A-LevelMathematicsNumerical methods9709/22 · May/June 2024 · Q69 Marks

The diagram shows the curve with equation y = ln(2x+1) / (x+3). The curve has a maximum point M. [Figure 6.1]

(a)Find an expression for dy/dx.[2]
(b)Show that the x-coordinate of M satisfies the equation x = (x+3)/ln(2x+1) - 0.5.[2]
(c)Show by calculation that the x-coordinate of M lies between 2.5 and 3.0 .[2]
(d)Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]

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Mark scheme

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

[M1] Attempt use of quotient rule Or equivalent method.
[A1] Obtain (x+3) * (2/(2x+1)) - ln(2x+1) / (x+3)² OE
[M1] Equate first derivative to zero and arrange as far as 2x+1=...
[A1] Confirm x = (x+3)/ln(2x+1) - 0.5 Answer given – necessary detail needed.
[M1] Consider sign of x - (x+3)/ln(2x+1) + 0.5 or equivalent for 2.5 and 3.0
[A1] Obtain -0.07 (0.0696...) and 0.4 (0.4166...) and justify conclusion Answer given – necessary detail needed.
[(M1)] Consider the values of f(x) = (x+3)/ln(2x+1) - 0.5 and obtain f(2.5) = 2.57 (2.5696...) and f(3) = 2.58 (2.58339...)
[(A1)] Conclude f(2.5)<3 and f(3)>2.5 so root lies in given interval Answer given – necessary detail needed.
[M1] Use iterative process correctly at least once
[A1] Obtain final answer 2.569 Answer required to exactly 4sf.
[A1] Show sufficient iterations to 6 sf to justify answer or show sign change in interval [2.5685, 2.5695]

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