Final answer:
The slope of the tangent line for the curve r = cos(3θ) at θ = π/3 is 0. The length of the polar curve r = θ² from 0 to √5 can be found by integrating the length element ds = √(r² + (dr/dθ)²) dθ.
Step-by-step explanation:
The slope of the tangent line for the curve r = cos(3θ) at θ = π/3 can be found by taking the derivative of the curve with respect to θ, and then evaluating it at θ = π/3. The derivative of r = cos(3θ) is dr/dθ = -3sin(3θ). Evaluating this at θ = π/3, we get dr/dθ = -3sin(π) which simplifies to -3*0 = 0. Therefore, the slope of the tangent line is 0.
The length of the polar curve r = θ² from 0 to √5 can be found by integrating the length element ds which is given by ds = √(r² + (dr/dθ)²) dθ. Substituting r = θ² and dr/dθ = 2θ, the length element becomes ds = √(θ⁴ + 4θ²) dθ. Integrating this from θ = 0 to θ = √5, we get the length of the polar curve as L = ∫0√5 √(θ⁴ + 4θ²) dθ. Solving this integral gives the length of the polar curve.