Textbook Question
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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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 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
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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08:02
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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06:03Parameterizing 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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06:49Differentiation of Parametric Curves
Related Practice
Textbook Question
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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Textbook Question
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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Textbook Question
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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Textbook Question
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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Textbook Question
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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