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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 12.1.33
Chapter 12, Problem 12.1.33

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=t,y= √(4−t²) a

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Start with the given parametric equations: \(x = t\) and \(y = \sqrt{4 - t^2}\).
Since \(x = t\), you can express \(t\) in terms of \(x\) as \(t = x\).
Substitute \(t = x\) into the equation for \(y\): \(y = \sqrt{4 - x^2}\).
To eliminate the square root, square both sides of the equation: \(y^2 = 4 - x^2\).
Rearrange the equation to express it in terms of \(x\) and \(y\): \(x^2 + y^2 = 4\).

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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, usually 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.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to remove the parameter t, resulting in a single equation relating x and y directly. This often requires solving one equation for t and substituting into the other.
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Using Algebraic Manipulation and Identities

To eliminate the parameter, algebraic techniques such as isolating variables, squaring both sides, or applying identities (like Pythagorean identities) are used. These steps help transform parametric forms into standard Cartesian equations.
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Related Practice
Textbook Question

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


The lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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Textbook Question

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=t ²−1, y=t ³ +t; t=2

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Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 2 cos t, y = 8 sin t; slope = -1

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Textbook Question

45–60. Areas of regions Find the area of the following regions.


The region inside the limaçon r = 4 - 2 cos θ

Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 2θ; (π/2, π/4)

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Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = 2 - 2 sin θ  b