Apply​ Newton's Method

Question

Apply​ Newton's method to ​f(x)=x^1/3 with x₀=1 and calculate x₁​, x₂, x₃, and x₄. Find a formula |x_n|.

What happens to |x_n| as n→∞​?

Draw a picture that shows what is going on.

Questions

calculate x₁​, x₂, x₃, and x₄

Find a formula |x_n|

What happens to |x_n| as n→∞​?

Answer 1

Setting f(x) = x^1/3, you would establish f'(x) as 3^-1x^(1/3-3/3)

or 3^-1x^-2/3.

Then write x − [f(x) / f'(x)] which goes to x − [x^1/3 / 3^-1x^-2/3]

and then to x − 3x or -2x.

Generate the table below beginning with x equal to 1:

x-----------------------(-2x)

1------------------------(-2)

(-2)-----------------------4

4-----------------------(-8)

(-8)---------------------16

16--------------------(-32)

The table shows that x starts at 1 and feeds each evaluation of -2x back

into -2x as "the new x". The resulting values of -2x are alternately negative

and positive which indicates that the series of results will forever move

between negative and positive values.

Of course, simple inspection of the equation x^1/3 = 0 gives solution as x = 0.

This solution is not obtained by Newton's Method Of Root Approximation which is the tabulation of results of -2x (starting at x = 1) as shown above. Otherwise, one would finally see a result of -2x that is identical to that result's input of x.

It is also seen that |x_n| doubles as n goes to infinity.