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  1. Consider tex2html_wrap_inline99 and let r = 25. First let tex2html_wrap_inline75 be 1.414. What root is found? Next let tex2html_wrap_inline75 be 1.145. What root is found? Why do two initial guesses so close together lead to different roots? Use your knowledge of the explanation of how Newton-Raphson works to answer this part of the problem.
  2. Use Newton-Raphson to find all points at which the graph of the function
    tex2html_wrap_inline107 has a horizontal tangent line. Explain why you can be sure that you have indeed found all such points.
  3. Consider tex2html_wrap_inline109 , let tex2html_wrap_inline111 and r = 30. Discuss what happens. Explain why this behavior occurs. Are there any values (that are not roots) that lead to a root? Find all roots of f(x) = 0. Again note that Newton-Raphson can be very sensitive to the initial guess that is used.
  4. Construct a polynomial function g(x) of degree 5 or greater that has no real roots. Explain why you are sure it has no real roots. Let r = 20. Select a value for tex2html_wrap_inline75 and run Newton. Try a different value for tex2html_wrap_inline75 (one not close to the original value) and run Newton. Discuss the output for these two runs. Describe what happens.

Sean O Anderson
Tue Oct 1 12:37:24 EDT 1996