Textbook Question
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r =3 − 6 cos θ
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.R.29
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
Verified step by step guidance1
Understand the given polar equation: \(r = 3 \cos 3\theta\). This represents a polar curve where the radius \(r\) depends on the angle \(\theta\).
Recognize that the equation is of the form \(r = a \cos n\theta\), which typically produces a rose curve with petals. The number of petals depends on \(n\):
- If \(n\) is odd, the rose has \(n\) petals.
- If \(n\) is even, the rose has \$2n$ petals.
Since \(n = 3\) (an odd number), the curve will have 3 petals. The amplitude \(3\) controls the length of each petal.
To graph the curve, create a table of values by choosing several values of \(\theta\) between \(0\) and \(2\pi\), calculate the corresponding \(r\) values using \(r = 3 \cos 3\theta\), and plot the points in polar coordinates.
Connect the plotted points smoothly, noting the symmetry and petal structure, to complete the graph of the rose curve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates
Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for graphing equations like r = 3 cos 3θ.
Recommended video:
05:32
Intro to Polar Coordinates
Graphing Polar Equations
Graphing polar equations involves plotting points by calculating r for various values of θ and then converting these to Cartesian coordinates if needed. Recognizing patterns such as petals or loops helps visualize curves like r = 3 cos 3θ, which produces a rose curve.
Recommended video:
3:47Introduction to Common Polar Equations
Rose Curves
Rose curves are a family of polar graphs defined by equations of the form r = a cos(nθ) or r = a sin(nθ). The number of petals depends on n: if n is odd, the curve has n petals; if even, it has 2n petals. For r = 3 cos 3θ, the graph has 3 petals.
Recommended video:
3:37Roses
Related Practice
Textbook Question
44–49. Areas of regions Find the area of the following regions.
The region inside the limaçon r=2+cosθ and outside the circle r=2
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Textbook Question
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
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Textbook Question
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r = 1 −sin θ
Textbook Question
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)
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Textbook Question
Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?
b. r=(½)+sinθ
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