nature, the sine function is widely used in mathematics, physics, and engineering.
Absolute Value
The absolute value of a function, denoted \|f(x)|\, measures the distance of the function's value from zero without considering direction. For instance, \|\text{sin}(x)|\ takes all the negative values of \(\text{sin}(x)\) and makes them positive. The periodic nature of \(\text{sin}(x)\) is retained when taking its absolute value, so \|\text{sin}(x)|\ has a period of \2\pi\ as well. This process does not affect the function's period, but it does transform it into a non-negative function.
Denominator Analysis
Denominator analysis is crucial when determining if a ratio of functions, such as \(\frac{\text{sin}(x)}{x}\), is periodic. Here, the numerator \(\text{sin}(x)\) is periodic with period \2\pi\. However, the denominator, x, is not periodic; it changes continuously without repeating at fixed intervals. This makes \(\frac{\text{sin}(x)}{x}\) non-periodic. To be periodic, both the numerator and the denominator need to satisfy the periodic condition simultaneously, which isn't the case here.

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