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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 12.2.33
Capitolo 12, Problema 12.2.33

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


(1, √3)

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1
Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\): \(r = \sqrt{x^2 + y^2}\) \(\theta = \arctan\left(\frac{y}{x}\right)\)
Calculate the radius \(r\) by substituting \(x = 1\) and \(y = \sqrt{3}\) into the formula: \(r = \sqrt{1^2 + (\sqrt{3})^2}\)
Calculate the angle \(\theta\) using the arctangent formula: \(\theta = \arctan\left(\frac{\sqrt{3}}{1}\right)\)
Express the polar coordinates as \((r, \theta)\) using the values found in steps 2 and 3.
For the second way, express the angle \(\theta\) in degrees instead of radians or use a known angle value (e.g., \(\theta = \frac{\pi}{3}\) radians or \(60^\circ\)) to write the polar coordinates as \((r, \theta)\).

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Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates describe points by their distance from the origin (r) and the angle (θ) from the positive x-axis. Understanding both systems is essential for converting between them.
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Intro to Polar Coordinates

Conversion Formulas Between Cartesian and Polar Coordinates

To convert from Cartesian (x, y) to polar (r, θ), use r = √(x² + y²) to find the radius and θ = arctan(y/x) to find the angle. Adjust θ based on the quadrant to get the correct direction.
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Intro to Polar Coordinates

Multiple Representations of Polar Coordinates

Polar coordinates are not unique; the same point can be represented by adding multiples of 2π to θ or by using negative r values with adjusted angles. Recognizing these alternatives allows expressing coordinates in at least two different valid ways.
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Intro to Polar Coordinates
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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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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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

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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75–76. Graphs to polar equations Find a polar equation for each conic section. Assume one focus is at the origin.

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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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