:y² + z² — ² by first sketching at least three cross sections in each of the 4. Graph the surface 1 = xy and yz planes.
:y² + z² — ² by first sketching at least three cross sections in each of the 4. Graph the surface 1 = xy and yz planes.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 4:**
**Task:** Graph the surface \( 1 = y^2 + z^2 - \frac{x^2}{4} \) by first sketching at least three cross sections in each of the \( xy \) and \( yz \) planes.
**Steps:**
1. **Identify the Equation:** The given equation represents a surface in three-dimensional space. Our goal is to analyze and plot this surface by examining its cross sections in the \( xy \) and \( yz \) planes.
2. **Cross Sections in the \( xy \)-Plane:**
- For different values of \( z \), we slice the surface parallel to the \( xy \)-plane, resulting in ellipse equations.
- Set \( z = k \) (where \( k \) is a constant), and substitute it into the surface equation:
\[
1 = y^2 + k^2 - \frac{x^2}{4}
\]
Rearranging gives:
\[
y^2 = 1 - k^2 + \frac{x^2}{4}
\]
3. **Cross Sections in the \( yz \)-Plane:**
- For different values of \( x \), we slice the surface parallel to the \( yz \)-plane, resulting in equations akin to hyperbolas.
- Set \( x = m \) (where \( m \) is a constant), and substitute it into the surface equation:
\[
1 = y^2 + z^2 - \frac{m^2}{4}
\]
Rearranging gives:
\[
y^2 + z^2 = 1 + \frac{m^2}{4}
\]
This describes an ellipse for each fixed \( m \).
**Summary:** By examining these cross sections, one can better understand and subsequently graph the surface \( 1 = y^2 + z^2 - \frac{x^2}{4} \). Analyzing cross sections in the \( xy \)-plane helps to understand the variation of \( y \) and \( x \) for different \( z \)-values, while doing so in the \( yz \)-plane helps in understanding the variation of \( y \) and \( z \) for different \( x \)-values.
Graphing these will provide visualization insight into the three-dimensional surface,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca574fc6-f355-4517-9427-4a1ce776d41c%2Fbec2ee6a-9662-454c-a23a-8824e93de864%2Fuqady4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
**Task:** Graph the surface \( 1 = y^2 + z^2 - \frac{x^2}{4} \) by first sketching at least three cross sections in each of the \( xy \) and \( yz \) planes.
**Steps:**
1. **Identify the Equation:** The given equation represents a surface in three-dimensional space. Our goal is to analyze and plot this surface by examining its cross sections in the \( xy \) and \( yz \) planes.
2. **Cross Sections in the \( xy \)-Plane:**
- For different values of \( z \), we slice the surface parallel to the \( xy \)-plane, resulting in ellipse equations.
- Set \( z = k \) (where \( k \) is a constant), and substitute it into the surface equation:
\[
1 = y^2 + k^2 - \frac{x^2}{4}
\]
Rearranging gives:
\[
y^2 = 1 - k^2 + \frac{x^2}{4}
\]
3. **Cross Sections in the \( yz \)-Plane:**
- For different values of \( x \), we slice the surface parallel to the \( yz \)-plane, resulting in equations akin to hyperbolas.
- Set \( x = m \) (where \( m \) is a constant), and substitute it into the surface equation:
\[
1 = y^2 + z^2 - \frac{m^2}{4}
\]
Rearranging gives:
\[
y^2 + z^2 = 1 + \frac{m^2}{4}
\]
This describes an ellipse for each fixed \( m \).
**Summary:** By examining these cross sections, one can better understand and subsequently graph the surface \( 1 = y^2 + z^2 - \frac{x^2}{4} \). Analyzing cross sections in the \( xy \)-plane helps to understand the variation of \( y \) and \( x \) for different \( z \)-values, while doing so in the \( yz \)-plane helps in understanding the variation of \( y \) and \( z \) for different \( x \)-values.
Graphing these will provide visualization insight into the three-dimensional surface,
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