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Laboratory I.8
Newton’s Method
Goals:
• The student will learn how to solve equations using Newton’s Method.
• The student will get an introduction to iteration and dynamical systems.
Before the Lab:
Newton’s Method is a technique which uses derivatives to solve equations that
you can’t solve by hand (algebraically). An example is cos
x
=
x
. There are no algebraic
tools which allow you to isolate
x
on only one side of that equation.
We’ll have you solve that particular equation in the lab; we’re going to work as
our example a simpler equation that we can solve by hand so that we know what we’re
doing. We’ll do
x
2
= 2, and in particular we’ll pretend we don’t know the positive
solution,
x
=
√
2 .
Solving
x
2
= 2 is the same as solving
x
2
– 2 = 0, so let’s graph
y
=
x
2
– 2 and look
where it crosses the
x
axis:
You can see the solution is just to the right of
x
= 1.4.
Newton’s Method works by taking an initial
guess and improving it. Let’s use as our initial
guess
x
0
= 2. We’re going to add a
tangent line to
y
=
x
2
– 2 to the picture:
The tangent line hits the
x
axis at
x
= 1.5. Notice
that this is much closer to the exact solution than
our original guess,
x
0
= 2. This, in essence, is
Newton’s Method: the tangent line to the curve at
a point will hit the
x
axis closer to the exact root
than the starting point.
Let’s check this with another picture. Our
original guess of
x
0
= 2 was “improved” to a guess of
x
1
= 1.5. Let’s draw the tangent line
to
y
=
x
2
– 2 at
x
= 1.5 and see what we get:
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The new tangent line is so close to the function that
we can hardly tell them apart. The intersection of
the new tangent line with the
x
axis is at
x
2
=
1.4167; the exact value of the solution is
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 Spring '10
 Moody
 Derivative, 3 pts, 2 pts, Rootfinding algorithm, 3pts

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