This notebook gives some examples that will help with the homework.
At any time, you can select a built-in Mathematica function and go to Help/Find Selected Function

First of all, FindRoot[ ]  does something like Newton’s method, with an initial guess you supply:

FindRoot[x^2 == 2, {x, 1.4}]  (* Note double equal sign ! *)

This gives a replacement rule.  You can define a variable with this value like this:

sqrt2 = x/.FindRoot[x^2 == 2, {x, 1.4}]  (* Note that /. has shape of a triangle; it's not ./ *)

UPDATED!
Here’s a trick to show all the decimal points of the approximate solution:

InputForm[sqrt2]

You can also solve (approximately) simultaneous equations:  For example

FindRoot[{y == x^2, y ==  x + 1}, {x, 2}, {y, 4}]

Another useful function is Nest[ ]

Clear[a];
f[x_] = -1 + a/2 (1 - x^2)
Nest[f, x, 2]

Mathematica can do symbolic differentiation too.  (This is helpful for problem 1(b) on the homework.)

D[x^2 + x y + y^2, x]

We can put these together to find the a-value of the 2 to 1 band merging.

a21 = a/.FindRoot[Nest[f, 0, 3] == Nest[f, 0, 4], {a, 3.67}]

Now we can go to the iteratedMap1.nb from January 27 on our web site.

Another very useful feature of computer algebra is factoring.  To find the fixed points we do this

Factor[f[x] -x]

So we see the fixed points of f are x = -1 and x = (2-a)/a.

Next, let’s think about the 4 to 2 band merging.