Graph the surface 1 = y2 + z2 –- by first sketching at least three cross sections in each of the xy and yz planes.
Graph the surface 1 = y2 + z2 –- by first sketching at least three cross sections in each of the 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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![### Graphing and Understanding Surfaces
#### Objective:
Graph the surface defined by the equation
\[ 1 = y^2 + z^2 - \frac{x^2}{4} \]
by first sketching at least three cross-sections in each of the \(xy\) and \(yz\) planes.
#### Instructions:
1. **Identify and Define the Cross Sections:**
- In the \(xy\)-plane, consider \(z\) at constant values and graph the resulting curves.
- In the \(yz\)-plane, consider \(x\) at constant values and graph the resulting curves.
2. **Analyze Cross Sections in the \(yz\)-Plane:**
- When \(x = 0\), the surface simplifies to \(1 = y^2 + z^2\), which represents a circle of radius 1.
- When \(x = \pm 2\), the surface simplifies to \(\frac{x^2}{4} = 1\), and thus \(y^2 + z^2 = 1 - \frac{x^2}{4} = 0\).
3. **Analyze Cross Sections in the \(xy\)-Plane:**
- When \(z = 0\), the surface simplifies to \(1 = y^2 - \frac{x^2}{4}\), representing hyperbolas.
- For different \(z\)-values, adjust the magnitude of \(\frac{z^2}{4}\) and therefore modify the resulting hyperbolas.
#### Detailed Steps:
1. **Step-by-Step Cross Sections in \(xy\)-Plane:**
- Set \(z = 0\): The equation becomes \(1 = y^2 - \frac{x^2}{4}\). Simplify and graph the hyperbola.
- Set \(z = 0.5\): The equation becomes \(1 = y^2 + 0.25 - \frac{x^2}{4}\). Simplify and graph the hyperbola.
- Set \(z = 1\): The equation becomes \(1 = y^2 + 1 - \frac{x^2}{4}\). Simplify and graph the new hyperbola.
2. **Step-by-Step Cross Sections in the \(yz\)-Plane:**
- Set \(x = 0\): The equation becomes \(1 = y^2 + z^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4bda451-1bcc-473c-9a91-77116df85d00%2Ff51a04fb-66ba-446f-8299-313b66c37a14%2Frryxxcl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing and Understanding Surfaces
#### Objective:
Graph the surface defined by the equation
\[ 1 = y^2 + z^2 - \frac{x^2}{4} \]
by first sketching at least three cross-sections in each of the \(xy\) and \(yz\) planes.
#### Instructions:
1. **Identify and Define the Cross Sections:**
- In the \(xy\)-plane, consider \(z\) at constant values and graph the resulting curves.
- In the \(yz\)-plane, consider \(x\) at constant values and graph the resulting curves.
2. **Analyze Cross Sections in the \(yz\)-Plane:**
- When \(x = 0\), the surface simplifies to \(1 = y^2 + z^2\), which represents a circle of radius 1.
- When \(x = \pm 2\), the surface simplifies to \(\frac{x^2}{4} = 1\), and thus \(y^2 + z^2 = 1 - \frac{x^2}{4} = 0\).
3. **Analyze Cross Sections in the \(xy\)-Plane:**
- When \(z = 0\), the surface simplifies to \(1 = y^2 - \frac{x^2}{4}\), representing hyperbolas.
- For different \(z\)-values, adjust the magnitude of \(\frac{z^2}{4}\) and therefore modify the resulting hyperbolas.
#### Detailed Steps:
1. **Step-by-Step Cross Sections in \(xy\)-Plane:**
- Set \(z = 0\): The equation becomes \(1 = y^2 - \frac{x^2}{4}\). Simplify and graph the hyperbola.
- Set \(z = 0.5\): The equation becomes \(1 = y^2 + 0.25 - \frac{x^2}{4}\). Simplify and graph the hyperbola.
- Set \(z = 1\): The equation becomes \(1 = y^2 + 1 - \frac{x^2}{4}\). Simplify and graph the new hyperbola.
2. **Step-by-Step Cross Sections in the \(yz\)-Plane:**
- Set \(x = 0\): The equation becomes \(1 = y^2 + z^
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