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8 marksRoots of Equations

CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2 · Question 4(b)(ii)

Let f(x) = x - 3\sin x - 1.

Use at least three iterations of the method of interval bisection to show that f(-0.538) \approx 0 in the interval [-0.7, -0.3].

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  1. 4(a)(i)Obtain the Maclaurin series expansion for g(x) up to and including the term in x^4.[6 marks]
  2. 4(a)(ii)Hence, estimate g(0.2) correct to three decimal places.[3 marks]
  3. 4(b)(i)Use the intermediate value theorem to show that f has at least one root in the interval [-2, 0].[3 marks]
  4. 4(c)Use the Newton–Raphson method with initial estimate x_1 = 5.5 to approximate the root of g(x) = \sin 3x in the interval [5, 6], correct to two decimal…[5 marks]

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