Newton-Raphson with verification

A question is this type if and only if it asks to apply Newton-Raphson and then verify the answer is correct to a specified number of decimal places using a change of sign or other method.

2 questions

Edexcel F1 2017 January Q6
6. $$f ( x ) = x ^ { 3 } - \frac { 1 } { 2 x } + x ^ { \frac { 3 } { 2 } } , \quad x > 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.6, 0.7].
  1. Taking 0.6 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
  2. Show that your answer to part (a) is correct to 3 decimal places.
Edexcel F1 2018 Specimen Q3
3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ - 2 , - 1 ]\).
  1. Taking - 1.5 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 2 decimal places.
  2. Show that your answer to part (a) gives \(\alpha\) correct to 2 decimal places.
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