Question

Determine whether the given function is periodic. If so, find its fundamental period. $$ \sinh 2 x $$

Short answer

If so, find its fundamental period. Answer: The function sinh(2x) is not periodic. Therefore, there is no fundamental period for this function.

Step-by-step solution

Understand hyperbolic functions

Hyperbolic functions are defined using exponential functions, unlike sine and cosine functions which are periodic by nature. The hyperbolic sine function, \(\sinh x\), can be represented as: $$ \sinh x = \frac{e^x - e^{-x}}{2} $$

Apply the property of hyperbolic functions on the given function

The given function is \(\sinh(2x)\). Therefore we can substitute \(2x\) in place of \(x\) in the above representation of hyperbolic sine function: $$ \sinh(2x) = \frac{e^{2x} - e^{-2x}}{2} $$

Examine periodicity

Now we need to verify if there exists a period T such that \(\sinh(2x) = \sinh(2(x+T))\) for all x. Let's substitute \(x+T\) into the equation: $$ \sinh(2(x+T)) = \frac{e^{2(x+T)} - e^{-2(x+T)}}{2} $$ For the function to be periodic, this must be equal to \(\sinh(2x)\) for all x. That is, we should have: $$ \frac{e^{2x} - e^{-2x}}{2} = \frac{e^{2(x+T)} - e^{-2(x+T)}}{2} $$ Simplifying, we get: $$ e^{2x} - e^{-2x} = e^{2x}e^{2T} - e^{-2x}e^{-2T} $$ We can see that the above equality doesn't hold for any particular non-zero value of T for all x. Therefore, the function \(\sinh(2x)\) is not periodic.

Conclusion

The given function, \(\sinh(2x)\), is not periodic. As a result, there is no fundamental period for this function.

Key concepts

Periodicity

Periodicity refers to the property of a function to repeat its values at regular intervals over its domain. A function is said to be periodic if there exists a non-zero constant, such as \(T\), known as the period, that satisfies \(f(x) = f(x+T)\) for all values of \(x\).
When a function is periodic, these periodic intervals hint at repeating patterns or cycles.
Real-world examples include the cycles of the moon or the oscillation of a pendulum.

• If a function meets the condition \(f(x) = f(x+T)\) for all \(x\), it's periodic.
• Common periodic functions include trigonometric functions like sine and cosine.

In contrast, not all mathematical functions exhibit periodicity. Hyperbolic functions, like \(\sinh x\), do not inherently have this property. This makes them unique since they often lack a natural repeating cycle and require specific exploration to understand their behavior.

Hyperbolic Sine

Hyperbolic sine, commonly written as \(\sinh x\), is a mathematical function related closely to the exponential function. It is defined by the expression:

• \(\sinh x = \frac{e^x - e^{-x}}{2}\)

This definition shows that \(\sinh x\) composes a mixture of exponential growth and decay. Unlike its trigonometric cousin, \(\sin x\), the hyperbolic sine behaves differently regarding periodicity.
The behavior of \(\sinh x\) differs markedly because, instead of oscillating like sine, it continuously grows or decreases without repeating values. This means there is no value \(T\) such that \(\sinh(x) = \sinh(x+T)\) for all \(x\). Consequently, \(\sinh x\) and expressions like \(\sinh(2x)\) do not exhibit periodicity.

• Hyperbolic functions like \(\sinh x\) help model hyperbolically occurring phenomena, such as the shape of cables in suspension bridges.
• Despite their name and symbolic similarity to trigonometric functions, hyperbolic functions follow different mathematical properties and behaviors.

Fundamental Period

In periodic functions, the fundamental period is the smallest positive value, \(T\), for which the function repeats itself. The smallest positive \(T\) indicates the precise length needed to observe a full cycle.
The determination of a fundamental period involves checking if such a smallest interval \(T\) exists where \(f(x) = f(x+T)\).

• If a function has a fundamental period, any multiple of \(T\) will also represent a period.
• Trigonometric actions like \(\sin(x)\) and \(\cos(x)\) are examples with well-known fundamental periods.

However, not all functions exhibit a fundamental period, as seen with \(\sinh(2x)\). During the exploration of \(\sinh(2x)\), it was found that no constant \(T\) satisfies the condition of periodicity. This indicates \(\sinh(2x)\) does not have a fundamental period.
Absence of a fundamental period in functions like \(\sinh(2x)\) suggests they continuously evolve without reset, demonstrating that the function never repeats its values identically at any fixed interval.