Periods

**xwrathbringerx** · August 17th 2009, 04:53 PM

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

**Haytham** · August 18th 2009, 12:23 AM

or prove that

$f(x)=f(x+\theta)$

**xwrathbringerx** · August 18th 2009, 12:31 AM

Hmmmm interesting

Could you please show me how to do it for this question as an example?

**Grandad** · August 18th 2009, 12:49 AM

Hello xwrathbringerx

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 $\sin(x)$ repeats every $2\pi$ , and we get the first complete cycle between $x = 0$ and $x = 2\pi$ . So the first cycle of $\sin(x^2)$ will occur between $(x^2) =0$ and $(x^2) = 2\pi$ ; i.e. $x = 0$ and $x = \sqrt{2\pi}$ .

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

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

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

Grandad

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