A-level · Numerical methods

Locating roots by change of signExam-style questions

Trapping a root of f(x) = 0 in an interval using the change-of-sign rule — why continuity is essential, what a sign change does and doesn't guarantee, and refining a root to a given number of decimal places.

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22 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    (a) By sketching a suitable pair of graphs, show that the equation cot⁡x=1+x\cot x = 1 + x has exactly one root, denoted by α\alpha, in the interval 0<x<12π0 < x < \tfrac{1}{2}\pi.

    (2 marks)
    (b) Verify by calculation that α\alpha lies between 0.50.5 and 11.

    (2 marks)
    (c) Show that if a sequence of values given by the iterative formula

    xn+1=tan⁡−1(11+xn)x_{n+1} = \tan^{-1}\left(\frac{1}{1 + x_{n}}\right)

    converges, then it converges to α\alpha.

    (1 mark)
    (d) Use this iterative formula with an initial value of 0.60.6 to determine α\alpha correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

    (3 marks)
    (e) Starting from this initial value, state the least number of iterations that must be calculated before α\alpha is established correct to 2 decimal places.

    (1 mark)
    Level 5 · 9 marksTry it
  2. 2
    f(x)=x3−4x+1\mathrm{f}(x) = x^3 - 4x + 1

    Figure 4 shows part of the curve with equation y=f(x)y = \mathrm{f}(x).-2-112-6-4-2246xyFigure 4
    (a) Show that the equation f(x)=0\mathrm{f}(x) = 0 has a root α\alpha in the interval [1.8,1.9][1.8, 1.9]. (2 marks)
    (b) Taking x0=1.9x_0 = 1.9, apply the Newton–Raphson procedure twice to f(x)\mathrm{f}(x) to obtain an approximation for α\alpha, giving your answer to 55 decimal places. (4 marks)
    (c) Explain, with reference to the gradient of the curve, why x0=1.15x_0 = 1.15 would be a poor choice of first approximation for the Newton–Raphson procedure. (2 marks)
    (d) An alternative approach uses the iterative formula

    xn+1=4xn−13x_{n+1} = \sqrt[3]{4x_n - 1}

    Using this formula with x0=1.9x_0 = 1.9, find the value of x2x_2, giving your answer to 44 decimal places. (2 marks)
    Level 5 · 10 marksTry it