Exercise 3: Newton'sMethod

Instructions:
You may email your notebook as an attachment to me at hknaust@utep.edu. Do only send unevaluated notebooks, do NOT send evaluated notebooks!

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 (x^2)-2 = 0 do indeed converge to (Sqrt[2]) (or - (Sqrt[2])).
2.) Use your algorithm to approximate a solution of the equation cos x = x.
3.) Investigate the behavior of Newton's Method applied to the equation (x^2)+1 = 0, which fails to have a real root. Create an illustration, 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.