A-LevelMathematicsNumerical methods9709/32 · Feb/Mar 2024 · Q78 Marks

The diagram shows the curve y = xe^(2x) - 5x and its minimum point M, where x = α. [Figure 7.1]

(a)Show that α satisfies the equation α = (1/2)ln(5 / (1 + 2α)).[3]
(b)Verify by calculation that α lies between 0.4 and 0.5.[2]
(c)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]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

[M1] Use correct product rule
[A1] Obtain correct derivative in any form
[A1] Equate derivative to zero and obtain α = (1/2)ln(5 / (1 + 2α))
[M1] Calculate the value of a relevant expression or values of a pair of expressions at x=0.4 and x = 0.5
[A1] Complete the argument correctly with correct calculated values
[M1] Use the iterative process α_(n+1) = (1/2)ln(5/(1+2α_n)) correctly at least twice anywhere in iteration process
[A1] Obtain final answer 0.47
[A1] Show sufficient iterations to 4 d.p. to justify 0.47 to 2 d.p. or show there is a sign change in the interval (0.465,0.475)

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