O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2012 · Q109 Marks

Variables x and y are such that y = e^(2x) + e^(-2x).

(i)Find dy/dx.[2]
(ii)By using the substitution u = e^(2x), find the value of y when dy/dx = 3.[4]
(iii)Given that x is decreasing at the rate of 0.5 units s⁻¹, find the corresponding rate of change of y when x = 1.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

[B1] (i) dy/dx = 2e^(2x)
[B1] (i) -2e^(-2x)
[M1] (ii) for attempt to obtain in 'quadratic' form (3 = 2e^(2x) - 2e^(-2x) => 2(e^(2x))² - 3(e^(2x)) - 2 = 0)
[DM1] (ii) for attempt to solve for e^(2x) ((2e^(2x) + 1)(e^(2x) - 2) = 0)
[M1] (ii) for using e^(2x) = 2 to find y (y = 2 + 1/2)
[A1] (ii) y = 5/2 (or 2.5)
[B1] (iii) for dx/dt = -0.5
[M1] (iii) for substitution of x = 1 into dy/dx (dy/dx = 2e² - 2e⁻²)
[M1] (iii) for correct application of chain rule (dy/dt = dy/dx * dx/dt)
[A1] (iii) -7.25 (3 s.f.)

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 9 marks.
Source: O-Level Additional Mathematics, May/June 2012 Paper 1 Variant 1, question 10.

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