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- 1.
- Consider
*f*(*x*) =*x*- 16^{4}*x*- 12 and let*r*= 32. First let*x*be 1.587. What root is found? Next let_{0}*x*be 1.588. What root is found? Is a root really 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._{0} - 2.
- Use Newton-Raphson to find all points at which the graph of the function

*f*(*x*) =*x*+ 3^{5}*x*- 17^{4}*x*+^{3}*x*- 8^{2}*x*+ 4 has a horizontal tangent line. Explain why you can be sure that you have indeed found all such points. - 3.
- Consider , let
*x*= 2 and_{0}*r*= 30. Discuss what happens. Explain why this behavior occurs. What is the signficance of the numbers that appear? Find all numbers which when used as*x*will lead to a root. Hand in appropriate printouts. Again note that Newton-Raphson can be very sensitive to the initial guess that is used._{0} - 4.
- Let
*f*(*x*) = 3*x*- 4^{4}*x*- 84^{3}*x*+ 288^{2}*x*+ 732. Let*x*= 2 and_{0}*r*= 10. Run**Newton**. What causes the result you get to occur? Explain this in terms of the Newton-Raphson process. - 5.
- 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*x*and run_{0}`Newton`. Try a different value for*x*(one not close to the original value) and run_{0}`Newton`. Discuss the output for these two runs. Describe what happens.

9/30/1997