You should always perform 3 or 4 iterations of my classroom example(s) yourself BEFORE attempting the homework.

1. Read and study this Maple example that illustrates Newton's method.
   (This is NOT the Maple code for Newton's method. It is simply an instructional demonstration of Newton's method.)

2. The function f(x) = 2sin(x) – 1/4*exp(x) has two zeros near x = –5. Apply 5 iterations of Newton's method by hand (and calculator):
   1. starting with x₀ = –5, and
   2. starting with x₀ = –4.5.

   Compare the results from parts (a) and (b). Explain what is happening. (Hint: look at the graph of f.)

3. Write a computer program in Maple for Newton's method to approximate a zero of a function f(x) starting with value x₀. Do NOT use Procedures (Proc)!
   (Use the bisection program and your false position code as templates.) Then use your code for Newton's method to approximate the first positive zero of

   f (x) = e^–x – cos x,

   starting with x₀ = 1 as we did in class.

   Make sure you use the printf command rather than the lprint command for printing.
   See these examples of Bad Tables.

   See the Newton's Method Algorithm (pseudo-code).

4. Once your Maple code for Newton's method works correctly in Problem 3, use it to do Problem 2 above. Perform 8 iterations.

5. Plot the functions f(x) = cos(x) and g(x) = x³ – 1 on a common plot.
   (See the examples around the second "Note" in Example 21 of the Maple Help Sheet entitled "Basics".)
   By inspection of the plot, select a starting value x₀ near the point of intersection. Using that starting value, apply 6 iterations of Newton's method to approximate the solution of the equation cos(x) = x³ – 1.

6. Use your Maple code for Newton's method to redo the previous problem using 8 iterations.

7. What can cause Newton's method to fail? Be clear and specific. Write complete and coherent sentences.

8. What rate of convergence does Newton's method give? (Linear? Quadratic? Cubic?)

9. Which statement is correct?
   1. Newton's method always gives 1-sided convergence toward the zero.
   2. Newton's method generally gives 1-sided convergence toward the zero.
   3. Newton's method always gives 2-sided convergence toward the zero.
   4. Newton's method generally gives 2-sided convergence toward the zero.
   5. None of the above are generally correct.

10. Do these problems (click).

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