A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2019 · Q59 Marks

The equation of a curve is y = e^(2x) / (4x + 1) and the point P on the curve has y-coordinate 10.

(i)Show that the x-coordinate of P satisfies the equation x = ½ ln(40x + 10).[2]
(ii)Use the iterative formula xn+1 = ½ ln(40xn + 10) with x₁ = 2.3 to find the x-coordinate of P correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]
(iii)Find the gradient of the curve at P, giving the answer correct to 3 significant figures.[4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2019). The mark or mark type is shown in brackets.

[M1] Attempt rearrangement of e^(2x) / (4x+1) = 10 to x = … involving ln
[A1] Confirm x = ½ ln(40x+10)
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 2.316
[A1] Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [2.3155, 2.3165]
[M1] Use quotient rule (or product rule) to find derivative
[A1] Obtain (2e^(2x)(4x + 1) − 4e^(2x)) / (4x+1)² or equivalent
[M1] Substitute answer from part (ii) (or more accurate value) into attempt at first derivative
[A1] Obtain 16.1

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