Textbook Question
Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π
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 12.4.58
57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.
r = 3/(2 + cos θ)
Verified step by step guidance1
Identify the form of the polar equation given: \(r = \frac{3}{2 + \cos \theta}\). This resembles the general form for conic sections in polar coordinates: \(r = \frac{ed}{1 + e \cos \theta}\) or \(r = \frac{ed}{1 + e \sin \theta}\), where \(e\) is the eccentricity and \(d\) is the distance from the focus to the directrix.
Rewrite the denominator to match the standard form. Notice that the denominator is \(2 + \cos \theta\), which can be expressed as \(2 + 1 \cdot \cos \theta\). Compare this with \(1 + e \cos \theta\) to identify \(e\) and \(d\) by factoring appropriately.
Determine the eccentricity \(e\) and the parameter \(d\) by equating the given equation to the standard form. This involves expressing \(r = \frac{ed}{1 + e \cos \theta}\) and solving for \(e\) and \(d\) using the constants in the numerator and denominator.
Once \(e\) and \(d\) are found, classify the conic section based on the value of \(e\): if \(e < 1\), it is an ellipse; if \(e = 1\), a parabola; if \(e > 1\), a hyperbola. This classification helps in understanding the shape and properties of the graph.
Identify and label the key features: vertices (points where \(r\) is minimum or maximum), focus (at the pole), directrix (a line related to \(d\) and \(e\)), and asymptotes if the conic is a hyperbola. Use these to sketch the graph and verify with a graphing utility.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Equations of Conic Sections
Polar equations describe conic sections using the radius r and angle θ from the pole. The general form r = ed / (1 + e cos θ) or r = ed / (1 + e sin θ) defines ellipses, parabolas, and hyperbolas based on the eccentricity e. Understanding this form helps identify the type of conic and its properties.
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Parabolas as Conic Sections
Eccentricity and Its Role
Eccentricity (e) measures how much a conic deviates from being circular. If e < 1, the conic is an ellipse; if e = 1, a parabola; and if e > 1, a hyperbola. It determines the shape and position of the conic relative to its focus and directrix.
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07:39Classifying Differential Equations
Graphing and Identifying Key Features
Graphing polar conics involves plotting points for various θ values and labeling vertices, foci, and directrices. Vertices are points closest or farthest from the pole, foci are fixed points defining the conic, and directrices are lines related to eccentricity. Asymptotes appear only for hyperbolas.
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6:22Graphs of Secant and Cosecant Functions
Related Practice
Textbook Question
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 4 cos 2θ; at the tips of the leaves
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.
Textbook Question
57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.
r = 1/(2 - 2 sin θ)
Textbook Question
90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
Let L be the latus rectum of the parabola y ² =4px for p>0. Let F be the focus of the parabola, P be any point on the parabola to the left of L, and D be the (shortest) distance between P and L. Show that for all P, D+|FP|+ is a constant. Find the constant.
Textbook Question
90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.
The length of the latus rectum of the parabola y ² =4px or x ² =4py is 4|p|.