Newton's Method

Example

f[x_]=Cos[Exp[x]]-Sin[x^2+4]

Here's a plot, so we can see that this function does have roots.

Plot[f[x],{x,-3,3}]

This is one way to set up Newton's method:

new[x_]:=SetPrecision[x-f[x]/f'[x],30]

Let's choose to start at 0.5, and see which root we find:

new[0.5]

new[%]

new[%]

new[%]

new[%]

new[%]

new[%]

new[%]

f[%]

Investigation

We can get a graphical picture of what Newton's Method is doing using the following commands.

f[x_]=Cos[Exp[x]]-Sin[x^2+4];
plotf=Plot[f[x],{x,0,2},PlotStyle->{Blue,Thick}];
new[x_]:=SetPrecision[x-f[x]/f'[x],30]

Clear[plot,line,dot]
a=0.5; (*change this value of a to change starting value*)
Do[
plot[i]=Plot[f[a]+f'[a](x-a),{x,0,2},PlotRange->{{0,2},{-2,2}},PlotStyle->{Red,Thick}];
line[i]=Graphics[Line[{{a,0},{a,f[a]}}]];
dot[i]=Graphics[Disk[{a,0},0.04]];

a=new[a];

,{i,1,5,1}]

Now evaluate the Manipulate command to get the pictures. As we vary i, we are stepping through Newton's Method.

Manipulate[Show[plotf,plot[i],line[i],dot[i],dot[i+1],AspectRatio->1],{i,1,4,1}]

Change the starting value a in the above to 0.25 and see what root is found. 
What if you start at a=0.15? 
What if you start at a=0.35?