AP Calculus BC || Parametric equations, polar coordinates, and vector-valued functions

Differentiating parametric equations

dy/dx = dy/dt / dx/dt

Second derivatives of parametric equations

d²y/dx² = (d/dt[dy/dx])/(dx/dt)

Parametric curve arc length on closed interval [a,b]

L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²)

Differentiating vector-valued functions

f'⟨x(t), y(t)⟩ = ⟨dx/dt, dx/dt⟩

Second derivative of vector-valued functions

f''⟨x(t), y(t)) = ⟨d²x/dt², d²y/dt²)

Magnitude of the velocity vector (given position)

||(v(t))|| = √((dx/dt)² + (dy/dt)²)

Magnitude of the displacement of a particle on closed interval [a,b] (given velocity)

||(f(t))|| = √((∫ₐᵇ x(t) dt) + (∫ₐᵇ y(t) dt))

x-coordinate of polar coordinates

x = r(θ) * cos(θ)

y-coordinate of polar coordinates

y = r(θ) * sin(θ)

Derivative of the x-coordinate (polar coordinates)

dx/dθ = dr/dθ cos(θ) + r(θ) -sin(θ)

Derivative of the y-coordinate (polar coordinates)

dy/dθ = dr/dθ sin(θ) + r(θ) cos(θ)

Derivative of the polar curve

dr/dθ = (dy/dθ)/(dx/dθ)

Area bounded by a polar curve

1/2 * ∫ₐᵇ [r(θ)]² dθ

Area between two polar curves

1/2 * ∫ₐᵇ ([R(θ)]² - [r(θ)]²) dθ
where R(θ) is the higher curve, and r(θ) is the lower curve.