AP Calculus BC Unit 9 Notes: Learning Parametric Curves from First Principles

Parametric curve

A curve described by giving both coordinates as functions of a parameter (usually t): $x = x(t), y = y(t)$.

Parameter (t)

A third variable that determines where you are on a parametric curve and can represent time, direction of travel, and speed along the curve.

Parameterization

A specific choice of functions x(t) and y(t) that traces a curve as t varies; different parameterizations can trace the same geometric curve with different directions or speeds.

Vertical line test (in relation to parametrics)

A test for whether y is a function of x; parametric equations are useful for curves that fail this test (loops, sideways curves).

Orientation (direction) of a parametric curve

The direction the curve is traced as t increases; the same curve can be traced forward or backward depending on the parameterization.

Eliminating the parameter

Algebraically removing t to get a relation between x and y (often to identify the curve’s shape), which typically loses direction and timing information.

Unit circle parameterization

The parametric form $x = \text{cos}(t), y = \text{sin}(t)$, which eliminates to $x^2 + y^2 = 1$.

Information lost when eliminating t

Which point corresponds to a given t, the direction of travel, and how fast the point moves along the curve.

Parametric derivative (first derivative)

The slope along a parametric curve: $\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}$, provided $\frac{dx}{dt} \neq 0$.

Chain Rule (parametric context)

The idea behind $\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}$: slope is a ratio of rates of change with respect to t.

Condition for dy/dx to exist (parametric)

dx/dt must be nonzero; otherwise dy/dx is undefined at that parameter value.

Tangent line to a parametric curve

A line touching the curve at t = a with slope $m = \frac{dy}{dx}|_{t=a}$ through the point $(x(a), y(a))$.

Point-slope form (for parametric tangent lines)

The tangent line equation at t = a: y − y(a) = m(x − x(a)), where m = (dy/dx)|_{t=a}.

Normal line

A line perpendicular to the tangent line; if the tangent slope is m $\neq 0$, the normal slope is $-\frac{1}{m}$.

Horizontal tangent (parametric test)

Occurs when dy/dt = 0 and dx/dt ≠ 0, making dy/dx = 0.

Vertical tangent (parametric test)

Occurs when $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$, making $\frac{dy}{dx}$ undefined.

Cusp / corner / stopping point warning

If dx/dt = 0 and dy/dt = 0 at the same t, you cannot conclude a vertical tangent; the point may be a cusp, corner, or momentary stop.

Second derivative (parametric form)

$\frac{d^2y}{dx^2}$ measures how $\frac{dy}{dx}$ changes with x and is computed by $\frac{d}{dt} \bigg( \frac{dy}{dx} \bigg) / \frac{dx}{dt}$.

Concavity (parametric interpretation)

Determined by the sign of $\frac{d^2y}{dx^2}$: positive means concave up, negative means concave down (locally, as x changes).

Where d^2y/dx^2 may be undefined

At parameter values where dx/dt = 0, because the formula for d^2y/dx^2 requires dividing by dx/dt.

Arc length (parametric curve)

The distance traveled along the curve from t = a to t = b: L = \bigg( \bigint_{a}^{b} \text{sqrt}\big((\frac{dx}{dt})^2 + (\frac{dy}{dt})^2\big) dt \bigg).

Differential arc length (ds)

A small piece of distance along the curve: $ds = \text{sqrt}\big((dx)^2 + (dy)^2\big)$.

Speed along a parametric curve (ds/dt)

The magnitude of velocity: $\frac{ds}{dt} = \text{sqrt}\big( \big( \frac{dx}{dt} \big)^2 + \big( \frac{dy}{dt} \big)^2 \big)$.

Arc length vs straight-line distance (chord length)

Arc length is total distance along the curve (integral of speed), not the endpoint distance $\text{sqrt}\big((x(b)−x(a))^2 + (y(b)−y(a))^2\big)$.

Absolute value issue in arc length simplification

When simplifying $\text{sqrt}(t^2)$, you must use $|t|$; dropping the absolute value can be wrong on intervals that include negative t.