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r = cos(0/7) %3D

r = cos(0/7) %3D

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question

what is the correct graph for this polar equation and how did you determine it?

The equation depicted is a polar equation: \[ r = \cos\left(\frac{\theta}{7}\right) \] This equation represents a polar coordinate system where \( r \), the radius, is defined as the cosine of the angle \( \theta \) divided by 7. The division by 7 results in a periodic function, often leading to interesting symmetrical patterns when graphed, resembling a multi-petaled rose. The graph for this equation would likely exhibit a symmetric, rosette pattern with several petals determined by the trigonometric properties of the cosine function and the divisor within its argument.
Transcribed Image Text:The equation depicted is a polar equation: \[ r = \cos\left(\frac{\theta}{7}\right) \] This equation represents a polar coordinate system where \( r \), the radius, is defined as the cosine of the angle \( \theta \) divided by 7. The division by 7 results in a periodic function, often leading to interesting symmetrical patterns when graphed, resembling a multi-petaled rose. The graph for this equation would likely exhibit a symmetric, rosette pattern with several petals determined by the trigonometric properties of the cosine function and the divisor within its argument.
### Graph Analysis for Educational Purposes This image contains six distinct plots demonstrating different mathematical curves, each labeled I through VI. #### Plot I This plot displays a simple closed curve resembling an asymmetric oval or an elongated circle, indicating a potential parametric or polar plot with varying coefficients affecting its symmetry and elongation. #### Plot II This plot shows a spiral originating from the center and expanding outward in a consistent, circular pattern. The equal spacing between loops suggests a logarithmic spiral, common in nature and mathematics to demonstrate growth and expansion. #### Plot III The plot illustrates a limaçon, evident from its characteristic small inner loop located inside a larger, rounded outer loop. This shape is often encountered in polar coordinate studies as it varies with trigonometric functions. #### Plot IV This plot highlights a rose curve with four distinct petals. Such curves arise from polar equations involving functions like sine or cosine, and the number of petals relates directly to factors within these functions. #### Plot V Here, another spiral is shown, with multiple loops tightly coiling into a central point. Similar to Plot II, this is likely a form of logarithmic spiral, but with more loops before reaching the outer boundary. #### Plot VI This plot presents a cardioid pattern. The shaped “heart-like” appearance is typical of cardioids, which are generated in polar coordinates with specific sine or cosine functions. Each plot represents unique mathematical equations and concepts explored in higher-level mathematics such as algebra, trigonometry, and calculus, serving as a practical visualization tool for students learning these advanced topics.
Transcribed Image Text:### Graph Analysis for Educational Purposes This image contains six distinct plots demonstrating different mathematical curves, each labeled I through VI. #### Plot I This plot displays a simple closed curve resembling an asymmetric oval or an elongated circle, indicating a potential parametric or polar plot with varying coefficients affecting its symmetry and elongation. #### Plot II This plot shows a spiral originating from the center and expanding outward in a consistent, circular pattern. The equal spacing between loops suggests a logarithmic spiral, common in nature and mathematics to demonstrate growth and expansion. #### Plot III The plot illustrates a limaçon, evident from its characteristic small inner loop located inside a larger, rounded outer loop. This shape is often encountered in polar coordinate studies as it varies with trigonometric functions. #### Plot IV This plot highlights a rose curve with four distinct petals. Such curves arise from polar equations involving functions like sine or cosine, and the number of petals relates directly to factors within these functions. #### Plot V Here, another spiral is shown, with multiple loops tightly coiling into a central point. Similar to Plot II, this is likely a form of logarithmic spiral, but with more loops before reaching the outer boundary. #### Plot VI This plot presents a cardioid pattern. The shaped “heart-like” appearance is typical of cardioids, which are generated in polar coordinates with specific sine or cosine functions. Each plot represents unique mathematical equations and concepts explored in higher-level mathematics such as algebra, trigonometry, and calculus, serving as a practical visualization tool for students learning these advanced topics.
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