Textbook Question
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 + √t, y = 2 - 4t; slope = -8
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 12.2.63
57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.
r = sin 3θ
Verified step by step guidance1
Understand the given polar equation: \(r = \sin 3\theta\). This represents a polar curve where the radius \(r\) depends on the angle \(\theta\) multiplied by 3 inside the sine function.
Recall that equations of the form \(r = \sin n\theta\) produce rose curves with petals. If \(n\) is an integer, the number of petals depends on whether \(n\) is odd or even: for odd \(n\), the rose has \(n\) petals; for even \(n\), it has \$2n$ petals.
To graph the curve, create a table of values by choosing several values of \(\theta\) between \(0\) and \(2\pi\), calculate \(r\) for each, and plot the points \((r, \theta)\) in polar coordinates. For example, evaluate \(r\) at \(\theta = 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \ldots\)
Plot the points on polar graph paper or using a graphing utility, connecting the points smoothly to reveal the petal shapes. Notice the symmetry and periodicity of the curve as \(\theta\) increases.
Use a graphing utility to check your hand-drawn graph. Input the equation \(r = \sin 3\theta\) and compare the plotted curve to your sketch, ensuring the number of petals and their orientation match.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Polar Equations
Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how r changes with θ is essential for graphing polar equations like r = sin 3θ, where the radius depends on the angle multiplied by a factor.
Recommended video:
05:32
Intro to Polar Coordinates
Graphing Polar Curves
Graphing polar curves involves plotting points for various values of θ and connecting them smoothly. Recognizing patterns such as petals or loops, especially in equations like r = sin nθ, helps predict the shape and symmetry of the graph.
Recommended video:
09:04Slope of Polar Curves
Use of Graphing Utilities
Graphing utilities, such as graphing calculators or software, allow for accurate visualization of complex polar curves. They help verify manual sketches and reveal detailed features like the number of petals and symmetry in curves like r = sin 3θ.
Recommended video:
06:15Graphing The Derivative
Related Practice
Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
25y² - 4x² = 100
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Textbook Question
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside one leaf of r = cos 3θ
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Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = x²
Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
10x² - 7y² = 140
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Textbook Question
53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.
The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.
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