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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 12.2.58
Capitolo 12, Problema 12.2.58

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

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Identify the given polar equation: \(r = 2 - 2 \sin \theta\). This represents a polar curve where \(r\) depends on the angle \(\theta\).
Recall that \(r\) is the distance from the origin to a point on the curve, and \(\theta\) is the angle measured from the positive x-axis. To graph, you will plot points for various values of \(\theta\) between \(0\) and \(2\pi\).
Create a table of values by choosing several \(\theta\) values (for example, \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\pi\), etc.) and compute the corresponding \(r\) values using the equation \(r = 2 - 2 \sin \theta\).
Plot each point in polar coordinates by moving \(r\) units from the origin at the angle \(\theta\). Connect these points smoothly to reveal the shape of the curve.
Use a graphing utility to input the equation \(r = 2 - 2 \sin \theta\) and compare the plotted points and shape with your manual graph to verify accuracy.

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Polar Coordinates and Polar Equations

Polar coordinates represent points in the plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Understanding how to interpret and plot equations given in polar form, like r = 2 - 2 sin θ, is essential for graphing polar curves.
Video consigliato:
05:32
Intro to Polar Coordinates

Graphing Polar Curves

Graphing polar curves involves plotting points for various values of θ and connecting them smoothly. Recognizing common shapes such as cardioids, limacons, and circles helps in sketching the curve accurately before verifying with a graphing utility.
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Percorso guidato
09:04
Slope of Polar Curves

Use of Graphing Utilities

Graphing utilities, such as graphing calculators or software, allow for precise visualization of polar curves. They help confirm the shape and key features of the graph, ensuring accuracy and aiding in understanding complex polar equations.
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Percorso guidato
06:15
Graphing The Derivative
Pratica correlata
Domanda del libro di testo

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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Domanda del libro di testo

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


The region common to the circles r = 2 sin θ and r = 1

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Domanda del libro di testo

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(1, √3)

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Domanda del libro di testo

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 upper half of the parabola x=y ², originating at (0, 0)

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Domanda del libro di testo

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

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Domanda del libro di testo

Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.

b. At what point does the tangency occur?

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