Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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) \
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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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