A-LevelMathematicsNumerical methods9709/21 · May/June 2024 · Q59 Marks

A curve has equation y = 1+e2x 1+3x . The curve has exactly one stationary point P.

(a)Find dy dx and hence show that the x-coordinate of P satisfies the equation x = 1 6 + 1 2 e⁻²x. [4]
(b)Show by calculation that the x-coordinate of P lies between 0.35 and 0.45 .[2]
(c)Use an iterative formula based on the equation in part (a) to find the x-coordinate of P correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]

Mark scheme

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

[*M1] Differentiate using quotient rule
[A1] Obtain ((1+3x) 2e²x – (1 + e²x) 3) / (1+3x)²
[DM1] Equate first derivative to zero and arrange as far as x = ...
[A1] Confirm x = 1/6 + 1/2 e⁻²x
[M1] Consider sign of x - 1/6 - 1/2 e⁻²x
[A1] Obtain -0.06... (– 0.064959...) and 0.08... (0.0800...) or equivalents and justify conclusion
[M1] Use iterative process correctly at least once
[A1] Obtain final answer 0.394
[A1] Show sufficient iterations to 5 sf to justify answer or show sign change in interval [0.3935, 0.3945]

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