Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.26

Chapter 12, Problem 12.R.26

24–26. Sets in polar coordinates Sketch the following sets of points.

4 ≤ r² ≤ 9

Verified step by step guidance

Understand that the inequality \(4 \leq r^{2} \leq 9\) describes a set of points in polar coordinates where \(r\) is the distance from the origin to the point.

Rewrite the inequality in terms of \(r\) by taking the square root of each part, remembering to consider both positive and negative roots: \(2 \leq r \leq 3\) (since \(r\) represents a radius, we consider only the non-negative values).

Interpret this as all points whose distance from the origin is at least 2 units and at most 3 units, which geometrically represents the region between two circles centered at the origin with radii 2 and 3.

To sketch the set, draw two concentric circles: one with radius 2 and another with radius 3, both centered at the origin.

Shade the annular region (ring-shaped area) between these two circles, as this represents all points satisfying \(4 \leq r^{2} \leq 9\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates System

Polar coordinates represent points in a plane using a radius and an angle, denoted as (r, θ). The radius r measures the distance from the origin, and θ is the angle from the positive x-axis. This system is useful for describing curves and regions that are circular or radial in nature.

Inequalities Involving r²

An inequality like 4 ≤ r² ≤ 9 restricts the radius r to values whose squares lie between 4 and 9. Since r² is always non-negative, this means r lies between 2 and 3, including both boundaries. This defines an annular region (ring-shaped) between two circles centered at the origin.

Graphing Regions in Polar Coordinates

To sketch sets defined by inequalities in polar coordinates, interpret the conditions on r and θ to identify the region. For 4 ≤ r² ≤ 9, the region is all points whose distance from the origin is between 2 and 3, forming a ring. The angle θ typically ranges over all real numbers unless otherwise restricted.

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Intro to Polar Coordinates

Related Practice

Textbook Question

44–49. Areas of regions Find the area of the following regions.

The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

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Textbook Question

7–8. Parametric curves and tangent lines

a. Eliminate the parameter to obtain an equation in x and y.

x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6

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Textbook Question

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)

Textbook Question

22–23. Arc length Find the length of the following curves.

x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A set of parametric equations for a given curve is always unique.

Textbook Question

42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

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