Find whether the function f(x) = sin(x^2) is periodic.
Ummm do I just draw a graph?
Hello xwrathbringerxTo 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 $\displaystyle 2\pi$.
Grandad