Lab Problems
When doing the lab, be sure to record all important output, along with your thoughts and observations. It will be useful to record observations onto graph paper. Be sure your submitted copy has all the plots and graphs you are asked to find during the course of the lab. The purpose of this lab is to learn how to use Newton's Method to solve equations, to understand the geometry of Newton's Method, and to see the importance of the initial guess.
1. This section will carefully walk you through deriving the Newton's Method formula and then using it on an example. Read carefully and answer the questions asked.
Very few equations f(x) = 0 can be solved exactly. You have learned methods and tricks in high school algebra for solving equations such as
(Spend a little time convincing yourself that you can solve these equations.) We also learned one method of approximating a solution using the bisection algorithm, and you have used MAPLE's fsolve command to approximate solutions. In this lab the method of approximating solutions used is attributed to Isaac Newton. It employs the idea that the tangent line to a curve closely approximates the curve near the point of tangency.
Suppose we want to solve f(x) = 0
where f(x) is defined. To use
Newton's Method, make an initial guess 
. The next
guess, 
, is found at the intersection of the x-axis with the tangent
line to y = f(x) at 
. See Figure 1
below.
We
need to find a formula for 
.
a) What is the slope of the tangent
line to 
b) What is the equation of this tangent line?
c) Since 
is the x-value
where the tangent line crosses the x-axis, 
is a point on
the tangent line.
Show the work necessary to conclude that
.
Once
we have 
, we repeat the process to get 
from 
, 
from 
, etc. If all
goes well, the
get closer and
closer to the root we seek.
d) Write a formula for 
in terms of 
.
e) Write a formula for 
in terms of 
.
f) This last section steps you through a graphical look at Newton's Method. Enter the commands as given and include the results in your report.
The problem is to find a zero of the function 
First define the function and
plot it.
> g:= x-> x^2-1;
> plot(g,0..4, -1.5..15);
Next define an initial guess 
.
> x0:=3;
Next define the equation of the
tangent line at the point 
then
plot the function and the tangent.
Finally redefine 
to the new
guess. To find the guess, click on
the graph where the tangent line crosses the x-axis. An approximation of that point will appear on the left of
the toolbar as shown below.
> x0:= 1.67;
Thus our first guess is 3, and our second guess is
1.67. Now on your own, repeat the
tangent plotting steps above and determine the third guess.
Computing Newton's Method by hand or even with a pocket calculator is tedious and error prone. Since there is a formula for each successive approximation for the root, the procedure is easily performed by a computer. We will use MAPLE to calculate roots using Newton's Method. For example, to quickly perform one step of Newton's Method (without the graphical work), simply enter
> newton(g, 3.0);
We can get multiple iterations at once by entering:
> newton(g, 3.0, 10);
2. Now it is your turn. The first application of Newton's Method is to solve the equation
a) Graph the function 
and explain how
you know that the above equation
has only one solution. Be sure to explain why there are no additional solutions beyond the
x-range you used to graph f(x).
b) We will use Newton's Method with 
. You may use
the Maple command "newton" in this section. You can choose between two uses of this command. The first is symbolic and has syntax newton(f,x0). This
gives one iteration of Newton's Method.
The second is numerical and gives decimal approximations to as many
iterations as you specify. Its
syntax is newton(f,x0,n). In both of these f is the name of a
function you have defined and x0 is the initial guess. In the last one, n is the number of
iterations you want to perform.
Continue until the answers you are getting are the same rounded to the nearest
.0001. What solution did you find?
c) Plot the first two iterations of Newton's Method. Use the method demonstrated in the example in problem 1.
3. The main difficulty in using
Newton's Method occurs in the choice of the initial guess, 
. A
poor
choice can lead to a sequence 
that does
not get close to the desired solution.
To
illustrate this use 
as in problem
2.
a) Let 
and repeat the
procedure for using Newton's Method that you used in
problem 2, part b. What happens? Are you getting close to the same answer you obtained
in problem 2, part b? Obtain another graph of f(x) and sketch the first two iterations of
Newton's Method, again using the method demonstrated in problem 1.
b) Let 
and again repeat
the procedure for using Newton's Method that you used in
problem 2, part b. What happens this time?
c) Find intervals (as large as possible) where you think Newton's Method converges to the
solution. Explain your reasoning.
4. This
problem examines the sensitivity of Newton's Method to the choice of 
. The ideas may
be clearer using the graphical approach demonstrated in Problem 1.
Let
. Clearly, the
equation 
has the three
solutions Ð1, 0, and 1. Before using MAPLE with this problem,
be sure to redefine the function f.
a) Argue that if 
, then Newton's Method will converge to the zero 1. Use a graph of f(x) as well as looking
at the derivative of f(x) for 
to justify your argument. By symmetry, if 
, Newton's Method will converge to the zero at Ð1. Be sure to discuss what happens if 
. Examine this
case by hand, or use MAPLE to perform the calculations. (Hint: If you use MAPLE, remember 
is 1/sqrt(3).)
b) Demonstrate that if we start with 
, then 
, and 
. Therefore, if
we start with 
, we do not converge to any of the three zeros. In this case, 
are called period
2 points.
c) The interesting behavior occurs
when 
, or by symmetry, when 
. For each
of the values given in the following table, use Newton's Method with MAPLE to
determine which zero you are finding starting with that particular 
. If it looks
like it is not finding any zero, indicate this.
Zero
found
.577
.578
.460
.466
.44722
.44723
5. Use Newton's Method to locate the root of
