1. Newton’s Method: Homework Create a Google Spreadsheet – that uses Newton’s Method – to find roots of x^2 + x = 1 Link to this page from your homepage Send link(s) to: – Kenneth.Church@jhu.edu and annirvine@gmail.com Kenneth.Church@jhu.eduannirvine@gmail.com Extra credit: – Multiple roots: There are two roots to this equation. You will get one root or the other depending on the initial value. Show that this is the case. That is, produce two spreadsheets with two initial values that converge on different roots. Which initial values converge on which root? – What happens if you try to use Newton’s Method to solve: x^2 + x = -1

2. Newton’s Method http://archives.math.utk.edu/visual.calculus/3/newton.5/ http://archives.math.utk.edu/visual.calculus/3/newton.5/

4. nx[n]x[n+1]sqrt(5) 13.0000002.3333332.236068 22.3333332.2380952.236068 32.2380952.2360692.236068 42.2360692.236068 5 6 7 8 9 102.236068 112.236068 122.236068 132.236068 142.236068

5. Symbolic Features (Bet you can’t do this with your favorite statistics package) Complex Numbers: Sqrt(-1) Roots (without approximations) Differentiation (without approximations) Integration (without approximations) The On-Line Encyclopedia of Integer Sequences Eval Symbolic Methods ≠ Numeric Methods – Newton’s Method: Approximation for Reals

6. Sqrt(-1)  Error (for many tools)

7. Roots (without approximations)

8. Numerical Methods: Approximations such as Newton’s Method

9. Complex Roots

10. Newton’s Method http://archives.math.utk.edu/visual.calculus/3/newton.5/ http://archives.math.utk.edu/visual.calculus/3/newton.5/

11. Newton’s Method http://archives.math.utk.edu/visual.calculus/3/newton.5/ http://archives.math.utk.edu/visual.calculus/3/newton.5/

12. Symbolic Alternative