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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.1.37
Capitolo 12, Problema 12.1.37

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


A circle centered at the origin with radius 4, generated counterclockwise

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Recall the standard parametric equations for a circle centered at the origin with radius \( r \): \( x = r \cos(t) \) and \( y = r \sin(t) \).
Since the radius is 4, substitute \( r = 4 \) into the equations to get \( x = 4 \cos(t) \) and \( y = 4 \sin(t) \).
The parameter \( t \) represents the angle in radians measured from the positive x-axis, and it controls the position on the circle.
To generate the circle counterclockwise starting from the point \( (4,0) \), let \( t \) vary over the interval \( [0, 2\pi] \).
Thus, the parametric equations are \( x = 4 \cos(t) \), \( y = 4 \sin(t) \), with \( t \in [0, 2\pi] \).

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
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Parameterizing Equations

Equation of a Circle

A circle centered at the origin with radius r satisfies x² + y² = r². To represent this circle parametrically, trigonometric functions sine and cosine are used, since they naturally describe circular motion.
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Parameterizing Equations of Circles & Ellipses

Parameter Interval and Direction

The parameter interval defines the portion of the curve traced by the parametric equations. For a full circle traced counterclockwise, t typically ranges from 0 to 2π, with x = r cos t and y = r sin t ensuring the correct orientation.
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Eliminating the Parameter
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