AP Calculus BC glossary

Parametric equations

Also called: Parameter

Parametric equations define the coordinates of a point separately as functions of a third variable called the parameter, usually time. They describe not just the shape of a curve but the direction and speed with which it is traced.

x = x(t), \quad y = y(t)

The parameter buys you curves that are not functions of $x$ , such as circles, and it records the traversal. A curve traced clockwise and the same curve traced counterclockwise are different parametrizations.

The slope comes from the chain rule as $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ , valid wherever $\frac{dx}{dt}$ is not zero. The second derivative differentiates that quotient with respect to $t$ , then divides by $\frac{dx}{dt}$ again.

The mistake

Computing the second derivative as $\frac{d^2y/dt^2}{d^2x/dt^2}$ . That is not the chain rule and gives the wrong answer.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)