Domanda del libro di testo
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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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 12.4.45
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2
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Identify the orientation of the hyperbola based on the vertices. Since the vertices are at (±2, 0), the hyperbola opens horizontally along the x-axis.
Write the standard form of the hyperbola equation centered at the origin with a horizontal transverse axis: \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\).
Use the vertices to find \(a\). The vertices are at (±a, 0), so from (±2, 0), we have \(a = 2\), which means \(a^{2} = 4\).
Use the slopes of the asymptotes to find \(b\). For a hyperbola with a horizontal transverse axis, the asymptotes are given by \(y = \pm \frac{b}{a} x\). Given the asymptotes \(y = \pm \frac{3}{2} x\), set \(\frac{b}{a} = \frac{3}{2}\) and solve for \(b\).
Substitute the values of \(a^{2}\) and \(b^{2}\) into the standard form equation \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) to write the equation of the hyperbola.
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Standard Form of a Hyperbola Centered at the Origin
A hyperbola centered at the origin with a horizontal transverse axis has the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex along the x-axis, and \(b\) relates to the shape of the hyperbola and its asymptotes.
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Percorso guidato
5:59
Graph Hyperbolas NOT at the Origin
Vertices of a Hyperbola
Vertices are the points where the hyperbola intersects its transverse axis. For a hyperbola centered at the origin with a horizontal transverse axis, the vertices are at \((\pm a, 0)\). Knowing the vertices helps determine the value of \(a\) in the equation.
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Percorso guidato
5:22Foci and Vertices of Hyperbolas
Asymptotes of a Hyperbola
The asymptotes of a hyperbola provide lines that the curve approaches but never touches. For a hyperbola centered at the origin with a horizontal transverse axis, the asymptotes are given by \( y = \pm \frac{b}{a} x \). Using the slopes of the asymptotes allows solving for \(b\) once \(a\) is known.
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Percorso guidato
5:50Asymptotes of Hyperbolas
Pratica correlata
Domanda del libro di testo
45–60. Areas of regions Find the area of the following regions.
The region outside the circle r = 1/2 and inside the circle r = cos θ
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Domanda del libro di testo
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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Domanda del libro di testo
Explain why the slope of the line θ=π/2 is undefined.
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Domanda del libro di testo
63–74. Arc length of polar curves Find the length of the following polar curves.
The spiral r = θ², for 0 ≤ θ ≤ 2π
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Domanda del libro di testo
23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.
A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.
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