Calculus/Parametric Introduction

Parametric and Polar Equations Parametric Differentiation Introduction Parametric equations are typically defined by two equations that specify both the coordinates of a graph using a parameter. They are graphed using the parameter (usually ) to figure out both the coordinates. Example 1 1.5em Note: This parametric equation is equivalent to the rectangular equation . Example 2 1.5em Note: This parametric equation is equivalent to the rectangular equation and the polar equation . Parametric equations can be plotted by using a -table to show values of for each value of . They can also be plotted by eliminating the parameter though this method removes the parameter's importance. Forms of Parametric Equations Parametric equations can be described in three ways: Parametric form Vector form An equality The first two forms are used far more often, as they allow us to find the value of the component at the given value of the parameter. The final form is used less often; it allows us to verify a solution to the equation, or find the parameter (or some constant multiple thereof). Parametric Form A parametric equation can be shown in parametric form by describing it with a system of equations. For instance: 1.5em Vector Form Vector form can be used to describe a parametric equation in a similar manner to parametric form. In this case, a position vector is given: 1.5em Equalities A parametric equation can also be described with a set of equalities. This is done by solving for the parameter, and equating the components. For example: 1.5em From here, we can solve for : 1.5em And hence equate the two right-hand sides: 1.5em Converting Parametric Equations There are a few common place methods used to change a parametric equation to rectangular form. The first involves solving for in one of the two equations and then replacing the new expression for with the variable found in the second equation. Example 1 1.5em becomes Example 2 1.5em Isolate the trigonometric functions 1.5em Use the identity 1.5em