교과서 질문
81–88. Arc length Find the arc length of the following curves on the given interval.
x = sin t, y = t - cos t; 0 ≤ t ≤ π/2
85
views
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
TOC.tableOfContents
- 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 12.4.31
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola that opens to the right with directrix x = -4
검증된 단계별 안내1
Recall the definition of a parabola: it is the set of all points equidistant from the focus and the directrix.
Since the directrix is given as the vertical line \(x = -4\) and the parabola opens to the right, the axis of symmetry is horizontal along the x-axis.
The vertex is at the origin \((0,0)\), so the focus must be on the positive x-axis, at some point \((p,0)\), where \(p > 0\).
The distance from the vertex to the directrix is \(|p| = 4\), so the focus is at \((4,0)\).
Use the standard form of a parabola that opens right: \(y^2 = 4px\). Substitute \(p = 4\) to get the equation \(y^2 = 16x\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m주요 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. This geometric definition helps derive the equation of the parabola by relating distances from any point on the curve to the focus and directrix.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
Orientation of Parabolas
The orientation of a parabola depends on the position of its focus and directrix. If the parabola opens right or left, its axis of symmetry is horizontal, and the equation involves x and y accordingly. For a parabola opening right, the directrix is vertical, and the equation typically has the form (y - k)^2 = 4p(x - h).
추천 영상:
7:42Properties of Parabolas
Vertex at the Origin and Directrix
When the vertex is at the origin, the parabola's equation simplifies since h = 0 and k = 0. Given the directrix x = -4, the focus lies on the opposite side of the vertex at x = 4, allowing calculation of the parameter p, which determines the distance from the vertex to the focus or directrix and shapes the parabola's equation.
추천 영상:
가이드 코스
5:59Graph Hyperbolas NOT at the Origin
관련 실천
교과서 질문
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 = 8 + 2t, y = 1; −∞ < t < ∞
78
views
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.
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.
78
views
교과서 질문
Given three polar coordinate representations for the origin.
89
views
교과서 질문
Second derivative Assume a curve is given by the parametric equations x=f(t) and y=g(t), where f and g are twice differentiable. Use the Chain Rule to show that y″x=(fʹ(t)g″(t)−gʹ(t)f″(t))/(fʹ(t))³.
43
views
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 = 1 - sin θ; (1/2, π/6)
82
views