A-LevelMathematicsNumerical methods9709/32 · Oct/Nov 2018 · Q58 Marks

The equation of a curve is y = x ln(8 – x). The gradient of the curve is equal to 1 at only one point, when x = a.

(i)Show that a satisfies the equation x = 8 - 8/ln(8-x)[3]
(ii)Verify by calculation that a lies between 2.9 and 3.1.[2]
(iii)Use an iterative formula based on the equation in part (i) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[M1] Use product rule on a correct expression
[A1] Obtain correct derivative in any form
[A1] Equate derivative to 1 and obtain x = 8 - 8/ln(8-x)
[M1] Calculate values of a relevant expression or pair of relevant expressions at x = 2.9 and x = 3.1
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative process Xn+1 = 8 - 8/ln(8-Xn) correctly to find at least two successive values.
[A1] Obtain final answer 3.02
[A1] Show sufficient iterations to at least 4 d.p. to justify 3.02 to 2 d.p., or show there is a sign change in the interval (3.015, 3.025)

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