Newton’s Method



How to enter into Calculator:

Find f ’(x)

Enter into calculator, press STO--> & store into x variable.

Now you can start entering above formula:  x - ((f(Xn))/(f ’(x)))

Then press STO-> & store into the x variable.
(this will automatically update the new x result into the old x).

Now press enter, and check result, write it down as answer of X(n+1).

Continue to do this until you get the exact same result twice.




Here is an example:

f(x)= cos(x)

You are told to use X0= 0.5 as the starting point.

Find the first derivative of f(x) --> f ’(Xn)= - sin(x). 

Now begin to enter into calculator but entering 0.5 

Then press STO-> & store into x variable, 

Now you can start entering above formula: x -((cos(x))/(-sin(x)))

Then press STO--> & store into the x variable
(this will automatically update the new x result into the old x)

Now press enter, and check result, which is 2.33048 something.

Write this answer down as answer of X(1).

The first was X0= 0.5 so this answer is X1=2.33048 something

to whatever significant digit the instructor specifies.

Now press enter, and check result, which is 1.3806 something.

Write this answer down as of X(2) = 1.3806 something.

Now press enter, and check result, which is 1.57312 something.

Write this answer down as of X(3) = 1.57312 something.

Now press enter, and check result, which is 1.570796 something. 
Write this answer down as of X(4) = 1.570796 something.

Continue to do this until you get the exact same result twice.

For this equation, you are done with this once you have gotten 1.57079632679 something twice.

That answer will occur at X(5) and at X(6).




Here is a link to a page that will do the whole problem

once you enter the original equation and the x value.


http://mss.math.vanderbilt.edu/~pscrooke/MSS/newtonnum.html

Slant Asymptote:

Exists ONLY if the degree of N is EXACTLY 1 more than the degree of Q.

* Example: N 4 / Q 3

 

Horizontal Asymptote


* N < Q, then y = the x-axis 

* N = Q, then y = the Leading Coefficients of N/Q. 

* N > Q, then the horizontal asymptote Does Not Exist.

Vertical Asymptote:

* Set Q(c)=0, Then x = c