Parametric Equations and Tangent Lines

Understanding Parametric Equations

Parametric equations express the coordinates of the points that make up a geometric object as functions of a variable, usually denoted as t (the parameter). This approach is especially useful for describing curves that are not functions in the traditional sense (i.e., they may fail the vertical line test).

• General Form:

• Parameter: The variable t traces the curve as it varies over an interval.

Tangent Lines to Parametric Curves

The tangent line to a parametric curve at a given value of t is the line that just touches the curve at the corresponding point and has the same direction as the curve at that point.

• Point of Tangency: for a specific .

• Slope of Tangent Line: The slope is given by , which for parametric equations is:

• Equation of Tangent Line: Using point-slope form:

, where at

• Vertical Tangent: Occurs when and .

• Horizontal Tangent: Occurs when and .

Example:

Given , , find the tangent line at .

• Compute

• Compute

• Compute derivatives: ,

• At : ,

• Slope:

• Equation:

Finding Vertical and Horizontal Tangents

• Vertical Tangent: Set and solve for .

• Horizontal Tangent: Set and solve for .

• Substitute these values into and to find the corresponding points.

Example:

• For , (vertical tangent).

• For , (horizontal tangents).

Area Enclosed by a Parametric Curve

Formula for Area

The area enclosed by a parametric curve , , as goes from to , is given by:

• This formula computes the signed area between the curve and the x-axis.

• If the curve is traced more than once as varies, the area may need to be adjusted accordingly.

Example:

Given , , to find the area enclosed by the curve and the x-axis, set up:

where and are the values of where the curve intersects the x-axis ().

Steps to Find the Area

1. Find the values of where (the curve crosses the x-axis).

2. Set up the integral .

3. Evaluate the integral to find the area.

Table: Tangent Line Conditions for Parametric Curves

Additional info: These notes expand on the homework questions by providing general methods, formulas, and examples for working with parametric equations, tangent lines, and area calculations, as relevant to Calculus II topics.