Problem 1 - Entrance Test

To find dy/dx for a polar curve, we use the formulas x = rcos(θ) and y = rsin(θ). Given r = 3sin(2θ): x = 3sin(2θ)cos(θ) y = 3sin(2θ)sin(θ) First, find dr/dθ: dr/dθ = d/dθ [3sin(2θ)] = 3 * 2cos(2θ) = 6cos(2θ). Next, find dx/dθ and dy/dθ: dx/dθ = (dr/dθ)cos(θ) - rsin(θ) dy/dθ = (dr/dθ)sin(θ) + rcos(θ) Evaluate r, dr/dθ, sin(θ), cos(θ) at θ = π/6: r = 3sin(2 * π/6) = 3sin(π/3) = 3(√3/2) = 3√3/2 dr/dθ = 6cos(2 * π/6) = 6cos(π/3) = 6(1/2) = 3 sin(π/6) = 1/2 cos(π/6) = √3/2 Now substitute these values into dx/dθ and dy/dθ: dx/dθ = (3)(√3/2) - (3√3/2)(1/2) = 3√3/2 - 3√3/4 = 6√3/4 - 3√3/4 = 3√3/4 dy/dθ = (3)(1/2) + (3√3/2)(√3/2) = 3/2 + 9/4 = 6/4 + 9/4 = 15/4 Finally, calculate dy/dx: dy/dx = (dy/dθ) / (dx/dθ) = (15/4) / (3√3/4) = 15 / (3√3) = 5/√3 = 5√3/3. The final answer is √(5√3)/3.