Manipulating the equation from the formulas for Newtons Method

For Question 4, what manipulations of I assume x_(k+1) = x_k - (a -1/x) / (1/x^2) result in the answer?

Also Question 5, how do we result in the answer from x_(k+1) = x_k - (log(x)) / (1/x) ?

Thanks!

TMosh
2
For Q4:

You’re given f(x) = a - 1/x and f’(x) = 1/x^2

Here I’ll use ‘x’ to represent x(k), for easier notation.

So you have:

x(k+1) = x - (a-1/x) / 1/x^2)

On the right side, the division by 1/x^2 is equivalent to multiplying by x^2

So this becomes:

x(k+1) = x - ((a - 1/x) * x^2)

Distribute the x^2, and you get:

x(k+1) = x - (ax^2 - x)

Simplify and you have:

x(k+1) = 2x - ax^2