Week 3

1
Week 3
- Bisection method - Newton's method -
Group work
2
Bisection
Overview
Keep taking midpoints to find a root
3
Bisection
Example
3x/2
3
f(x) 5ln x 2 - x - 8
f(4.4312) 12.6
f(0.1) -18.4
f(4.1219) 1.7
f(10) 31771.5
f(3.9672) -1.7
f(5.05) 62.0
f(4.0446) -0.1
f(2.575) -5.8
f(4.0832) 0.7
f(3.8125) -4.1
4
Newton's
Overview
Keep using the derivative to guess where the root
is
f(x )?
n
x x -
n1
n
f(x )?
n
5
Newton's
Example
3x/2
3
f(x) 5ln x 2 - x - 8
3x/2
2
f'(x) 5/x 3/2 ln 2 2 - 3x
6
3x/2
3
f(x) 5ln x 2 - x - 8
7
Newton's
Example
3x/2
3
f(x) 5ln x 2 - x - 8
3x/2
2
f'(x) 5/x 3/2 ln 2 2 - 3x
x 7.2
x 5.0
x 10
6
3
0
x 6.4
x 4.5
x 9.1
4
1
7
x 4.2
x 5.7
x 8.1
8
5
2
8
Newton's
Example
½
f(x) x
-½
f'(x) ½x
f(x )?
0
x x -
-x
1
0
0
f'(x )?
0
9
Newton's
Problems
- Derivative is zero - You don't always converge