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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh.12 - Parametric and Polar Curves문제 12.4.33
12장, 문제 12.4.33

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola with focus at (3, 0)

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Recall that a parabola with its vertex at the origin and focus on the x-axis has the standard form equation \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus.
Identify the value of \(p\) from the given focus. Since the focus is at \((3, 0)\), the distance \(p\) is 3.
Substitute \(p = 3\) into the standard form equation to get \(y^2 = 4 \times 3 \times x\).
Simplify the equation to \(y^2 = 12x\), which represents the parabola with vertex at the origin and focus at \((3, 0)\).
Verify the orientation: since the focus is on the positive x-axis, the parabola opens to the right, consistent with the equation \(y^2 = 4px\) where \(p > 0\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this geometric definition is essential to derive the equation of a parabola given its focus and vertex.
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가이드 코스
05:43
Definition of the Definite Integral

Standard Form of a Parabola with Vertex at the Origin

When the vertex is at the origin, a parabola with a horizontal axis of symmetry has the form x² = 4py (vertical) or y² = 4px (horizontal). The parameter p represents the distance from the vertex to the focus, which helps in writing the equation.
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가이드 코스
5:18
Circles in Standard Form

Relationship Between Focus and Parameter p

The distance p is the distance from the vertex to the focus along the axis of symmetry. For a focus at (3, 0), p = 3, indicating the parabola opens horizontally. This value is used directly in the standard form equation y² = 4px.
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05:23
Finding Area Between Curves on a Given Interval
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교과서 질문

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.


A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of 50 m, completing one lap in 24 seconds.

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교과서 질문

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 4 + sin θ; (4, 0) and (3, 3π/2)

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교과서 질문

9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

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교과서 질문

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 a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

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교과서 질문

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r cos θ = -4

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교과서 질문

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The left half of the parabola y=x ² +1, originating at (0, 1)

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