AP Calculus BC Unit 9 Study Guide: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Parametric equation

A way to describe a curve by writing both coordinates as functions of a parameter: x=f(t), y=g(t).

Parameter

An independent variable (often t, sometimes time) used to generate points (x(t), y(t)) on a curve.

Dependent variables (parametric)

In parametric form, x and y are dependent because they both depend on the parameter t.

Independent variable (parametric)

In parametric equations, the parameter t is the independent variable.

Direction of tracing

The direction a curve is drawn as the parameter increases; it is part of the parametric description.

Eliminating the parameter

Solving for t from one parametric equation and substituting into the other to get a Cartesian relationship between x and y.

Information lost when eliminating the parameter

Eliminating t may hide which portion is traced, the direction of motion, and whether the curve is traced multiple times.

Standard circle parametric form

x=a cos(t), y=a sin(t), which traces the circle x^2+y^2=a^2 (typically counterclockwise as t increases).

Frequency change (e.g., cos(2t), sin(2t))

Replacing t by kt in trig parametrics changes how fast the angle runs and can cause the curve to be traced multiple times.

Multiple tracing

When a parametric curve passes over the same points more than once for a given parameter interval (e.g., x=3cos(2t), y=3sin(2t) traces the circle twice for 0≤t≤2π).

Parametric slope (dy/dx)

For x=x(t), y=y(t), the slope is dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Horizontal tangent (parametric)

Occurs at t=a when dy/dt(a)=0 and dx/dt(a)≠0.

Vertical tangent (parametric)

Occurs at t=a when dx/dt(a)=0 and dy/dt(a)≠0 (dy/dx is undefined/infinite).

Cusp/corner behavior (parametric)

A potential sharp point or non-smooth behavior that can occur when dx/dt=0 and dy/dt=0 at the same parameter value (needs further analysis).

Tangent line (parametric)

At t=a, compute point (x(a),y(a)) and slope m=(y'(a))/(x'(a)), then use y−y(a)=m(x−x(a)).

Normal line

A line perpendicular to the tangent line; if tangent slope is m (nonzero), normal slope is −1/m.

Second derivative for parametric curves

d^2y/dx^2 = [d/dt(dy/dx)] / (dx/dt).

Concavity (parametric)

Determined by the sign of d^2y/dx^2; it measures how the slope dy/dx changes as x changes.

Signed area

Area computed by an integral that can be negative depending on orientation (e.g., if x decreases, dx/dt<0).

Area under a parametric curve

On t from a to b, A = ∫_a^b y(t)·(dx/dt) dt (this is ∫ y dx written in terms of t).

Geometric area vs. signed area

Signed area may be negative; if a problem asks for “area,” you usually report a positive value (may require absolute value or splitting intervals).

Arc length (parametric)

For t from a to b, L=∫_a^b √[(dx/dt)^2+(dy/dt)^2] dt.

Differential arc length element (ds)

For a small dt, ds ≈ √[(dx/dt)^2+(dy/dt)^2] dt.

Polar coordinates

A coordinate system using (r, θ), where r is the (directed) distance from the origin and θ is the angle from the positive x-axis.

Polar equation

An equation of a curve in polar form, typically r=f(θ).

Polar-to-Cartesian conversion

Convert (r,θ) to (x,y) using x=r cos(θ) and y=r sin(θ).

Cartesian-to-polar conversion

Relates r and (x,y) by r^2 = x^2 + y^2.

Equivalent polar representations

The same point can be written as (r,θ) or (r,θ+2π) (and other angle shifts by 2πk).

Negative r equivalence

(−r, θ) represents the same point as (r, θ+π); negative r “flips” the point across the origin.

Circle in polar form (r=a)

The equation r=a represents a circle of radius a centered at the origin.

Shifted circle form (r=2a cosθ)

A common polar circle form; for example r=4cosθ converts to (x−2)^2+y^2=4 (circle centered at (2,0) with radius 2).

Rose curve

A polar curve of the form r=a cos(nθ) or r=a sin(nθ), producing a petal-like “rose” shape.

Cardioid

A heart-shaped polar curve commonly of the form r=a±b cosθ or r=a±b sinθ (e.g., r=1+cosθ).

Limacon

A polar curve family often written r=a±b cosθ or r=a±b sinθ; depending on parameters it may have an inner loop or dimple.

Polar graphing interpretation of r

For each θ, r tells how far from the origin to plot the point (with negative r reversing direction by π).

Polar symmetry tests

Helpful checks: f(−θ)=f(θ) (symmetry about polar axis); f(π−θ)=f(θ) (symmetry about θ=π/2); f(θ+π)=−f(θ) (symmetry about the pole).

Slope of a polar curve (dy/dx)

Treat θ as the parameter: dy/dx=(dy/dθ)/(dx/dθ), where x=r cosθ and y=r sinθ.

dx/dθ for a polar curve

If r=f(θ), then dx/dθ = (dr/dθ)cosθ − r sinθ.

dy/dθ for a polar curve

If r=f(θ), then dy/dθ = (dr/dθ)sinθ + r cosθ.

Horizontal/vertical tangents in polar

Horizontal tangent when dy/dθ=0 and dx/dθ≠0; vertical tangent when dx/dθ=0 and dy/dθ≠0.

Area enclosed by a polar curve

From θ=α to β, A= (1/2)∫_α^β r^2 dθ.

Area between two polar curves

If outer radius is R(θ) and inner is r(θ), then A=(1/2)∫_α^β (R(θ)^2−r(θ)^2) dθ (outer minus inner).

Arc length in polar form

From θ=α to β, L=∫_α^β √[r^2+(dr/dθ)^2] dθ.

Vector-valued function

A function that outputs a vector; in the plane it often has the form r(t)=⟨x(t),y(t)⟩.

Position vector r(t)

The vector ⟨x(t),y(t)⟩ giving the location of a particle at time/parameter t.

Velocity vector v(t)

The derivative of position: v(t)=r′(t)=⟨x′(t),y′(t)⟩.

Speed

The magnitude of velocity (a nonnegative scalar): |v(t)|=√[(x′(t))^2+(y′(t))^2].

Acceleration vector a(t)

The derivative of velocity: a(t)=v′(t)=r″(t).

Displacement

The net change in position from t=a to t=b: r(b)−r(a), equivalently ∫_a^b v(t) dt.

Total distance traveled

The total length of the path from t=a to t=b: ∫_a^b |v(t)| dt (not the same as displacement).