A-LevelMathematicsDifferentiation9709/33 · May/June 2019 · Q47 Marks
The equation of a curve is y = (1 + e⁻ˣ) / (1 - e⁻ˣ), for x > 0.
- (i)Show that dy/dx is always negative.[3]
- (ii)The gradient of the curve is equal to −1 when x = a. Show that a satisfies the equation e^(2a) – 4e^a + 1 = 0. Hence find the exact value of a.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.
- [M1] Use the quotient or product rule
- [A1] Obtain correct derivative in any form
- [A1] Reduce to -2e⁻ˣ / (1-e⁻ˣ)² or equivalent, and explain why this is always negative
- [B1] Equate derivative to –1 and obtain the given equation
- [B1] State or imply u² – 4u + 1 = 0, or equivalent in eᵃ
- [M1] Solve for a
- [A1] Obtain answer a = ln (2+√3) and no other
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