# Periods

• Aug 17th 2009, 04:53 PM
xwrathbringerx
Periods
Find whether the function f(x) = sin(x^2) is periodic.
Ummm do I just draw a graph?
• Aug 18th 2009, 12:23 AM
Haytham
or prove that

$\displaystyle f(x)=f(x+\theta)$
• Aug 18th 2009, 12:31 AM
xwrathbringerx
Hmmmm interesting

Could you please show me how to do it for this question as an example?
• Aug 18th 2009, 12:49 AM
Hello xwrathbringerx
Quote:

Originally Posted by xwrathbringerx
Find whether the function f(x) = sin(x^2) is periodic.
Ummm do I just draw a graph?

To be periodic a function has to repeat at regular intervals. Now we know that $\displaystyle \sin(x)$ repeats every $\displaystyle 2\pi$, and we get the first complete cycle between $\displaystyle x = 0$ and $\displaystyle x = 2\pi$. So the first cycle of $\displaystyle \sin(x^2)$ will occur between $\displaystyle (x^2) =0$ and $\displaystyle (x^2) = 2\pi$; i.e. $\displaystyle x = 0$ and $\displaystyle x = \sqrt{2\pi}$.

The next cycle for $\displaystyle \sin(x)$ is between $\displaystyle x = 2\pi$ and $\displaystyle x = 4\pi$. So for $\displaystyle \sin(x^2)$ it's between $\displaystyle x = \sqrt{2\pi}$ and $\displaystyle x = \sqrt{4\pi}$.

So the question is: Is the difference between $\displaystyle 0$ and $\displaystyle \sqrt{2\pi}$ the same as between $\displaystyle \sqrt{2\pi}$ and $\displaystyle \sqrt{4\pi}$? Work it out and see.

If it helps, here's the graph of the function between $\displaystyle x = 0$ and about $\displaystyle 2\pi$.