[-p,
p].
Determine the x
and y
values of the maximum and minimum of h(x)
closest to the highest and lowest points of intersection with k(x),
respectively. 1. a. This problem uses Newton's method to find the third root of 43 by solving
where n = 3 and m = 43. You may use either the Excel spreadsheet (that you download) or follow the directions on the Help page and have Maple help you obtain the Newton iterates for this problem.
Let x0 = 3 and perform 5 Newton iterations. Show these iterations in your Lab report (report only the iteration number, x, and f(x) values) and determine how many iterations were required for 4 decimal places of accuracy. So what is the third root of 43?
b. Now consider the functions:
Find the derivative of h(x).
What is the period of h(x)
? Graph h(x)
and k(x)
for x
[-p,
p].
Determine the x
and y
values of the maximum and minimum of h(x)
closest to the highest and lowest points of intersection with k(x),
respectively.
c. This part of the problem uses Newton's method to find the points of intersection seen in the graphs in Part b. We could form a new function f(x) = h(x) - k(x), which is zero at the points of intersection. Use the following different initial guesses: x0 = -0.5, 0.5, 1, and -0.75. Create one or more tables showing the first 5 Newton iterations for each of these starting values. (Once again, report only the iteration number, x, and f(x) values.) From this information, give the x and y values for these points of intersection. Describe what is happening with the last guess in the sequence and write a brief explanation of why these iterates are so different from the others.