1. ## Periods

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

2. or prove that

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

3. Hmmmm interesting

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

4. 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 $2\pi$.