Parametric Equations and Tangent Lines

Understanding Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, typically denoted as t. This approach is especially useful for describing curves that are not functions in the traditional sense (i.e., those that fail the vertical line test).

• Parametric Form: , , where varies over an interval.

• Curve Representation: As changes, the point traces out a curve in the plane.

Tangent Lines to Parametric Curves

The tangent line to a parametric curve at a given value of is found by computing the derivatives of and with respect to $t$ and using them to find the slope of the tangent.

• Derivative Formulas:

  • Slope of Tangent: (provided )

• Equation of Tangent Line: At , the tangent line at is:

where

Vertical and Horizontal Tangents

• Horizontal Tangent: Occurs when and .

• Vertical Tangent: Occurs when and .

To find points where the tangent is horizontal or vertical, solve the respective derivative equations for and substitute back into the parametric equations to find the corresponding points.

Example 1: Tangent Lines for , ,

• Given: ,

• At :

• Derivatives:

  • (for )

• Tangent Line at :

  • Slope

  • Point:

  • Equation:

Finding Vertical and Horizontal Tangents

• Horizontal Tangent:

  • At : ,

• Vertical Tangent:

  • At : ,

• Note: Both tangents occur at the same point in this example.

Example 2: Area Enclosed by a Parametric Curve and the x-axis

• Given: ,

• Area Formula for Parametric Curves:

• To find the area, determine the interval where the curve is above the x-axis (i.e., ) and compute the definite integral.

Steps to Find the Area

1. Find for :

2. Set up the integral:

3. Determine and by solving :

   • or

4. Evaluate the definite integral from to :

Summary Table: Tangent Line Conditions

Additional info: For graphing, use a graphing calculator or Desmos to plot the parametric equations and visually confirm the points where the tangent is vertical or horizontal. Label these points and the tangent lines as required.