The graph is as follows:

Finding the root of a function.

Try x0 = 1.5

Then

`x_1=x_0-(f(x_0))/(f’(x_0))`

` = 1.5-(f(1.5))/(f’(1.5))`

Now f(1.5) = 2(1.5)2 − 1.5 − 2 = 1

f '(x) = 4x − 1 and f '(1.5) = 6 − 1 = 5

So

`x_1=1.5-1/5=1.3`

So `1.3` is a better approximation.

Continuing the process,

`x_2=x_1-(f(x_1))/(f’(x_1))`

` = 1.3- (f(1.3))/(f’(1.3))`

` = 1.3-0.08/4.2`

`=1.2809524`

We can continue for as many steps as necessary to give the required accuracy.

Check: Using some Computer Algebra Systems, (eg Mathcad) we also need to give an initial guess (say, x = 2) and the result is:

root(2x2x − 2, x) = 1.2807764064044

The software just keeps applying the algorithm until the decimal digits don't change any more. It then returns the answer.

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