(*  That's how you write comments  *)
(*  To execute a command, type it, 
     then press the SHIFT key, 
     and, while holding it down, press RETURN 
       - below I typed  aa = Cos[1.4] and pressed RETURN 
       to get the value of cos(1.4)  *)
aa = Cos[1.4]

(*  To define the function  f(x) = ln(cos(x)), 
     type the following:  f[x_] = Log[Cos[x]] 
       and, while holding SHIFT down, press RETURN; 
     the underscore after x tels Mathematica that 
      x is just a dummy variable  *)
f[x_] = Log[Cos[x]]
(*  Now you can compute the derivative of f(x) defined above 
        by executing the command  D[f[x],x]  - try below  *)
D[f[x],x]

(*  Let us see whether the computer knows elementary functions  *)
Exp[f[x]]
(*  Evaluate the value of f(1.4) 
      and assign its value to the variable p  *)
p = f[1.4]

(*  Now let us see the first 30 term of the Taylor series 
       of f(x) around 0 - how long will it take you to do this by hand?  *)
Series[f[x],{x,0,30}]

(*  Below you see a fixed point iteration 
       of the function  g(x) = cos(x)  with initial value 1 
       - that's an example of a control of the execution 
        using "For";  note that the command  p=N[1,3000] 
       tells Mathematica that the variable  p  is equal to 1 
       with precision of 3000 digits;  
     - the iterates converge to a fixed point of cos(x), 
     approximately equal to  0.739085...; 
     - as you can see, the convergence is not very vast  *)
p=N[1,3000];  g[x_] :=Cos[x];
For[i=1, i<=40, i++, 
  {p=g[p], 
    Print[i], Print[N[p,20]],  Print[ ], 
    }
  ]
(*  Below is the Newton method for finding the root 
       of the equation  cos(x)=x;  
     we first write it as an equation of the form 
       f(x) = cos(x) - x = 0 , 
     and then apply Newton's method with initial value 1  
      - note that after only 6 steps we know more than 
       70 correct digits of the answer!!!  *)
p=N[1,3000];  g[x_] :=Cos[x]-x;
For[i=1, i<=8, i++, 
  {p=p - g[p]/g'[p], 
    Print[i], Print[N[p,70]],  Print[ ], 
    }
  ]