Newton-Raphson with derivative given or simple

2. (a) Show that \(\mathrm { f } ( x ) = x ^ { 4 } + x - 1\) has a real root \(\alpha\) in the interval [0.5, 1.0].
[0pt] (b) Starting with the interval [0.5, 1.0], use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\).
(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\). Give your answer to 3 decimal places.

8. $$f ( x ) = x ^ { 3 } - 2 x - 3$$

1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval \([ 1,2 ]\).
2. Starting with the interval \([ 1,2 ]\), use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
3. Using \(x _ { 0 } = 1.8\) 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 3 significant figures.

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.

7 The equation $$24 x ^ { 3 } + 36 x ^ { 2 } + 18 x - 5 = 0$$ has one real root, \(\alpha\).

1. Show that \(\alpha\) lies in the interval \(0.1 < x < 0.2\).
2. Starting from the interval \(0.1 < x < 0.2\), use interval bisection twice to obtain an interval of width 0.025 within which \(\alpha\) must lie.
3. Taking \(x _ { 1 } = 0.2\) 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 decimal places.
   (4 marks)

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)

8 Given that the equation \(x ^ { 3 } + 2 x - 7 = 0\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { \mathrm { o } } = 1\) to find this root correct to 3 decimal places.

6 The equation \(x ^ { 3 } - x - 1 = 0\) has exactly one real root in the interval \(0 \leq x \leq 3\).

1. Denoting this root by \(\alpha\), find the integer \(n\) such that \(n < \alpha < n + 1\).
2. Taking \(n\) as a first approximation, use the Newton-Raphson method to find \(\alpha\), correct to 2 decimal places. You must show the result of each iteration correct to an appropriate degree of accuracy.

$$f(x) = \frac{1}{2}x^4 - x^3 + x - 3$$

1. Show that the equation \(f(x) = 0\) has a root \(\alpha\) between \(x = 2\) and \(x = 2.5\) [2]
2. Starting with the interval \([2, 2.5]\) use interval bisection twice to find an interval of width \(0.125\) which contains \(\alpha\). [3]

The equation \(f(x) = 0\) has a root \(\beta\) in the interval \([-2, -1]\).

1. Taking \(-1.5\) as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(f(x)\) to obtain a second approximation to \(\beta\). Give your answer to 2 decimal places. [5]

You are given that \(f(x) = \ln(2x - 5) + 2x^2 - 30\), for \(x > 2.5\).

1. Show that \(f(x) = 0\) has a root \(\alpha\) in the interval \([3.5, 4]\). [2]

A student takes 4 as the first approximation to \(\alpha\). Given \(f(4) = 3.099\) and \(f'(4) = 16.67\) to 4 significant figures,

1. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures. [2]
2. Show that \(\alpha\) is the only root of \(f(x) = 0\). [2]

Let \(y = (x - 1)\left(\frac{2}{x^2} + t\right)\) define \(y\) as a function of \(x\) (\(x > 0\)), for each value of the real parameter \(t\).

1. When \(t = 0\),
   1. determine the set of values of \(x\) for which \(y\) is positive and an increasing function, [3]
   2. locate the stationary point of \(y\), and determine its nature. [2]
2. It is given that \(t = 2\) and \(y = -2\).
   1. Show that \(x\) satisfies \(f(x) = 0\), where \(f(x) = x^3 + x - 1\). [1]
   2. Prove that \(f\) has no stationary points. [2]
   3. Use the Newton-Raphson method, with \(x_0 = 1\), to find \(x\) correct to 4 significant figures. [4]