Use the guidelines of this section to sketch the curve. y = e-X sin(x), 0 < x < 2 y y 1.0 1.5 0.5 1.0 0.5 37 -0.5 -0.5 -1.0 -1.0 -1.5 y y 0.4 0.30 0.25 0.3 0.20 0.2 0.15 0.10 0.1 0.05
Use the guidelines of this section to sketch the curve. y = e-X sin(x), 0 < x < 2 y y 1.0 1.5 0.5 1.0 0.5 37 -0.5 -0.5 -1.0 -1.0 -1.5 y y 0.4 0.30 0.25 0.3 0.20 0.2 0.15 0.10 0.1 0.05
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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![**Guidance for Sketching Curves on an Educational Website**
**Example Function:**
\[ y = e^{-x} \sin(x), \quad 0 \leq x \leq 2\pi \]
**Graphs and Diagrams:**
1. **First Graph (Top Left Quadrant):**
- **Axes:** The horizontal axis (x-axis) ranges from 0 to \( 2\pi \). The vertical axis (y-axis) ranges approximately from \(-1.0\) to \( 1.75\).
- **Curve Description:** This graph features one complete cycle of a damped sine wave. The amplitude decreases exponentially with increasing \( x \), due to the factor \( e^{-x} \). The function crosses the x-axis multiple times within the given range.
2. **Second Graph (Top Right Quadrant):**
- **Axes:** The x-axis ranges from 0 to \( 2\pi \). The y-axis ranges approximately from \(-1.5\) to \( 1.0\).
- **Curve Description:** This graph displays one full cycle of a sine wave that is modulated by the exponential decay factor \( e^{-x} \). It has visible oscillations that diminish in amplitude as \( x \) increases.
3. **Third Graph (Bottom Left Quadrant):**
- **Axes:** The x-axis ranges from 0 to \( 2\pi \). The y-axis ranges from 0 to approximately 0.4.
- **Curve Description:** This graph depicts a positive exponential decay. Initially, there's a sharp peak that decays rapidly to approach zero as \( x \) approaches \( 2\pi \). This represents the positive portion of the damped sine wave where \( \sin(x) \) remains positive.
4. **Fourth Graph (Bottom Right Quadrant):**
- **Axes:** The x-axis ranges from 0 to \( 2\pi \). The y-axis ranges from 0 to approximately 0.3.
- **Curve Description:** Similar to the third graph, but this graph shows the decay in a positive direction with a different scaling factor on the y-axis that emphasizes the damping effect more clearly. The function again approaches zero as \( x \) approaches \( 2\pi \).
These graphs collectively illustrate the behavior of the function \( y = e^{-x} \sin(x) \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6653f5ea-9ae4-4965-89e5-4546e31fc625%2F2ca81213-fad5-49bf-8430-47095a8f0965%2F480ltf.png&w=3840&q=75)
Transcribed Image Text:**Guidance for Sketching Curves on an Educational Website**
**Example Function:**
\[ y = e^{-x} \sin(x), \quad 0 \leq x \leq 2\pi \]
**Graphs and Diagrams:**
1. **First Graph (Top Left Quadrant):**
- **Axes:** The horizontal axis (x-axis) ranges from 0 to \( 2\pi \). The vertical axis (y-axis) ranges approximately from \(-1.0\) to \( 1.75\).
- **Curve Description:** This graph features one complete cycle of a damped sine wave. The amplitude decreases exponentially with increasing \( x \), due to the factor \( e^{-x} \). The function crosses the x-axis multiple times within the given range.
2. **Second Graph (Top Right Quadrant):**
- **Axes:** The x-axis ranges from 0 to \( 2\pi \). The y-axis ranges approximately from \(-1.5\) to \( 1.0\).
- **Curve Description:** This graph displays one full cycle of a sine wave that is modulated by the exponential decay factor \( e^{-x} \). It has visible oscillations that diminish in amplitude as \( x \) increases.
3. **Third Graph (Bottom Left Quadrant):**
- **Axes:** The x-axis ranges from 0 to \( 2\pi \). The y-axis ranges from 0 to approximately 0.4.
- **Curve Description:** This graph depicts a positive exponential decay. Initially, there's a sharp peak that decays rapidly to approach zero as \( x \) approaches \( 2\pi \). This represents the positive portion of the damped sine wave where \( \sin(x) \) remains positive.
4. **Fourth Graph (Bottom Right Quadrant):**
- **Axes:** The x-axis ranges from 0 to \( 2\pi \). The y-axis ranges from 0 to approximately 0.3.
- **Curve Description:** Similar to the third graph, but this graph shows the decay in a positive direction with a different scaling factor on the y-axis that emphasizes the damping effect more clearly. The function again approaches zero as \( x \) approaches \( 2\pi \).
These graphs collectively illustrate the behavior of the function \( y = e^{-x} \sin(x) \
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