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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.23
Chapter 12, Problem 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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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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Related Practice
Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

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Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = sin t, y = t - cos t; 0 ≤ t ≤ π/2

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Textbook Question

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

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Textbook Question

Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

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Textbook Question

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

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Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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