Parametric & Polar: every key term you need

Parametric Equations

Curve defined by x=f(t), y=g(t) where t is parameter (often time). Single parameter traces 2D curve.

Parametric Derivatives

dy/dx = (dy/dt)/(dx/dt). Second derivative d²y/dx² = d/dt(dy/dx) / (dx/dt). Used for finding slopes, tangent lines.

Speed & Velocity (Parametric)

Velocity vector = (dx/dt, dy/dt). Speed = √[(dx/dt)² + (dy/dt)²]. Arc length = ∫√[(dx/dt)² + (dy/dt)²] dt.

Concavity (Parametric)

Curve concave up if d²y/dx² > 0. Requires careful handling since x increases nonlinearly.

Eliminating Parameter

Solve one equation for t, substitute into other to get y=f(x). Useful for identifying curve type (circle, ellipse, etc.).

Polar Coordinates

Point location: (r, θ) where r = distance from origin, θ = angle from positive x-axis. Convert: x=r cos θ, y=r sin θ.

Polar to Rectangular

x=r cos θ, y=r sin θ, r²=x²+y², tan θ=y/x. Convert when needed for integration or analysis.

Rectangular to Polar

r=√(x²+y²), θ=arctan(y/x). Important for curves like spirals, roses, lemniscates.

Polar Derivatives

dy/dx = (r' sin θ + r cos θ)/(r' cos θ - r sin θ) where r' = dr/dθ. Used for tangent line slopes.

Polar Curve Types

r=a (circle), r=a cos θ (circle), r=a sin nθ (rose), r=ae^(bθ) (spiral), r²=a² cos 2θ (lemniscate).

Polar Area

Area = ½∫r² dθ from θ₁ to θ₂. Accounts for sector area swept by radius vector.

Unit 9 Summary

Parametric equations describe motion; dy/dx = (dy/dt)/(dx/dt). Polar coordinates useful for circular/radial motion; area = ½∫r² dθ.

More AP Calculus BC guides

• Limits & Continuity
• Applications of Derivatives
• Infinite Series
• Differential Equations