Unit 9 · Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Parametric equations ○ — A way of describing a curve with two functions of a parameter: x = x(t) and y = y(t). As the parameter t varies, the pair (x(t), y(t)) traces out the curve.

Derivative of a parametric function ● (core concept) — For a curve given parametrically, dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. Its value at a point is the slope of the line tangent to the curve there.

Second derivative of a parametric function ● (core concept) — d²y/dx² = [d/dt(dy/dx)] / (dx/dt): differentiate the first parametric derivative with respect to t, then divide by dx/dt.

Arc length of a parametric curve ● (core concept) — For x = x(t), y = y(t) from t₁ to t₂, the curve's length is the definite integral of √((dx/dt)² + (dy/dt)²) dt.

Vector-valued function ○ — A function r(t) = ⟨x(t), y(t)⟩ whose output is a vector; each component is an ordinary function of the parameter t, and it traces a curve in the plane.

Derivative of a vector-valued function ● (core concept) — Differentiate component by component: r′(t) = ⟨x′(t), y′(t)⟩. The same derivative rules used for real-valued functions apply to each component.

Integral of a vector-valued function ● (core concept) — Integrate component by component, the same way real-valued functions are integrated. With a rate vector and initial conditions, this produces the particular position function of a particle.

Position from a velocity vector ● (core concept) — Given a particle's velocity vector and its initial position, its position function is the antiderivative of the velocity vector that satisfies the initial conditions — an initial value problem in the plane.

Velocity vector ● (core concept) — For a particle in planar motion, v(t) = ⟨x′(t), y′(t)⟩ — the derivative of the position vector. It gives both the rate of motion and its direction.

Speed of a particle in planar motion ● (core concept) — The magnitude of the velocity vector: |v(t)| = √((x′(t))² + (y′(t))²). Speed is a nonnegative scalar, unlike velocity.

Acceleration in planar motion ● (core concept) — a(t) = v′(t) = ⟨x″(t), y″(t)⟩ — the rate of change of the velocity vector of a particle moving in the plane.

Displacement vs. distance traveled ● (core concept) — Over a time interval, displacement (net change in position) is the definite integral of the velocity vector, while total distance traveled is the definite integral of speed.

Polar coordinates ○ — A way of locating a point by (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Conversions: x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x.

Derivative in polar form ● (core concept) — For a curve given by r = f(θ), dy/dx = (dy/dθ)/(dx/dθ). Derivatives of r, x, and y with respect to θ — and first and second derivatives of y with respect to x — give information about the curve's behavior.

Area of a polar region ● (core concept) — For r = f(θ) swept from θ = α to θ = β, the area is ½ times the definite integral of r² dθ — the polar analogue of accumulating area.

Area between two polar curves ● (core concept) — A = ½∫(r_outer² − r_inner²) dθ, integrated over the θ-interval where one curve lies outside the other. The curves' intersection points determine the limits of integration.