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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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.60a
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
Verified step by step guidance1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos\theta\), \(y = r \sin\theta\), and \(r^2 = x^2 + y^2\).
Given the equation \(r^2 = \sec^2\theta\), rewrite \(\sec^2\theta\) as \(\frac{1}{\cos^2\theta}\), so the equation becomes \(r^2 = \frac{1}{\cos^2\theta}\).
Multiply both sides by \(\cos^2\theta\) to get \(r^2 \cos^2\theta = 1\).
Substitute \(r \cos\theta\) with \(x\) (since \(x = r \cos\theta\)), so the equation becomes \(x^2 = 1\).
Interpret the Cartesian equation \(x^2 = 1\) as two vertical lines \(x = 1\) and \(x = -1\), which are the Cartesian form of the given polar equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar to Cartesian Coordinate Conversion
This involves expressing polar coordinates (r, θ) in terms of Cartesian coordinates (x, y) using the relationships x = r cos θ and y = r sin θ. Converting polar equations to Cartesian form helps in identifying and analyzing curves using familiar algebraic methods.
Recommended video:
05:32
Intro to Polar Coordinates
Trigonometric Identities
Trigonometric identities, such as sec²θ = 1/cos²θ and the Pythagorean identity sin²θ + cos²θ = 1, are essential for manipulating and simplifying equations involving trigonometric functions. These identities enable rewriting the given polar equation into a more workable form.
Recommended video:
7:17Verifying Trig Equations as Identities
Identification of Conic Sections
Conic sections include ellipses, parabolas, and hyperbolas, each with characteristic Cartesian equations. After converting the polar equation, recognizing the standard form of these curves allows one to classify the curve correctly based on its algebraic equation.
Recommended video:
5:33Parabolas as Conic Sections
Related Practice
Textbook Question
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)
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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
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
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
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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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