뒤로Study Notes: Tangent Lines and Enclosed Area for Parametric Curves
스터디 가이드 - 스마트 노트
자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.
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
Find for :
Set up the integral:
Determine and by solving :
or
Evaluate the definite integral from to :
Summary Table: Tangent Line Conditions
Condition | Equation | Interpretation |
|---|---|---|
Horizontal Tangent | Slope of tangent is zero | |
Vertical Tangent | Tangent is vertical (undefined slope) |
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.
