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Create a table of values of sin(1/x) for the following values of x: 2/pi, 2/3pi,2/5pi, 2/7pi, 2/9pi, 2/11pi

Create a table of values of sin(1/x) for the following values of x: 2/pi, 2/3pi,2/5pi, 2/7pi, 2/9pi, 2/11pi

Calculus: Early Transcendentals
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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Create a table of values of sin(1/x) for the following values of x: 2/pi, 2/3pi,2/5pi, 2/7pi, 2/9pi, 2/11pi

### Analyzing the Behavior of \( \sin \left( \frac{1}{x} \right) \) Near \( x = 0 \) Consider the function \( \sin \left( \frac{1}{x} \right) \) and its behavior as \( x \) approaches 0. The graph of this function is provided for visual analysis, and we will address the following parts: (a), (b), and (c). #### Graph Explanation The graph illustrates \( \sin \left( \frac{1}{x} \right) \) against \( x \), focusing on the interval around \( x = 0 \). 1. **Axes Labels & Orientation**: - The x-axis (horizontal) ranges from \( -\infty \) to \( \infty \). - The y-axis (vertical) ranges from -1 to 1, aligning with the amplitude of a sine function. 2. **Key Features Near \( x = 0 \)**: - As \( x \) approaches 0, the frequency of oscillations of the sine function increases significantly. This is due to the \( \frac{1}{x} \) term inside the sine function, causing rapid changes in the angle fed into the sine function as \( x \) gets smaller. - The function appears to oscillate infinitely between -1 and 1 in a very narrow range near \( x = 0 \). 3. **Visible Markings**: - Specific points are marked on the x-axis corresponding to certain values of \( x \): \( \frac{2}{9\pi} \), \( \frac{2}{7\pi} \), and \( \frac{2}{5\pi} \). These points show where the function crosses the x-axis again as the value of \( x \) increases from zero. The behavior of \( \sin \left( \frac{1}{x} \right) \) as \( x \to 0 \) includes increasingly frequent oscillations, creating an appearance of a "dense" area on the graph near \( x = 0 \). The function is not defined at \( x = 0 \) because \( \frac{1}{x} \) is undefined there. ### Task Instructions To fully grasp the function's behavior and implications, follow the exercises in parts (a), (b), and (c)
Transcribed Image Text:### Analyzing the Behavior of \( \sin \left( \frac{1}{x} \right) \) Near \( x = 0 \) Consider the function \( \sin \left( \frac{1}{x} \right) \) and its behavior as \( x \) approaches 0. The graph of this function is provided for visual analysis, and we will address the following parts: (a), (b), and (c). #### Graph Explanation The graph illustrates \( \sin \left( \frac{1}{x} \right) \) against \( x \), focusing on the interval around \( x = 0 \). 1. **Axes Labels & Orientation**: - The x-axis (horizontal) ranges from \( -\infty \) to \( \infty \). - The y-axis (vertical) ranges from -1 to 1, aligning with the amplitude of a sine function. 2. **Key Features Near \( x = 0 \)**: - As \( x \) approaches 0, the frequency of oscillations of the sine function increases significantly. This is due to the \( \frac{1}{x} \) term inside the sine function, causing rapid changes in the angle fed into the sine function as \( x \) gets smaller. - The function appears to oscillate infinitely between -1 and 1 in a very narrow range near \( x = 0 \). 3. **Visible Markings**: - Specific points are marked on the x-axis corresponding to certain values of \( x \): \( \frac{2}{9\pi} \), \( \frac{2}{7\pi} \), and \( \frac{2}{5\pi} \). These points show where the function crosses the x-axis again as the value of \( x \) increases from zero. The behavior of \( \sin \left( \frac{1}{x} \right) \) as \( x \to 0 \) includes increasingly frequent oscillations, creating an appearance of a "dense" area on the graph near \( x = 0 \). The function is not defined at \( x = 0 \) because \( \frac{1}{x} \) is undefined there. ### Task Instructions To fully grasp the function's behavior and implications, follow the exercises in parts (a), (b), and (c)
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