A-LevelMathematicsIntegration9709/31 · Oct/Nov 2022 · Q910 Marks

The diagram shows part of the curve y = (3 - x)e^(−x/3) for x ≥ 0, and its minimum point M. [Figure 9.1]

(a)Find the exact coordinates of M.[5]
(b)Find the area of the shaded region bounded by the curve and the axes, giving your answer in terms of e.[5]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[M1] (a) Use correct product or quotient rule
[A1] (a) Obtain correct derivative in any form
[DM1] (a) Equate their derivative to zero and solve for x
[A1] (a) Obtain x = 6
[A1] (a) Obtain y = -3e⁻² or exact equivalent.
[M1] (b) Commence integration and reach a(3-x)e^(-x/3) + b∫e^(-x/3) dx, where ab≠0
[A1] (b) Obtain -3(3-x)e^(-x/3) - 3∫e^(-x/3) dx, or equivalent
[A1] (b) Complete integration and obtain 3xe^(-x/3) or equivalent
[DM1] (b) Substitute limits x = 0 and x = 3, having integrated twice
[A1] (b) Obtain answer 9/e or exact equivalent

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