Textbook Question
Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.
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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 12.3.67
63–74. Arc length of polar curves Find the length of the following polar curves.
The complete cardioid r = 4 + 4 sin θ
Verified step by step guidance1
Recall the formula for the arc length \( L \) of a polar curve \( r = r(\theta) \) from \( \theta = a \) to \( \theta = b \):
\[
L = \int_{a}^{b} \sqrt{r(\theta)^2 + \left(\frac{d r}{d \theta}\right)^2} \, d\theta
\]
Identify the given polar curve: \( r = 4 + 4 \sin \theta \). We need to find \( \frac{d r}{d \theta} \), the derivative of \( r \) with respect to \( \theta \).
Compute the derivative:
\[
\frac{d r}{d \theta} = 4 \cos \theta
\]
Determine the interval for \( \theta \) that traces the complete cardioid. Since cardioids are typically traced once as \( \theta \) goes from \( 0 \) to \( 2\pi \), set the limits of integration as \( a = 0 \) and \( b = 2\pi \).
Set up the integral for the arc length:
\[
L = \int_{0}^{2\pi} \sqrt{(4 + 4 \sin \theta)^2 + (4 \cos \theta)^2} \, d\theta
\]
This integral can then be simplified and evaluated to find the length of the cardioid.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Polar Curves
Polar coordinates represent points in the plane using a radius and an angle (r, θ). Polar curves are defined by equations relating r and θ, such as r = 4 + 4 sin θ. Understanding how to interpret and plot these curves is essential for analyzing their properties, including arc length.
Recommended video:
05:32
Intro to Polar Coordinates
Arc Length Formula for Polar Curves
The arc length of a polar curve r(θ) from θ = a to θ = b is given by the integral ∫ from a to b of √(r(θ)² + (dr/dθ)²) dθ. This formula accounts for changes in both radius and angle, allowing calculation of the curve's length in polar form.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Polar Functions
To apply the arc length formula, you must differentiate r(θ) with respect to θ. This involves using standard differentiation rules on trigonometric functions like sin θ. Accurate computation of dr/dθ is crucial for evaluating the integral correctly.
Recommended video:
05:32Intro to Polar Coordinates
Related Practice
Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
2 ≤ r ≤ 8
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Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
x² = 12y
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Textbook Question
53–56. Simple curves Tabulate and plot enough points to sketch a graph of the following equations.
r = 1 - cos θ
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Textbook Question
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
An ellipse with vertices (±5, 0), passing through the point (4, 3/5)
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Textbook Question
37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.
A circle centered at the origin with radius 4, generated counterclockwise
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