# Newton's Method

• Dec 11th 2007, 04:54 PM
FalconPUNCH!
Newton's Method
Use Newton's Method to approximate the given number correct to eight decimal places.

$^{100}\sqrt{100}$

$f(x) = x^{100} - 100$

$f'(x) = 100x^{99}$

I know how to use Newton's Method but is there a faster way, a shortcut, to find it or am I going to have to start from $x1 = 1$?

am I going to have to plug in numbers until I get two to match eight decimals?

Second Problem:
Use Newton's Method to approximate the indicated root of the equation correct to six decimal places.

The positive root of $2cosx = x^4$

I know how to do Newton's Method but these two were giving me problems.
• Dec 11th 2007, 07:19 PM
FalconPUNCH!
For the second one I get

$f(x) = 2cosx - x^4$

$f'(x) = -2sinx - 4x^3$

I get $X1 = 2$ as my initial approximation but when I plug what I get into Newton's Formula I get some weird numbers and none of them are close to each other.

For my second approximation I get 1.5022769 and I don't know but I feel that it's wrong and I don't think I should continue with the wrong answer. :confused:
• Dec 11th 2007, 08:04 PM
FalconPUNCH!
I just did the first one using Newton's Method and got what I was looking for. The second problem I posted seems to be a little tougher and I need help with that one.

Edit: My only problem with the first one is I don't know what to use an initial approximation. It took me almost two pages to get the answer. Can anyone tell me a faster way to get an approximate number eight decimal places?
• Dec 11th 2007, 08:21 PM
Quote:

Edit: My only problem with the first one is I don't know what to use an initial approximation. It took me almost two pages to get the answer. Can anyone tell me a faster way to get an approximate number eight decimal places?
get a computer to do the calculating for you. I can't imagine why anyone would be learning Newton's method without some mathematical programming environment, eg. Haskell interpreter, R gui.
• Dec 11th 2007, 08:28 PM
FalconPUNCH!
Quote:

get a computer to do the calculating for you. I can't imagine why anyone would be learning Newton's method without some mathematical programming environment, eg. Haskell interpreter, R gui.

Yeah I can do that but I want to know how to find an initial approximation. Whatever I use is always far away from the actual root.
• Dec 11th 2007, 09:02 PM
earboth
Quote:

Originally Posted by FalconPUNCH!
I just did the first one using Newton's Method and got what I was looking for. The second problem I posted seems to be a little tougher and I need help with that one.

Edit: My only problem with the first one is I don't know what to use an initial approximation. It took me almost two pages to get the answer. Can anyone tell me a faster way to get an approximate number eight decimal places?

Hello,

I would draw a rough sketch of the 2 graphs. For instance with #1:
Draw the graph $f(x)=2\cos(x)\ ,~-\pi < x < \pi$ anf $p(x)=x^4$. Use the estimated x-value of the intersection as an initial value.
When I used $x_0 = 1$ I got the result with 8 decimals exact after 3 steps.

to #2. If n is a large number then $\sqrt[n]{n} \approx 1$. If you use a value a little bit larger than 1 as an initial value it will be sufficient. But with your example the sequence of x-value converges very slowly. So when I used $x_0 = 1.5$ it took me more than 20 cycles to get a nearly exact value.
• Dec 11th 2007, 09:08 PM
FalconPUNCH!
Quote:

Originally Posted by earboth
Hello,

I would draw a rough sketch of the 2 graphs. For instance with #1:
Draw the graph $f(x)=2\cos(x)\ ,~-\pi < x < \pi$ anf $p(x)=x^4$. Use the estimated x-value of the intersection as an initial value.
When I used $x_0 = 1$ I got the result with 8 decimals exact after 3 steps.

to #2. If n is a large number then $\sqrt[n]{n} \approx 1$. If you use a value a little bit larger than 1 as an initial value it will be sufficient. But with your example the sequence of x-value converges very slowly. So when I used $x_0 = 1.5$ it took me more than 20 cycles to get a nearly exact value.

Thanks for helping me. Yeah for $\sqrt[n]{n}$ it took me around 20 cycles. I'm going to probably get a number closer to one so I can fit it on half a page. Thanks for your assistance. :)
• Dec 12th 2007, 08:53 AM
topsquark
Quote: