Parametric/polar - Calculus BC AP Study Notes

Parametric equations describe a curve using a third variable (parameter t) where x = f(t) and y = g(t). Unlike standard y = f(x) form, parametric form captures motion and direction, essential for modeling paths through space.

Key Parametric Formulas:

• Derivative: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0
• Second derivative: d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)]/(dx/dt)
• Arc length: L = ∫[a to b] √[(dx/dt)² + (dy/dt)²] dt
• Surface area (revolution about x-axis): S = ∫[a to b] 2πy√[(dx/dt)² + (dy/dt)²] dt

Polar coordinates represent points as (r, θ) where r is the distance from origin and θ is the angle from the positive x-axis. Conversion formulas: x = r cos θ, y = r sin θ; conversely, r² = x² + y², tan θ = y/x.

Key Polar Formulas:

• Slope of tangent: dy/dx = (dr/dθ · sin θ + r cos θ)/(dr/dθ · cos θ - r sin θ)
• Area enclosed: A = ½∫[α to β] r² dθ
• Arc length: L = ∫[α to β] √[r² + (dr/dθ)²] dθ

Memory Aid (PARAMETRIC): Parameter Allows Richer Analysis: Motion, Experiments, Trajectories Reflected In Curves

Command words: Calculate (find numerical answer), Sketch (draw with key features labeled), Find (determine using appropriate method), Show that (prove with clear working).

Parametric curves model real motion brilliantly. Consider a projectile launched at angle α with initial velocity v₀: x(t) = (v₀ cos α)t and y(t) = (v₀ sin α)t - ½gt². The parameter t represents time, capturing both horizontal and vertical motion simultaneously—impossible with standard functions. At the trajectory's peak, dy/dt = 0, directly modeling when vertical velocity vanishes.

Cycloid motion (path traced by a point on a rolling wheel) uses x = r(t - sin t), y = r(1 - cos t). This appears in architectural arches and brachistochrone problems (fastest descent path). The parametric form elegantly handles the loop structure that would require multiple y-values for single x-values.

Polar coordinates excel in rotational symmetry. A lighthouse beam sweeps with r = 2 + cos 3θ creating a three-petaled rose. Planetary orbits follow r = a(1-e²)/(1 + e cos θ) (elliptical paths), where polar form naturally expresses distance from a focal point (the sun).

Naval navigation uses polar coordinates: a ship at bearing 045° and distance 50 km is immediately r = 50, θ = 45°. The Archimedean spiral r = aθ models vinyl record grooves and spider web patterns.

Analogy: Parametric equations are like GPS coordinates over time—tracking both longitude and latitude as time advances. Polar coordinates resemble radar displays—distance and angle from center.

Converting between forms reveals hidden properties: the parametric circle x = cos t, y = sin t becomes the identity x² + y² = 1, while the polar line r = 2 sec θ converts to the simple vertical line x = 2.