교과서 질문
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)
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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
12장, 문제 12.1.33
31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.
x=t,y= √(4−t²) a
검증된 단계별 안내1
Start with the given parametric equations: \(x = t\) and \(y = \sqrt{4 - t^2}\).
Since \(x = t\), you can express \(t\) in terms of \(x\) as \(t = x\).
Substitute \(t = x\) into the equation for \(y\): \(y = \sqrt{4 - x^2}\).
To eliminate the square root, square both sides of the equation: \(y^2 = 4 - x^2\).
Rearrange the equation to express it in terms of \(x\) and \(y\): \(x^2 + y^2 = 4\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually 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.
추천 영상:
가이드 코스
08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves rewriting the parametric equations to remove the parameter t, resulting in a single equation relating x and y directly. This often requires solving one equation for t and substituting into the other.
추천 영상:
가이드 코스
05:59Eliminating the Parameter
Using Algebraic Manipulation and Identities
To eliminate the parameter, algebraic techniques such as isolating variables, squaring both sides, or applying identities (like Pythagorean identities) are used. These steps help transform parametric forms into standard Cartesian equations.
추천 영상:
4:42Solve Trig Equations Using Identity Substitutions
관련 실천
교과서 질문
45–60. Areas of regions Find the area of the following regions.
The region inside one leaf of the rose r = cos 5θ
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교과서 질문
Golden Gate Bridge Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in a parabola (see figure). Assuming the origin is midway between the towers on the deck of the bridge, find an equation that describes the cables. How long is a guy wire that hangs vertically from the cables to the roadway 500 m from the center of the bridge?
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교과서 질문
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola symmetric about the y-axis that passes through the point (2, -6)
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교과서 질문
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(1, 2π/3)
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교과서 질문
45–60. Areas of regions Find the area of the following regions.
The region inside the limaçon r = 4 - 2 cos θ
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