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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.R.17
Capitolo 12, Problema 12.R.17

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise

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Recall that the standard parametric equations for a circle centered at the origin with radius \(r\) are \(x = r \cos(t)\) and \(y = r \sin(t)\), where \(t\) is the parameter representing the angle in radians.
Since the given circle is \(x^2 + y^2 = 9\), the radius \(r\) is \(3\). So the standard parametric form is \(x = 3 \cos(t)\) and \(y = 3 \sin(t)\).
Note that the standard parametric form traces the circle counterclockwise as \(t\) increases. To generate the circle clockwise, we need to reverse the direction of traversal.
To reverse the direction, replace \(t\) by \(-t\) in the parametric equations. This gives \(x = 3 \cos(-t)\) and \(y = 3 \sin(-t)\).
Use the even and odd properties of cosine and sine: \(\cos(-t) = \cos(t)\) and \(\sin(-t) = -\sin(t)\). So the parametric equations for the circle traced clockwise are \(x = 3 \cos(t)\) and \(y = -3 \sin(t)\).

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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 more flexible descriptions of curves, including direction and speed.
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Parameterizing Equations

Equation of a Circle

The standard equation x² + y² = r² represents a circle centered at the origin with radius r. For r = 3, the circle includes all points (x, y) satisfying x² + y² = 9, forming a closed curve equidistant from the center.
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Parameterizing Equations of Circles & Ellipses

Orientation and Direction in Parametric Curves

Parametric curves have an orientation determined by how the parameter t increases. For a circle, the direction can be counterclockwise or clockwise, which affects the sign and order of the parametric functions, such as using sine and cosine with positive or negative signs.
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Percorso guidato
06:49
Differentiation of Parametric Curves
Pratica correlata
Domanda del libro di testo

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

r =3 − 6 cos θ

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65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (±4, 0) and directrices x = ±2

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19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve

The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2

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58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.

x²/16 - y²/9 = 1; (20/3, -4)

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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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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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