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5 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2 · Question 4(a)(ii)

Use the Newton-Raphson method to show that if x_1 is a first approximation to \alpha in the interval [1, 2], then a second approximation to \alpha in the interval [1, 2] is given by x_2 = \frac{2x_1^3 - 1}{3x_1^2 - 3}.

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  1. 4(a)(i)Show that the function f(x) = x^3 - 3x + 1 has a root \alpha in the closed interval [1, 2].[3 marks]
  2. 4(b)(i)Use the binomial theorem or Maclaurin's theorem to expand (1 + x)^{-1/2} in ascending powers of x as far as the term in x^3, stating the values of x for which…[4 marks]
  3. 4(b)(ii)Obtain a similar expansion for (1 - x)^{1/2}.[4 marks]
  4. 4(b)(iii)Prove that if x is so small that x^3 and higher powers of x can be neglected, then \sqrt{\frac{1 - x}{1 + x}} \approx 1 - x + \frac{1}{2}x^2.[5 marks]
  5. 4(b)(iv)Hence, by taking x = \frac{1}{17}, show, without using calculators or tables, that \sqrt{2} is approximately equal to \frac{1635}{1156}.[4 marks]

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