Problem 68 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. The function \(f(x)=\sin \left(x^{2}\right)\) is periodic, with period \(2 \pi\)

Step-by-Step Solution

Answer

The statement is false; \( f(x) = \sin(x^2) \) is not periodic.

Step 1: Understanding the Definition of a Periodic Function

A function \( f(x) \) is periodic with period \( T \) if for all \( x \) in the domain of \( f \), \( f(x + T) = f(x) \). This means that the function repeats its values regularly after a certain interval \( T \).

Step 2: Determine if \( f(x) = \sin(x^2) \) Has a Period

For \( f(x) = \sin(x^2) \), check if there exists a \( T \) such that \( \sin((x+T)^2) = \sin(x^2) \) for all \( x \). Begin by expanding \( (x + T)^2 = x^2 + 2xT + T^2 \). The condition becomes \( \sin(x^2 + 2xT + T^2) = \sin(x^2) \). This requires \( \sin(2xT + T^2) \) to be zero for all \( x \).

Step 3: Analyze the Equation \( \sin(2xT + T^2) = 0 \)

The equation \( \sin(2xT + T^2) = 0 \) implies that \( 2xT + T^2 \) must be a multiple of \( \pi \), meaning \( 2xT + T^2 = n\pi \) for some integer \( n \). However, for a fixed \( T \), this cannot hold true for all \( x \). The variable \( x \) makes the terms quadratic and linear, which can't all satisfy this equation for every \( x \).

Step 4: Conclusion about the Periodicity

Since no specific \( T \) can satisfy \( f(x+T) = f(x) \) for all \( x \), \( f(x) = \sin(x^2) \) has no period. Therefore, the statement that \( f(x) = \sin(x^2) \) is periodic with period \( 2\pi \) is false.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental aspect of mathematics, often utilized in various fields such as physics and engineering. These functions include sine, cosine, and tangent, and they relate the angles of a triangle to the lengths of its sides.

Key properties of trigonometric functions include:

• They are periodic, meaning they repeat their values over regular intervals.
• They are defined for all real numbers.
• They are based on the unit circle, where the sine of an angle is the y-coordinate of the corresponding point on the circle.

The sine function, in particular, is a smooth and continuous function. It plays a vital role in describing phenomena such as sound waves and light. Always remember that the nature of these functions mimics natural patterns, making them excellent tools for modeling physical behaviors. With this understanding, analyzing their periodic nature becomes essential.

Function Periodicity

Function periodicity is a core concept in studying functions that repeat at regular intervals. A function is said to be periodic if there exists a constant period, such that the function's values repeat over this interval.

Mathematically, for a function \( f(x) \) to be periodic with period \( T \), the equation \( f(x + T) = f(x) \) must hold true for all values of \( x \) within the domain.

Given a trigonometric function like \( \sin(x) \), its periodicity plays a crucial role, often with a known period of \( 2\pi \). However, it's crucial to verify the periodic nature of more complex or altered functions. Not every trigonometric form retains the typical sinusoidal period.

When examining the function \( f(x) = \sin(x^2) \), the challenge is determining if a single period exists that satisfies the periodicity condition, which, unlike the standard sine function, does not simply repeat every \( 2\pi \). Thus, the complexity of \( x^2 \) in the sine argument alters the function's behavior significantly.

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the most essential trigonometric functions. It originates from the study of angles and is defined as the y-coordinate of a point on the unit circle.

Key aspects of the sine function include:

• Its range is limited between -1 and 1.
• It is periodic with a standard period of \( 2\pi \).
• The function is continuous and smooth.

A fascinating feature of the sine function is its symmetry and repetition over intervals, making it a perfect model for oscillatory motion. However, when transformed – such as introducing an exponent in its argument like \( \sin(x^2) \) – the function loses its regular periodic attributes.

With \( \sin(x^2) \), each integer multiplication in its phase becomes non-linear, and discovering a universal period becomes improbable. Thus, while the standard \( \sin(x) \) is predictably periodic, modifications in its argument form lead to more complex and non-periodic behavior.