Domanda del libro di testo
44–49. Areas of regions Find the area of the following regions.
The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ
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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.R.10a
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = t² + 4, y = -t, for -2 < t < 0; (5, 1)
Guida verificata passo dopo passo1
Identify the given parametric equations: \(x = t^{2} + 4\) and \(y = -t\), with the parameter \(t\) in the interval \(-2 < t < 0\).
Express the parameter \(t\) in terms of \(y\) from the second equation: since \(y = -t\), then \(t = -y\).
Substitute \(t = -y\) into the first equation to eliminate the parameter: \(x = (-y)^{2} + 4\).
Simplify the expression: \(x = y^{2} + 4\).
Write the final Cartesian equation relating \(x\) and \(y\): \(x = y^{2} + 4\), which represents the curve without the parameter \(t\).
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often 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.
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08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other to find an explicit or implicit equation of the curve.
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05:59Eliminating the Parameter
Domain Restrictions on the Parameter
The parameter t is often restricted to a specific interval, which limits the portion of the curve described. Understanding these bounds is essential to correctly interpret the curve's segment and verify points like (5, 1) lie within the given parameter range.
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05:59Eliminating the Parameter
Pratica correlata
Domanda del libro di testo
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6
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Domanda del libro di testo
Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.
a. For what value of p is P tangent to H?
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Domanda del libro di testo
Conic parameters: A hyperbola has eccentricity e = 2 and foci (0, ±2). Find the location of the vertices and directrices.
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Domanda del libro di testo
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
4x² + 8y² = 16
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Domanda del libro di testo
Cartesian conversion Write the equation x=y ² in polar coordinates and state values of θ that produce the entire graph of the parabola.
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