A-level · Numerical methods

Iterative methods (x = g(x))Exam-style questions

Solving f(x) = 0 by rearranging it into x = g(x) and iterating x_(n+1) = g(x_n) from a starting value — why some rearrangements converge and others diverge, and reading staircase and cobweb diagrams.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The equation

$\mathrm{e}^x = 3x^2$

has three real roots. The largest of these roots is $\alpha$
(a) Use a suitable change of sign to show that $\alpha$ lies between $3$ and $4$

(2 marks)
(b) Show that, for $x > 0$ , the equation $\mathrm{e}^x = 3x^2$ can be rearranged to give

$x = \ln 3 + 2\ln x$

(3 marks)
(c) (i) Taking $x_1 = 2$ , find the values of $x_2$ , $x_3$ and $x_4$ given by the iterative formula

$x_{n+1} = \ln 3 + 2\ln x_n$

Give each value to three decimal places.

(ii) Figure 1 shows sketches of the curve $y = \ln 3 + 2\ln x$ and the line $y = x$

Figure 1Illustrate the iteration in part (c)(i) on Figure 1 by drawing a staircase or cobweb diagram.

Show the positions of $x_2$ , $x_3$ and $x_4$ on the $x$ -axis.

(iii) Explain why the iteration $x_{n+1} = \ln 3 + 2\ln x_n$ does not converge to $\alpha$ if $x_1 = 0.5$

(5 marks)
Level 5 · 10 marksTry it