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Given the function $F(x)=\frac{\pi}{2}\sin(x)$. Find the fixed points and, if they exist, determine if the points are attractors or repulsors without differentiation.

I already found the fixed points but I need help in the second part.

My doubt is if there is any way to determinate if the points are attractors or repulsors without using differentiation (In the course that I’m in, I can't use differentiation yet).

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  • $\begingroup$ Can you do so visually (e.g., drawing spiderweb diagrams)? $\endgroup$ – kccu Apr 21 at 23:50
  • $\begingroup$ Yes, i think i can use anything but differentiation. How does the spiderweb diagrams work? $\endgroup$ – Rodrigo Pizarro Apr 22 at 0:07
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For a fixed point $x$, consider a point $x+\varepsilon$ close to $x$ and try to determine whether the map $x+\varepsilon\mapsto \frac{\pi}{2}\sin(x+\varepsilon)$ takes you nearer to $x$ or further from it. For instance, if $x=0$, then $\epsilon\mapsto\frac{\pi}{2}\sin\epsilon\approx\frac{\pi}{2}\epsilon$, which is further away from $0$. So $0$ is a repulsor.

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  • $\begingroup$ maybe expand a little bit on the final approximation withoud differentiation? $\endgroup$ – Andres Mejia Apr 22 at 0:20

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