Math 131		Lab 9 Newton's Method		15 Mar.

The objectives of lab 9 are to use Newton's method to solve the equation 
f( x ) = 0 and to obtain more practice with rules for derivatives.

Appendix C of your textbook on pages 619-622 describes Newton's method.
You should read those pages carefully.

One way to use your TI calculator for Newton's method follows.
Enter f( x ) for y1 on the Y= screen.
Enter f'( x ) for y2.
Enter x - f( x ) / f'( x ) as x - y1( x ) / y2( x ) for y3.
Then on the HOME screen use the following two commands to set an initial
approximation and then to perform one step of Newton's method.
	
	initialApproximation STO A
	y3( A ) STO A

Repeat the previous command as needed.

Do parts a, b and c for each of exercises 1 - 4.
a. Identify the function used as y1 in Newton's method.  Graph that function
   in an appropriate window.  Copy your graph accurately on graph paper.  
   Label the axes and the function.  Identify some key points.
b. Use the graph to determine the number of solutions.
c. Find each solution with Newton's method with accuracy to 6 decimals.
   Write the equation xNew = x - f( x ) / f'( x ) for particular f and f' in
   the exercise.  For each solution make a table similar to Table C.6 with 
   2 columns labeled step number and approximate root.


1. Do page 622 #7.
2. Do page 622 #10.  For part b also do page 164 #2b.
3. From lab 2 solve 0.01e^( 0.01x ) = x^2 with Newton's method.
4a. Explain why the equation cos( cos( x ) ) = sin( sin( 2x ) ) / sin( x )
    has an infinite number of solutions.
 b. Determine the first two positive solutions of the equation in part a.
    Provide a graph that shows these two points.
 c. Explain how to obtain the other solutions from the ones in part b.

Group Lab Report
On Wednesday 23 Mar. please submit a group report containing the results of 
Lab 9.  Submit one report for each group.  Write a lookback.  Discuss your 
observations and your results.  Reflect on what you learned from doing this lab.
Discuss the impact the lab had on helping you understand Newton's method.
Indicate whether real group work occurred.

The lab report is due Wednesday 23 Mar. before 9 AM.

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