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.12a
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)
Guida verificata passo dopo passo1
Start with the given parametric equations: \(x = \ln t\) and \(y = 8 \ln t^{2}\), where \(1 \leq t \leq e^{2}\).
Recall the logarithm property: \(\ln t^{2} = 2 \ln t\). Use this to rewrite \(y\) as \(y = 8 \times 2 \ln t = 16 \ln t\).
Since \(x = \ln t\), substitute \(\ln t\) in the expression for \(y\) to get \(y = 16x\).
This gives the Cartesian equation relating \(x\) and \(y\) without the parameter \(t\): \(y = 16x\).
Note the domain for \(t\) translates to \(x\) because \(x = \ln t\). Since \(1 \leq t \leq e^{2}\), then \(0 \leq x \leq 2\).
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
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08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove the parameter 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 in x and y.
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05:59Eliminating the Parameter
Logarithmic Functions and Their Properties
Logarithmic functions, like ln(t), are the inverses of exponential functions and have properties such as ln(a^b) = b ln(a). Understanding these properties is essential for simplifying expressions and eliminating parameters when the parametric equations involve logarithms.
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Percorso guidato
06:21Properties of Functions
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
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
x² - y²/2 = 1
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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
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)
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Domanda del libro di testo
61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.
r = 3/(1 - 2 cos θ)
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