Given the function z= x² sin(y), match the following cross-sections with the following sketches: I II H III
Given the function z= x² sin(y), match the following cross-sections with the following sketches: I II H III
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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Question
- **Description:** The graph is a parabolic curve opening upwards. The curve is symmetric about the vertical axis and intersects the origin (0,0).
**Sketch II:**
- **Description:** The graph contains oscillatory behavior with varying amplitude. It exhibits an oscillating sine wave pattern with higher amplitudes as it moves away from the center. The oscillations are symmetric around the vertical axis.
**Sketch III:**
- **Description:** This graph shows a wavy pattern characterized by regular sine wave oscillations. The curve is symmetric and periodic, reflecting the nature of the sine function.
**Explanation:**
To determine which sketch corresponds to which cross-section of the given function \( z = x^2 \sin(y) \), consider the behavior of the function along different axes:
1. **Cross-section parallel to the \(z-x\) plane (i.e., fixing \( y \) and varying \( x \)):** When \( y \) is constant, \( \sin(y) \) is a constant, and the equation simplifies to \( z = kx^2 \), where \( k \) is a constant. This results in a parabolic shape. Hence, Sketch I represents this cross-section.
2. **Cross-section parallel to the \(z-y\) plane (i.e., fixing \( x \) and varying \( y \)):** With \( x \) constant, the equation becomes \( z = k \sin(y) \), where \( k \) is a constant \( x^2 \). This generates a sine wave. If \( x \) is non-zero, the amplitude of the sine wave is larger; hence, Sketch III represents this cross-section.
3. **Cross-section when \( x = 0 \) (i.e., along the \( y \)-axis):** The function simplifies to \( z = 0 \), leading to the flatline or approach to the plane. Sketch II illustrates the significant changes in amplitudes of sine waves symmetrically about the y-axis, representing cross-sections away from the \( x \)-axis.
By examining the sketches and the given equation, we can conclude the following](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d3d5d5c-da0a-4026-882c-d4c3428233cc%2F65062a81-f5c6-4ea3-bd0e-3050ada9d676%2F946012_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matching Cross-Sections with Sketches**
Given the function \( z = x^2 \sin(y) \), match the following cross-sections with the following sketches:
**Sketch I:**
- **Description:** The graph is a parabolic curve opening upwards. The curve is symmetric about the vertical axis and intersects the origin (0,0).
**Sketch II:**
- **Description:** The graph contains oscillatory behavior with varying amplitude. It exhibits an oscillating sine wave pattern with higher amplitudes as it moves away from the center. The oscillations are symmetric around the vertical axis.
**Sketch III:**
- **Description:** This graph shows a wavy pattern characterized by regular sine wave oscillations. The curve is symmetric and periodic, reflecting the nature of the sine function.
**Explanation:**
To determine which sketch corresponds to which cross-section of the given function \( z = x^2 \sin(y) \), consider the behavior of the function along different axes:
1. **Cross-section parallel to the \(z-x\) plane (i.e., fixing \( y \) and varying \( x \)):** When \( y \) is constant, \( \sin(y) \) is a constant, and the equation simplifies to \( z = kx^2 \), where \( k \) is a constant. This results in a parabolic shape. Hence, Sketch I represents this cross-section.
2. **Cross-section parallel to the \(z-y\) plane (i.e., fixing \( x \) and varying \( y \)):** With \( x \) constant, the equation becomes \( z = k \sin(y) \), where \( k \) is a constant \( x^2 \). This generates a sine wave. If \( x \) is non-zero, the amplitude of the sine wave is larger; hence, Sketch III represents this cross-section.
3. **Cross-section when \( x = 0 \) (i.e., along the \( y \)-axis):** The function simplifies to \( z = 0 \), leading to the flatline or approach to the plane. Sketch II illustrates the significant changes in amplitudes of sine waves symmetrically about the y-axis, representing cross-sections away from the \( x \)-axis.
By examining the sketches and the given equation, we can conclude the following
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