Newton's Method
Lesson 4-9

Root being found: r

1^st approximation:

(Found from guessing/rough sketch of graph)

Tangent line: L (to curve y=f(x) at points (,f(x))

X-intercept of L: x

Approximation number: n

*the idea of Newton’s method is that the tangent line is close to the curve and so it’s x-intercept x, is close to the x-intercept of the curve

Example 1:

1. y-f(= f’()(x- )
2. Because the x-intercept of L is x, set y=0

0-f()=f’()( x- )

3. If f”( x)0, solve for x

x= x-

Use xas a second approximation to r and repeat procedure with x replaced by xtangent the line at (x, f(x0 = 3^rd approximation

x= x-

Follow with N th approximation

*If the number xbecomes closer and closer to r as n becomes larger, then the sequence converges to r and equals

Example 1:

F(x) = x-2x –5

F’(x)= 3x-2

1. x= x
2. Use n=1 x= x

=

3. Use n=2

= 2.1

Example 2:

Use Newton’s method to find correct to eight decimal places

1.

use f(x) =

f’(x)=

Newton’s Method:

2. 1 as initial approximation to get

3. Because x and x are the same we can conclude that 1.2246205

Example 3:

Find, correct to 6 decimal places, the root of the equation cosx = x

1. cosx –x= 0
1. f(x) = cos x-1
2. f’(x)= sin x-1
2. Formula =
1. 
3. Approximate:
1. 
4. Answer: The root of the equation is 0.73908513