# Newton's Method of Approximation

• Jun 13th 2011, 06:42 AM
purplec16
Newton's Method of Approximation
Use Newton's Method of Approximation to approximate the zero of the function in the indicated interval to six decimal places

$\displaystyle f(x)={x}^{3 }+3{x}^{ 2}-3$ in the interval $\displaystyle [-2,0]$

I know how to use Newton's method of Approximation but what I am confused with is 2 things:

1. What number do I use in my approximation (i.e my $\displaystyle {x}_{0 }$ )

2. When it says to six decimal places does that mean I have to do the approximation six times?
• Jun 13th 2011, 07:05 AM
TheEmptySet
Quote:

Originally Posted by purplec16
Use Newton's Method of Approximation to approximate the zero of the function in the indicated interval to six decimal places

$\displaystyle f(x)={x}^{3 }+3{x}^{ 2}-3$ in the interval $\displaystyle [-2,0]$

I know how to use Newton's method of Approximation but what I am confused with is 2 things:

1. What number do I use in my approximation (i.e my $\displaystyle {x}_{0 }$ )

2. When it says to six decimal places does that mean I have to do the approximation six times?

For question 1 you need to guess. Since

$\displaystyle f(-2)=1$ and $\displaystyle f(-1)=-1$ I would guess the midpoint $\displaystyle x=-\frac{3}{2}$

For number two you need to keep using newtons method until the 6th digit after the decimal place no longer changes it could be more of less than six times.

P.S it will not take 6 iterations.
• Jun 13th 2011, 07:32 AM
purplec16
Ok thank you would -.5 be a good guess?
• Jun 13th 2011, 09:43 AM
Also sprach Zarathustra
You can use the following to "know when to stop":

$\displaystyle |x_0-x_n|< \frac{M}{2m}|x_{n+1}-x_n}|$

Where $\displaystyle M=sup\{f''(x) | x\in I\}$ and $\displaystyle m=inf\{f'(x)| x\in I\}$
• Jun 13th 2011, 09:44 AM
Also sprach Zarathustra
You can use the following to "know when to stop":

$\displaystyle |x_0-x_n|< \frac{M}{2m}|x_{n+1}-x_n}|$

Where $\displaystyle M=sup\{f''(x) x\in I\}$ and $\displaystyle m=inf\{f'(x) x\in I\}$