Derive Newton-Raphson formula

A question is this type if and only if it requires showing that the Newton-Raphson iterative formula can be written in a specific given form, starting from f(x) = 0.

4 questions

OCR FP2 2010 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{63afce50-e15f-4634-b2f1-ad5d78ab8bf5-2_597_1006_973_571} A curve with no stationary points has equation \(y = \mathrm { f } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has one real root \(\alpha\), and the Newton-Raphson method is to be used to find \(\alpha\). The tangent to the curve at the point \(\left( x _ { 1 } , \mathrm { f } \left( x _ { 1 } \right) \right)\) meets the \(x\)-axis where \(x = x _ { 2 }\) (see diagram).
  1. Show that \(x _ { 2 } = x _ { 1 } - \frac { \mathrm { f } \left( x _ { 1 } \right) } { \mathrm { f } ^ { \prime } \left( x _ { 1 } \right) }\).
  2. Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation \(x = x _ { 1 }\), gives a sequence of approximations approaching \(\alpha\).
  3. Use the Newton-Raphson method, with a first approximation of 1 , to find a second approximation to the root of \(x ^ { 2 } - 2 \sinh x + 2 = 0\).
OCR FP2 2013 June Q5
5 You are given that the equation \(x ^ { 3 } + 4 x ^ { 2 } + x - 1 = 0\) has a root, \(\alpha\), where \(- 1 < \alpha < 0\).
  1. Show that the Newton-Raphson iterative formula for this equation can be written in the form $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 4 x _ { n } ^ { 2 } + 1 } { 3 x _ { n } ^ { 2 } + 8 x _ { n } + 1 } .$$
  2. Using the initial value \(x _ { 1 } = - 0.7\), find \(x _ { 2 }\) and \(x _ { 3 }\) and find \(\alpha\) correct to 5 decimal places.
  3. The diagram shows a sketch of the curve \(y = x ^ { 3 } + 4 x ^ { 2 } + x - 1\) for \(- 1.5 \leqslant x \leqslant 1\).
    \includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-3_602_926_749_566} Using the copy of the diagram in your answer book, explain why the initial value \(x _ { 1 } = 0\) will fail to find \(\alpha\).
Edexcel Paper 2 2018 June Q5
  1. The equation \(2 x ^ { 3 } + x ^ { 2 } - 1 = 0\) has exactly one real root.
    1. Show that, for this equation, the Newton-Raphson formula can be written
    $$x _ { n + 1 } = \frac { 4 x _ { n } ^ { 3 } + x _ { n } ^ { 2 } + 1 } { 6 x _ { n } ^ { 2 } + 2 x _ { n } }$$ Using the formula given in part (a) with \(x _ { 1 } = 1\)
  2. find the values of \(x _ { 2 }\) and \(x _ { 3 }\)
  3. Explain why, for this question, the Newton-Raphson method cannot be used with \(x _ { 1 } = 0\)
OCR H240/01 Q9
9 The equation \(x ^ { 3 } - x ^ { 2 } - 5 x + 10 = 0\) has exactly one real root \(\alpha\).
  1. Show that the Newton-Raphson iterative formula for finding this root can be written as $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - x _ { n } ^ { 2 } - 10 } { 3 x _ { n } ^ { 2 } - 2 x _ { n } - 5 }$$
  2. Apply the iterative formula in part (a) with initial value \(x _ { 1 } = - 3\) to find \(x _ { 2 } , x _ { 3 } , x _ { 4 }\) correct to 4 significant figures.
  3. Use a change of sign method to show that \(\alpha = - 2.533\) is correct to 4 significant figures.
  4. Explain why the Newton-Raphson method with initial value \(x _ { 1 } = - 1\) would not converge to \(\alpha\).