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.