A parameter is a variable that is used to describe a set of equations, often defining a curve or surface. It allows for the expression of coordinates as functions of one or more independent variables.
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In parametric equations, each coordinate (x and y) is expressed as a function of an independent parameter, commonly denoted as $t$.
Parametric equations can describe curves that are not functions in the Cartesian coordinate system, such as circles and ellipses.
To eliminate the parameter from parametric equations, solve one equation for the parameter and substitute it into the other equation.
The derivative $\frac{dy}{dx}$ in parametric form is found using $\frac{dy/dt}{dx/dt}$ where $x = f(t)$ and $y = g(t)$.
Arc length of a curve defined by parametric equations $x=f(t)$ and $y=g(t)$ from $t=a$ to $t=b$ is given by $$\int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt.$$
Review Questions
How do you express coordinates in terms of a parameter?
What is the process to eliminate the parameter from parametric equations?
How do you find the derivative $\frac{dy}{dx}$ for parametric equations?
Equations where the coordinates are expressed as functions of one or more parameters.
$\frac{dy}{dx}$: The derivative representing the rate of change of y with respect to x; for parametric equations, it is found using $\frac{dy/dt}{dx/dt}$.