10/2007

Newton's Method

Objective:

To become familiar with Newton's Method, how to use it and when it works.

The method we will use today is called Newton's Method, after Sir Isaac Newton.

It is a method for finding the zeros of a function.

Part 1:

We will illustrate the method on the function.

We note that there is

a zero near t = 5.

1. Find the derivative of f and enter it in the box below:

3. Ask Mathcad to display the value of f at this point:

4. At that point, we want to draw the tangent line to the graph. (That's why we need the derivative.) We hope that the function f behaves like its tangent line, at least for a while. If we find where the tangent line crosses the t-axis, maybe the graph of f crosses the t-axis nearby.

Read that sentence again, out loud, to your lab partner. Got it?

That new t-value, called t₁, ought to be closer to a zero of f.

What is the slope of the tangent line to the curve at t = 5?

Now find an equation for the tangent line through the point (5,-22), and enter the

value of the y-intercept below:

If you have found the correct equation, you should see a tangent line to the curve in

the graph below.

Now we have a new guess, t₁ -- the point where the tangent line crosses the t-axis.

Using pencil and paper (and perhaps your calculator), find where the tangent line crosses the t-axis.

The tangent line crosses the t-axis at t =

We could go through the whole process again to find another guess, t₂, and so forth. In fact, we could come up with a recursive formula. Each new value of t depends on the function itself and its derivative (slope) at the old value.

We are just finding the tangent line at t_n

and finding where it crosses the t-axis.

Let's look at the first 11 values of t, and see if they settle down to a zero.

We will also examine the value of f at those points.

How can you tell that we have found a zero of f?

Part 2:

Let's try Newton's Method again, to find all the zeros of the polynomial

Now Mathcad knows a "derivative" of this function. (It actually computes a difference

quotient DP/Dt, with a very small Dt, at each point t.)

If we change to a different function P(t), Mathcad will automatically update the derivative, too.

The graph of P is displayed on the interval [-2,8].

We note that, for this function, there are four roots. Now we will find them, as quickly and

as accurately as possible, using Newton's method.

Enter your initial guess here:

It looks as though there is a

zero at

We will go through 10 iterations

(steps) of the Newton process.

Values of t Values of P(t)

Make a note (in text) of what your starting value for t₀ was, and how many iterations of Newton's Method it took before getting three decimal places of accuracy.

Locate all four zeros of P(t) by choosing different initial values t₀ and record your results in the table below (in text):

Initial Guess Number of Iterations Actual Zero

1.

2.

3.

4.

Try to find one of the zeros by zooming in on the graph. How long does it take you to get

three-decimal place accuracy? Which method do you prefer? Why?

Switch typists now.

Determine the number of solutions of the equation P(t) = 0, and find them using Newton's Method. Record your results here.

Using that same polynomial, try a starting value of t₀ = 0.001.

What happens? Can you explain why?

As a group, use Microsoft Word (and the Equation Editor if necessary) to write a maximum of one page discussing the topics covered in this lab. Explain what you have learned; be sure to address the following:

1. What does Newton's Method find?

2. Describe briefly the process Newton's Method uses. (You start with a guess, then...)

3. Compare Newton's Method with the process of zooming. Which is faster? Which do you prefer?

4. Does Newton's Method always work? Do you have to be careful using Newton's Method? Explain.

Hand in the writeup with your lab.