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
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...
Related questions
Question
Which one is the correct graph and how did you find it.
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.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning