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1.12 I: Numerical methods (A-level only)

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Question 39

The equation x3−3x2+5=0x^3 - 3x^2 + 5 = 0x3−3x2+5=0 has exactly one real root.

a.

Show that, for this equation, the Newton-Raphson formula can be written

xn+1=2xn3−3xn2−53xn2−6xn\displaystyle x_{n+1} = \frac{2x_n^3 - 3x_n^2 - 5}{3x_n^2 - 6x_n}xn+1​=3xn2​−6xn​2xn3​−3xn2​−5​

[3]
b.

Using the formula in part (a) with x1=−1x_1 = -1x1​=−1, find the values of x2 x_2\,x2​ and x3x_3x3​. Give your answers to four decimal places.

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c.

Explain why, for this equation, the Newton-Raphson method cannot be used with x1=0x_1 = 0x1​=0.

[1]
Markscheme

1.12 I: Numerical methods (A-level only) Questions

  1. A Level
  2. /Maths
  3. /1.12 I: Numerical methods (A-level only)

125 exam-style questions on AQA A Level Maths 1.12 I: Numerical methods (A-level only), covering 1.12.1 Locating roots by change of sign (A-level only), 1.12.2 Iterative methods and Newton-Raphson (A-level only), 1.12.3 Numerical integration (A-level only), and 1.12.4 Numerical methods in context (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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