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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![**Sketch the Curve**
Given the equation for the curve:
\[
(x^2 + y^2)^3 = 4x^2y^2
\]
To sketch this curve:
1. **Identify Symmetries**: The equation is symmetric in both x and y, suggesting that the curve should be symmetric about the x-axis, y-axis, and the origin.
2. **Analyze Behavior**:
- At the origin (0, 0), both sides of the equation are zero, so (0, 0) is a point on the curve.
- The degree of the equation suggests a closed and possibly looping structure.
3. **Plot Key Points:**
- For small values of \(x\) and \(y\), evaluate the equation to find intercepts and behavior near the axes.
- Check points where \(x = 0\) and \(y = 0\) to simplify plotting.
4. **Consider Special Cases:**
- If \(x = y\), the equation simplifies to \((2x^2)^3 = 4x^4x^2\), helping to find further intersection or symmetric points.
5. **Graph Features:**
- The shape and intersections of the graph can hint at possible loops, cusps, or singular points.
Analyzing these features will help visualize and plot the curve accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff57c5c9b-acb3-4a3d-a39b-55eeff7ee4ad%2F12eeae8a-d343-406f-a239-fca839f08f31%2F9x6mh9s_processed.png&w=3840&q=75)
Transcribed Image Text:**Sketch the Curve**
Given the equation for the curve:
\[
(x^2 + y^2)^3 = 4x^2y^2
\]
To sketch this curve:
1. **Identify Symmetries**: The equation is symmetric in both x and y, suggesting that the curve should be symmetric about the x-axis, y-axis, and the origin.
2. **Analyze Behavior**:
- At the origin (0, 0), both sides of the equation are zero, so (0, 0) is a point on the curve.
- The degree of the equation suggests a closed and possibly looping structure.
3. **Plot Key Points:**
- For small values of \(x\) and \(y\), evaluate the equation to find intercepts and behavior near the axes.
- Check points where \(x = 0\) and \(y = 0\) to simplify plotting.
4. **Consider Special Cases:**
- If \(x = y\), the equation simplifies to \((2x^2)^3 = 4x^4x^2\), helping to find further intersection or symmetric points.
5. **Graph Features:**
- The shape and intersections of the graph can hint at possible loops, cusps, or singular points.
Analyzing these features will help visualize and plot the curve accurately.
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