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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 12.2.37
Capitolo 12, Problema 12.2.37

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r cos θ = -4

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Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\).
Given the equation \(r \cos \theta = -4\), substitute \(x\) for \(r \cos \theta\) to rewrite the equation in Cartesian form.
After substitution, the equation becomes \(x = -4\).
Recognize that \(x = -4\) represents a vertical line in the Cartesian coordinate plane where all points have an \(x\)-coordinate of \(-4\).
Therefore, the curve described by the polar equation \(r \cos \theta = -4\) is a vertical line located 4 units to the left of the \(y\)-axis.

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Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations between them.
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Intro to Polar Coordinates

Conversion Formulas Between Polar and Cartesian Coordinates

The key formulas are x = r cos θ and y = r sin θ. These allow conversion from polar to Cartesian form by expressing r and θ in terms of x and y, enabling the rewriting of polar equations into Cartesian equations.
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Intro to Polar Coordinates

Interpreting Cartesian Equations to Identify Curves

Once converted, the Cartesian equation can be analyzed to identify the type of curve it represents, such as lines, circles, or parabolas. Recognizing standard forms helps describe the geometric nature of the curve.
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Introduction to Parametric Equations
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11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 4 + sin θ; (4, 0) and (3, 3π/2)

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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 with focus at (3, 0)

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9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

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90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

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


A hyperbola with vertices (±4, 0) and foci (±6, 0)

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63–74. Arc length of polar curves Find the length of the following polar curves.


{Use of Tech} The complete limaçon r=4−2cosθ

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