A-LevelMathematicsNumerical methods9709/32 · Feb/Mar 2021 · Q911 Marks

Let f(x) = (e^(2x) + 1) / (e^(2x) - 1), for x > 0.

(a)The equation x = f(x) has one root, denoted by α. Verify by calculation that α lies between 1 and 1.5.[2]
(b)Use an iterative formula based on the equation in part (a) to determine α correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]
(c)Find f'(x). Hence find the exact value of x for which f'(x) = −8.[6]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.

[M1] Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 1.5
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative formula x_(n+1) = (e^(2xn) + 1) / (e^(2xn) - 1), or equivalent, correctly at least once
[A1] Obtain final answer 1.20
[A1] Show sufficient iterations to 4 dp to justify 1.20 to 2 dp, or show there is a sign change in the interval (1.195,1.205)
[M1] Use quotient rule
[A1] Obtain correct derivative in any form
[M1] Equate derivative to – 8 and obtain a quadratic in e^(2x)
[A1 OE] Obtain 2(e^(2x))² - 5e^(2x) + 2 = 0
[M1] Solve a 3-term quadratic in e^(2x) for x
[A1] Obtain answer x = ½ ln 2, or exact equivalent, only

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