Textbook Question
29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.
r = 2 cos θ and r = 1 + cos θ
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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 12.1.23
15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π
Verified step by step guidance1
Start with the given parametric equations: \(x = \cos t\) and \(y = 1 + \sin t\), where \(0 \leq t \leq 2\pi\).
To eliminate the parameter \(t\), recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\) from the parametric equations.
From \(x = \cos t\), we have \(\cos t = x\). From \(y = 1 + \sin t\), isolate \(\sin t\) as \(\sin t = y - 1\).
Substitute these into the Pythagorean identity: \((y - 1)^2 + x^2 = 1\). This is the Cartesian equation relating \(x\) and \(y\) without the parameter \(t\).
Recognize that this equation represents a circle centered at \((0,1)\) with radius \(1\). The parameter \(t\) increases from \(0\) to \(2\pi\), so the positive orientation corresponds to moving counterclockwise around the circle starting at the point \((1,1)\) when \(t=0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
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08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a single equation relating x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.
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Guided course
05:59Eliminating the Parameter
Curve Orientation and Description
The orientation of a parametric curve refers to the direction in which the curve is traced as the parameter increases. Describing the curve includes identifying its shape (e.g., circle, ellipse) and indicating the direction of traversal, which is important for understanding motion or integration along the curve.
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11:41Summary of Curve Sketching
Related Practice
Textbook Question
Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2.
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Textbook Question
Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by
x = 80t, y = −4.9t² + 3000, t ≥ 0
where the origin is the point on the ground directly beneath the plane at the moment of the release (see figure). Graph the trajectory of the packet and find the coordinates of the point where the packet lands.
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Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = sin θ sec² θ
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Textbook Question
29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.
r = 1 and r = 2 sin 2θ
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Textbook Question
Cartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r² = a² cos 2θ, where a is a real number.