Parametric/polar

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.

Example 1: Given parametric equations x = t² - 2t, y = t³ - 3t, find dy/dx when t = 2.

Solution: First find derivatives with respect to t:

• dx/dt = 2t - 2
• dy/dt = 3t² - 3

Apply formula: dy/dx = (dy/dt)/(dx/dt) = (3t² - 3)/(2t - 2)

At t = 2: dy/dx = (3(4) - 3)/(2(2) - 2) = 9/2

Examiner note: Always state both derivatives before dividing. Check dx/dt ≠ 0.

Example 2: Find the area enclosed by one petal of r = 3 sin 2θ.

Solution: First identify petal boundaries. One petal exists where r ≥ 0 and 2θ goes from 0 to π, so θ ∈ [0, π/2].

Apply area formula: A = ½∫[0 to π/2] (3 sin 2θ)² dθ = ½∫[0 to π/2] 9 sin² 2θ dθ = (9/2)∫[0 to π/2] ½(1 - cos 4θ) dθ [using power reduction] = (9/4)[θ - (sin 4θ)/4]₀^(π/2) = (9/4)[(π/2 - 0) - (0 - 0)] = 9π/8

Examiner note: Sketch the curve first to identify integration bounds. The identity sin² u = (1 - cos 2u)/2 is essential.

Example 3: Find arc length of x = e^t cos t, y = e^t sin t from t = 0 to t = π.

Solution: dx/dt = e^t cos t - e^t sin t = e^t(cos t - sin t) dy/dt = e^t sin t + e^t cos t = e^t(sin t + cos t)

(dx/dt)² + (dy/dt)² = e^(2t)[(cos t - sin t)² + (sin t + cos t)²] = e^(2t)[2cos²t + 2sin²t] = 2e^(2t)

L = ∫[0 to π] √(2e^(2t)) dt = √2 ∫[0 to π] e^t dt = √2[e^t]₀^π = √2(e^π - 1)