Newton-Raphson with complex derivative required

Questions where the function involves fractional powers, negative powers, or combinations requiring non-trivial differentiation (chain rule, product rule, or quotient rule) before applying Newton-Raphson.

18 questions · Standard +0.2

1.09d Newton-Raphson method
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Edexcel F1 2015 January Q2
7 marks Standard +0.8
2. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + \frac { 1 } { 2 \sqrt { x ^ { 5 } } } + 2 , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 2,3 ]\).
  2. Taking 3 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2016 January Q2
7 marks Standard +0.3
2. $$\mathrm { f } ( x ) = x ^ { 2 } - \frac { 3 } { \sqrt { x } } - \frac { 4 } { 3 x ^ { 2 } } , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.6,1.7]
  2. Taking 1.6 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2017 June Q5
9 marks Moderate -0.3
5. $$f ( x ) = 30 + \frac { 7 } { \sqrt { x } } - x ^ { 5 } , \quad x > 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [2,2.1].
[0pt]
  1. Starting with the interval [2,2.1], use interval bisection twice to find an interval of width 0.025 that contains \(\alpha\).
  2. Taking 2 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 2 decimal places.
Edexcel F1 2020 June Q1
7 marks Standard +0.8
1. $$f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1.4,1.5]
  2. Determine \(\mathrm { f } ^ { \prime } ( x )\) .
  3. Using \(x _ { 0 } = 1.4\) as a first approximation to \(\alpha\) ,apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to calculate a second approximation to \(\alpha\) ,giving your answer to 3 decimal places. \(f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0\)
    1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1.4,1.5]
    2. Determine \(\mathrm { f } ^ { \prime } ( x )\) .
Edexcel F1 2022 June Q2
9 marks Standard +0.3
2. $$f ( x ) = 10 - 2 x - \frac { 1 } { 2 \sqrt { x } } - \frac { 1 } { x ^ { 3 } } \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [0.4, 0.5]
  2. Determine \(\mathrm { f } ^ { \prime } ( x )\).
  3. Using \(x _ { 0 } = 0.5\) 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 3 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval [4.8, 4.9]
    [0pt]
  4. Use linear interpolation once on the interval [4.8, 4.9] to find an approximation to \(\beta\), giving your answer to 3 decimal places.
Edexcel F1 2024 June Q3
7 marks Standard +0.3
3. $$\mathrm { f } ( x ) = x ^ { 3 } - 5 \sqrt { x } - 4 x + 7 \quad x \geqslant 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 0.25,1 ]\)
  1. Use linear interpolation once on the interval [ \(0.25,1\) ] to determine an approximation to \(\alpha\), giving your answer to 3 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval [1.5, 2.5]
  2. Determine \(\mathrm { f } ^ { \prime } ( x )\)
  3. Hence, using \(x _ { 0 } = 1.75\) as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to determine a second approximation to \(\beta\), giving your answer to 3 decimal places.
Edexcel FP1 2009 January Q5
9 marks Moderate -0.3
5. $$f ( x ) = 3 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 20$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.1,1.2].
  2. Find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Using \(x _ { 0 } = 1.1\) 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.
Edexcel FP1 2010 January Q2
9 marks Moderate -0.3
2. $$f ( x ) = 3 x ^ { 2 } - \frac { 11 } { x ^ { 2 } }$$
  1. Write down, to 3 decimal places, the value of \(\mathrm { f } ( 1.3 )\) and the value of \(\mathrm { f } ( 1.4 )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between 1.3 and 1.4
    [0pt]
  2. Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains \(\alpha\).
  3. Taking 1.4 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel FP1 2013 January Q3
6 marks Moderate -0.3
3. $$\mathrm { f } ( x ) = 2 x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } - 5 , \quad x > 0$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4.5, 5.5].
  2. Using \(x _ { 0 } = 5\) 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 3 significant figures.
Edexcel FP1 2014 January Q4
5 marks Standard +0.3
4. $$f ( x ) = 2 x ^ { \frac { 1 } { 2 } } - \frac { 6 } { x ^ { 2 } } - 3 , \quad x > 0$$ A root \(\beta\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,4 ]\).
Taking 3.5 as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\beta\). Give your answer to 3 decimal places.
Edexcel FP1 2010 June Q3
10 marks Standard +0.3
3. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 7 } { x } + 2 , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between 1.4 and 1.5
    [0pt]
  2. Starting with the interval [1.4,1.5], use interval bisection twice to find an interval of width 0.025 that contains \(\alpha\).
  3. Taking 1.45 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x ) = x ^ { 3 } - \frac { 7 } { x } + 2\) to obtain a second approximation to \(\alpha\), giving your answer to 3 decimal places.
Edexcel FP1 2011 June Q4
6 marks Moderate -0.3
4. $$f ( x ) = x ^ { 2 } + \frac { 5 } { 2 x } - 3 x - 1 , \quad x \neq 0$$
  1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.7, 0.9].
  2. Taking 0.8 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.
Edexcel FP1 2012 June Q3
6 marks Standard +0.8
3. $$f ( x ) = x ^ { 2 } + \frac { 3 } { 4 \sqrt { } x } - 3 x - 7 , \quad x > 0$$ A root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,5 ]\).
Taking 4 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 2 decimal places.
Edexcel FP1 2014 June Q2
7 marks Standard +0.3
2. $$\mathrm { f } ( x ) = x ^ { 3 } - \frac { 5 } { 2 x ^ { \frac { 3 } { 2 } } } + 2 x - 3 , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [1.1,1.5].
  2. Find f'(x).
  3. Using \(x _ { 0 } = 1.1\) 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 decimal places.
Edexcel FP1 2016 June Q2
6 marks Standard +0.3
2. $$f ( x ) = 3 x ^ { \frac { 3 } { 2 } } - 25 x ^ { - \frac { 1 } { 2 } } - 125 , \quad x > 0$$
  1. Find \(f ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [12, 13].
  2. Using \(x _ { 0 } = 12.5\) 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 3 decimal places.
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Pre-U Pre-U 9794/1 2012 Specimen Q7
4 marks Standard +0.3
7 Given that the equation \(x = 2 - \frac { 1 } { ( x + 1 ) ^ { 2 } }\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { 0 } = 2\) to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2014 June Q7
4 marks Standard +0.3
7 Taking \(x = 2\) as a first approximation, use the Newton-Raphson process to find a root of the equation \(\frac { 1 } { x ^ { 2 } } - 0.119 - 0.018 x = 0\). Give your answer correct to 3 significant figures.
Edexcel FP1 Q5
9 marks Moderate -0.3
$$\text{f}(x) = 3\sqrt{x} + \frac{18}{\sqrt{x}} - 20.$$
  1. Show that the equation f\((x) = 0\) has a root \(a\) in the interval \([1.1, 1.2]\). [2]
  2. Find \(f'(x)\). [3]
  3. Using \(x_0 = 1.1\) as a first approximation to \(a\), apply the Newton-Raphson procedure once to f\((x)\) to find a second approximation to \(a\), giving your answer to 3 significant figures. [4]