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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 12.1.33
Capitolo 12, Problema 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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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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Eliminating the Parameter

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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Solve Trig Equations Using Identity Substitutions
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57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.


r = 3/(2 + cos θ)

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


The region inside one leaf of the rose r = cos 5θ

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Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge? 

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31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola symmetric about the y-axis that passes through the point (2, -6)

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25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(1, 2π/3)

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


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

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