Newton’s Method
1 2
Finding a root for: f  x   x  3
2
5
4
3
We will use Newton’s
Method to find the
root between 2 and 3.
2
1
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-3

1 2
f  x  x  3
2
1.5
f   x  x
1.5
2
z 3
(not drawn to scale)
3
1 2
f  3   3  3  1.5
2
Guess:
mtangent  f  3  3
1.5
3
z
1.5
3
 2.5
(new guess)
3
1.5
z
3

1 2
f  x  x  3
2
1.5
f   x  x
2.5
1
2
f  2.5    2.5   3  .125
2
Guess:
2
z
3
mtangent  f   2.5  2.5
.125
2.5 
 2.45
2.5 (new guess)
.125
z
2.5

1 2
f  x  x  3
2
1.5
f   x  x
Guess: 2.45
f  2.45  .00125
z
2
3
mtangent  f   2.45  2.45
.00125
2.45 
 2.44948979592
2.45
.00125
z
2.45
(new guess)

Guess:
2.44948979592
f  2.44948979592  .00000013016
Amazingly close to zero!
This is Newton’s Method of finding roots. It is an example
of an algorithm (a specific set of computational steps.)
It is also called the Newton-Raphson method
This is a recursive algorithm because a set of steps are
repeated with the previous answer put in the next
repetition. Each repetition is called is called an iteration.

Guess:
2.44948979592
f  xn 
f  2.44948979592

.00000013016
 xn1  xn 
Newton’s Method:
f   xn 
Amazingly close to zero!
This is Newton’s Method of finding roots. It is an example
of an algorithm (a specific set of computational steps.)
It is sometimes called the Newton-Raphson method
This is a recursive algorithm because a set of steps are
repeated with the previous answer put in the next
repetition. Each repetition is called an iteration.

3
Find where y  x  x crosses y  1 .
1  x3  x
n
xn
0  x3  x  1 f  x   x3  x 1
f  xn 
0
1
1
1
1.5
.875
f  xn 
xn1  xn 
f   xn 
f   xn 
2
1
1
 1.5
2
5.75
.875
1.5 
 1.3478261
5.75
2 1.3478261 .1006822 4.4499055
1.3252004 
3
f   x   3x2 1
1.3252004
 1.3252004  1.0020584  1

There are some limitations to Newton’s method:
Looking for this root.
Bad guess.
Wrong root found
Failure to converge

Acknowledgement
I wish to thank Greg Kelly from Hanford High School, in Richland,
Washington, USA for his hard work in contributing towards this PowerPoint.
http://online.math.uh.edu/
Greg has kindly given permission for this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit the Western Australian
Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au