Exercise 6: Newton'sMethod

Instructions:
You may email your notebook as an attachment to me at helmut@math.utep.edu. Do only send unevaluated notebooks, do NOT send evaluated notebooks!
Alternatively you may post your notebook on your webpage and let me know that you have done so.

1.) Newton's Method, a standard procedure to find a real root of an equation, can be found in nearly every Calculus book.
Implement Newton's Method for Mathematica. Do not use the built-in command in an add-on package. Validate your implementation by checking that the approximations for the equation f (x) = (x^2)-2 do indeed converge to (Sqrt[2]).
2.) Investigate the behavior of Newton's Method applied to the function f (x) = (x^2)+1, which fails to have a real root. Create a notebook, which conveys to "your students" the chaotic behavior encountered. Your notebook should use numerical results as well as visualizations (an animation?). Convince "your students" of the fact that the Newton approximations for f (x) = (x^(2 )) + 1 are dense on the real line.