Domanda del libro di testo
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
75
views
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 12.4.21
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
4x² - y² = 16
Guida verificata passo dopo passo1
Rewrite the given equation 4x² - y² = 16 in a standard form by dividing both sides by 16 to isolate the terms: \(\frac{4x^{2}}{16} - \frac{y^{2}}{16} = 1\).
Simplify the fractions to get \(\frac{x^{2}}{4} - \frac{y^{2}}{16} = 1\), which matches the standard form of a hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) where \(a^{2} = 4\) and \(b^{2} = 16\).
Identify the center of the hyperbola at the origin (0,0), and note that since the \(x^{2}\) term is positive, the hyperbola opens left and right along the x-axis.
Find the vertices by using \(a = \sqrt{4} = 2\), so the vertices are at \((\pm 2, 0)\).
Calculate the foci using \(c = \sqrt{a^{2} + b^{2}} = \sqrt{4 + 16} = \sqrt{20}\), so the foci are at \((\pm \sqrt{20}, 0)\), and find the equations of the asymptotes using \(y = \pm \frac{b}{a} x = \pm \frac{4}{2} x = \pm 2x\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Identification of Conic Sections from Equations
Conic sections are curves obtained by intersecting a plane with a double-napped cone. Their equations typically involve quadratic terms in x and y. By analyzing the signs and coefficients of x² and y² terms, one can classify the curve as a parabola, ellipse, or hyperbola. For example, if the equation has both squared terms with opposite signs, it represents a hyperbola.
Video consigliato:
Percorso guidato
3:08
Geometries from Conic Sections
Standard Forms and Properties of Conic Sections
Each conic section has a standard form equation revealing key features like vertices, foci, and axes. For hyperbolas, the standard form helps identify the center, vertices, foci, and asymptotes. Understanding how to rewrite the given equation into standard form is essential for graphing and labeling these elements accurately.
Video consigliato:
Percorso guidato
5:33Parabolas as Conic Sections
Graphing and Key Features of Hyperbolas
Graphing a hyperbola involves plotting its vertices and foci, and drawing its asymptotes, which guide the curve's shape. The vertices are points where the hyperbola intersects its principal axis, and the foci are fixed points defining the curve. Asymptotes are straight lines that the hyperbola approaches but never touches, determined by the equation's coefficients.
Video consigliato:
Percorso guidato
4:20Graph Hyperbolas at the Origin
Pratica correlata
Domanda del libro di testo
41–44. Intersection points and area Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves
r = 1 + sin θ and r = 1 + cos θ
43
views
Domanda del libro di testo
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the circle r = 8 sin θ
52
views
Domanda del libro di testo
63–66. Tracing hyperbolas and parabolas Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as θ increases from 0 to 2π.
r = 3/(1 - cos θ)
45
views
Domanda del libro di testo
45–60. Areas of regions Find the area of the following regions.
The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ
64
views
Domanda del libro di testo
63–74. Arc length of polar curves Find the length of the following polar curves.
{Use of Tech} The complete limaçon r=4−2cosθ
96
views