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Sketch the curve with the given polar equation by first sketching the graph of r as a function of 0 in Cartesian coordinates. r = 1 - cos(0) y 1.5 y 0.5 1.0 X -1.5 1.0 -0.5 0.5 1.0 1.5 0.5 -0.5 -1.0 2.0 -1.5 -1.0 -0.5 0.5 -1.5 -0.5 -1.0 -1.5 y 1.5

Sketch the curve with the given polar equation by first sketching the graph of r as a function of 0 in Cartesian coordinates. r = 1 - cos(0) y 1.5 y 0.5 1.0 X -1.5 1.0 -0.5 0.5 1.0 1.5 0.5 -0.5 -1.0 2.0 -1.5 -1.0 -0.5 0.5 -1.5 -0.5 -1.0 -1.5 y 1.5

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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Which one is the correct graph and how did you find it.

**Title: Sketching Polar Curves in Cartesian Coordinates** **Introduction:** In this exercise, we explore the process of sketching a polar curve by converting it into Cartesian coordinates. The given polar equation is \( r = 1 - \cos(\theta) \). **Objective:** Sketch the curve represented by the given polar equation by first sketching the graph of \( r \) as a function of \( \theta \) in Cartesian coordinates. **Graph Overview:** Four graphs are presented, each illustrating potential Cartesian coordinate interpretations of the given polar equation. 1. **Top Left Graph:** - The graph is a complete circle centered at \((-0.5, 0)\) with a radius of 1.5. - The x-axis ranges from -1.5 to 1.5, while the y-axis ranges from -2.0 to 0.5. - This shape is not consistent with the expected polar plot. 2. **Top Right Graph:** - A cardioid shape is evident, centered at the origin. - The x-axis ranges from -2.0 to 0.5, while the y-axis ranges from -1.5 to 1.5. - This graph accurately reflects the polar equation \( r = 1 - \cos(\theta) \). 3. **Bottom Left Graph:** - This graph resembles an inverted cardioid. - The x-axis ranges from -0.5 to 2.0, and the y-axis ranges from -1.5 to 1.5. - While similar to the correct form, this graph is flipped horizontally. 4. **Bottom Right Graph:** - A circle is centered at \((0, -0.5)\) with a radius of 1.5. - The x-axis and y-axis range from -1.5 to 1.5. - This shape is not consistent with the expected polar curve. **Conclusion:** The top right graph correctly illustrates the polar curve \( r = 1 - \cos(\theta) \) as a cardioid. When converting polar equations to Cartesian coordinates, verifying the transformation through sketching can ensure an accurate representation of the function.
Transcribed Image Text:**Title: Sketching Polar Curves in Cartesian Coordinates** **Introduction:** In this exercise, we explore the process of sketching a polar curve by converting it into Cartesian coordinates. The given polar equation is \( r = 1 - \cos(\theta) \). **Objective:** Sketch the curve represented by the given polar equation by first sketching the graph of \( r \) as a function of \( \theta \) in Cartesian coordinates. **Graph Overview:** Four graphs are presented, each illustrating potential Cartesian coordinate interpretations of the given polar equation. 1. **Top Left Graph:** - The graph is a complete circle centered at \((-0.5, 0)\) with a radius of 1.5. - The x-axis ranges from -1.5 to 1.5, while the y-axis ranges from -2.0 to 0.5. - This shape is not consistent with the expected polar plot. 2. **Top Right Graph:** - A cardioid shape is evident, centered at the origin. - The x-axis ranges from -2.0 to 0.5, while the y-axis ranges from -1.5 to 1.5. - This graph accurately reflects the polar equation \( r = 1 - \cos(\theta) \). 3. **Bottom Left Graph:** - This graph resembles an inverted cardioid. - The x-axis ranges from -0.5 to 2.0, and the y-axis ranges from -1.5 to 1.5. - While similar to the correct form, this graph is flipped horizontally. 4. **Bottom Right Graph:** - A circle is centered at \((0, -0.5)\) with a radius of 1.5. - The x-axis and y-axis range from -1.5 to 1.5. - This shape is not consistent with the expected polar curve. **Conclusion:** The top right graph correctly illustrates the polar curve \( r = 1 - \cos(\theta) \) as a cardioid. When converting polar equations to Cartesian coordinates, verifying the transformation through sketching can ensure an accurate representation of the function.
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