Fixed Points and Graphical Analysis

Question

For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$.

I have no problem finding the fixed points, but I don't even know what the second part of the problem is asking for or even what to do. Can someone please help?

Answer 1

Start with a graph of $f(x)=\sin(2x)$ together with the line $y=x$:

It appears that there is a fixed point round about $x_0\approx 0.95$. Since $$f'(0.95) \approx 2\cos(2\cdot 0.95) \approx -0.65,$$ we expect this point to be attractive under iteration. To perform graphical analysis, simply place your pencil on the yellow line $y=x$ at the point whose $x$-coordinate is $x_1$, namely the point where you'd like to start. Now, move vertically to the blue graph and then horizontally back to the yellow line. You've successfully moved from $x_1$ to $x_2$.

Now simply repeat this process until you feel you see the convergence. You can mouse-over the region below to see twenty iterates.

Incidentally, you say you had no problem finding the fixed points, i.e. solving $\sin(2x)=x$ for $x$. The easiest way I see to do that is to simply perform the iteration. In Mathematica, something like

f[x_] = Sin[2 x];
NestList[f, 0.5, 20]

(* Out: {0.5, 0.841471, 0.993718, 0.914454, 0.966874, 0.934853, 
  0.955658, 0.942581, 0.950993, 0.945656, 0.949073, 0.946898, 0.948288, 
  0.947402, 0.947967, 0.947607, 0.947837, 0.94769, 0.947784, 0.947724, 
  0.947762} *)

Thus, we see a fixed point at about $0.9477$. This is exactly the technique I advocated in response to your question here.