Newton-Raphson with derivative given or simple

1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$

1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.

8

1. The equation $$x ^ { 3 } + 2 x ^ { 2 } + x - 100000 = 0$$ has one real root. Taking \(x _ { 1 } = 50\) as a first approximation to this root, use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to the root.
2. 1. Given that \(S _ { n } = \sum _ { r = 1 } ^ { n } r ( 3 r + 1 )\), use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$S _ { n } = n ( n + 1 ) ^ { 2 }$$
   2. The lowest integer \(n\) for which \(S _ { n } > 100000\) is denoted by \(N\). Show that $$N ^ { 3 } + 2 N ^ { 2 } + N - 100000 > 0$$
3. Find the value of \(N\), justifying your answer.

1 The equation $$x ^ { 3 } - x ^ { 2 } + 4 x - 900 = 0$$ has exactly one real root, \(\alpha\). Taking \(x _ { 1 } = 10\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\). Give your answer to four significant figures.
(3 marks)

8 [Figure 2, printed on the insert, is provided for use in this question.]
The diagram shows a part of the graph of \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 3 } - 2 x - 1$$ The point \(P\) has coordinates \(( 1 , - 2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{a77cc9c3-5ff6-4abc-931e-e811740267f2-05_606_565_717_740}

1. Taking \(x _ { 1 } = 1\) as a first approximation to a root of the equation \(\mathrm { f } ( x ) = 0\), use the NewtonRaphson method to find a second approximation, \(x _ { 2 }\), to the root.
2. On Figure 2, draw a straight line to illustrate the Newton-Raphson method as used in part (a). Mark \(x _ { 1 }\) and \(x _ { 2 }\) on Figure 2
3. By considering \(f ( 2 )\), show that the second approximation found in part (a) is not as good as the first approximation.
4. Taking \(x _ { 1 } = 1.6\) as a first approximation to the root, use the Newton-Raphson method to find a second approximation to the root. Give your answer to three decimal places.
   (2 marks)