A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2024 · Q59 Marks

The diagram shows part of the curve with equation y = x³/(x+2). At the point P, the gradient of the curve is 6. [Diagram]

(a)Show that the x-coordinate of P satisfies the equation x = ³√12x+12.[4]
(b)Show by calculation that the x-coordinate of P lies between 3.8 and 4.0.[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. Show the result of each iteration to 5 significant figures.[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] (a) Differentiate using quotient rule [OE]
[A1] (a) Obtain (x+2)3x²-x³/(x+2)² [OE]
[DM1] (a) Equate first derivative to 6 and simplify at least as far as x³ =...
[A1] (a) Confirm x = ³√12x+12 [Answer given – necessary detail needed.]
[M1] (b) Consider sign of x - ³√12x+12, or equivalent, for 3.8 and 4.0
[A1] (b) Obtain -0.06... and 0.08..., or equivalents, and justify conclusion [Answer given – necessary detail needed.]
[M1] (c) Use iterative process correctly at least once
[A1] (c) Obtain final answer 3.88 [Answer required to exactly 3 sf.]
[A1] (c) Show sufficient iterations to 5 sf to justify answer or show sign change in the interval [3.875, 3.885]

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