Problem 15 - Entrance Test

x = a(θ - sin θ), y = a(1 + cos θ). dx/dθ = a(1 - cos θ), dy/dθ = -a sin θ. dy/dx = (dy/dθ) / (dx/dθ) = (-a sin θ) / (a(1 - cos θ)) = -sin θ / (1 - cos θ). d²y/dx² = d/dx (dy/dx) = (d/dθ (dy/dx)) / (dx/dθ) = d/dθ (-sin θ / (1 - cos θ)) / (a(1 - cos θ)). d/dθ (-sin θ / (1 - cos θ)) = -[(cos θ(1 - cos θ) - sin θ(sin θ)) / (1 - cos θ)²] = -[(cos θ - cos²θ - sin²θ) / (1 - cos θ)²] = -[(cos θ - 1) / (1 - cos θ)²] = 1 / (1 - cos θ). d²y/dx² = [1 / (1 - cos θ)] / [a(1 - cos θ)] = 1 / [a(1 - cos θ)²]. At θ = π/2, d²y/dx² = 1 / [a(1 - cos(π/2))²] = 1 / [a(1 - 0)²] = 1 / a.