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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.1.23
Capitolo 12, Problema 12.1.23

15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.


x = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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Start with the given parametric equations: \(x = \cos t\) and \(y = 1 + \sin t\), where \(0 \leq t \leq 2\pi\).
To eliminate the parameter \(t\), recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\) from the parametric equations.
From \(x = \cos t\), we have \(\cos t = x\). From \(y = 1 + \sin t\), isolate \(\sin t\) as \(\sin t = y - 1\).
Substitute these into the Pythagorean identity: \((y - 1)^2 + x^2 = 1\). This is the Cartesian equation relating \(x\) and \(y\) without the parameter \(t\).
Recognize that this equation represents a circle centered at \((0,1)\) with radius \(1\). The parameter \(t\) increases from \(0\) to \(2\pi\), so the positive orientation corresponds to moving counterclockwise around the circle starting at the point \((1,1)\) when \(t=0\).

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as 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

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a single equation relating x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.
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Eliminating the Parameter

Curve Orientation and Description

The orientation of a parametric curve refers to the direction in which the curve is traced as the parameter increases. Describing the curve includes identifying its shape (e.g., circle, ellipse) and indicating the direction of traversal, which is important for understanding motion or integration along the curve.
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Summary of Curve Sketching
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