Variables x and y are related by the equation y = 2+tan(1-x) where 0 ≤ x ≤ π/2. Given that x is increasing at a constant rate of 0.04 radians per second, find the corresponding rate of change of y when y = 3.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B2] dy/dx = −sec²(1−x) oe must be seen
2. [B1] for dy/dx = k sec²(1−x) oe, k≠ −1
3. [SC1] for dy/dx = -sec²(1-x) or dy/dx = -sec²(1-x) + c
4. [M1] Solves 3 = 2 + tan(1-x) as far as 1 - x = tan⁻¹1
5. [A1] 1-x = π/4 isw or 0.7853[98...] x = 1 - π/4 isw or 0.2146[01...]
6. [M1] Correct use of chain rule and correctly writes in terms of cosine or tangent: (their(-1) / cos²(their π/4)) × 0.04 oe, soi or (their(-1){1+tan²(their π/4)}) × 0.04 oe soi dep on at least B1 and an attempt to solve 3 = 2 + tan(1-x)
7. [A1] -0.08 oe, nfww dep on all previous marks awarded