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 = cos t, y = 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: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Chain Rule (parametric context)

The idea behind dy/dx = (dy/dt)/(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 = (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 ≠ 0, the normal slope is −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 dx/dt = 0 and dy/dt ≠ 0, making 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)

d^2y/dx^2 measures how dy/dx changes with x and is computed by d^2y/dx^2 = (d/dt(dy/dx)) / (dx/dt).

Concavity (parametric interpretation)

Determined by the sign of 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 = ∫[a,b] sqrt((dx/dt)^2 + (dy/dt)^2) dt.

Differential arc length (ds)

A small piece of distance along the curve: ds = sqrt((dx)^2 + (dy)^2).

Speed along a parametric curve (ds/dt)

The magnitude of velocity: ds/dt = sqrt((dx/dt)^2 + (dy/dt)^2).

Arc length vs straight-line distance (chord length)

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

Absolute value issue in arc length simplification

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