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1.12 I: Numerical methods (A-level only)

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Question 62

A structural beam's vertical deflection D(x)D(x)D(x) (in mm) at a distance xxx (in metres) from the left support is modeled by the function

D(x)=x4−6x3+5x+12,x≥0 D(x) = x^4 - 6x^3 + 5x + 12, \quad x \ge 0 D(x)=x4−6x3+5x+12,x≥0
a.

Show that the equation D(x)=0D(x) = 0D(x)=0 has a root, α\alphaα, in the interval [5.5,6.0][5.5, 6.0][5.5,6.0].

[2]
b.

Show that the equation D(x)=0D(x) = 0D(x)=0 can be rewritten in the form

x=6x2−5−12x3 x = \sqrt[3]{6x^2 - 5 - \frac{12}{x}} x=36x2−5−x12​​

The iterative formula

xn+1=6xn2−5−12xn3 x_{n+1} = \sqrt[3]{6x_n^2 - 5 - \frac{12}{x_n}} xn+1​=36xn2​−5−xn​12​​

is used to find α\alphaα.

[2]
c.

Starting with x1=5.7x_1 = 5.7x1​=5.7 and using the iterative formula,

(i) find, to 4 decimal places, the value of x2x_2x2​.

(ii) find, to 4 decimal places, the value of α\alphaα.

[4]
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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