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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![**Exercise: Graphing a Function**
**Objective:** Sketch a graph of the function \( f(x, y) = x \sin(y) \).
**Instructions:** To understand the behavior of the function \( f(x, y) = x \sin(y) \) and visualize it, you can follow these steps:
**Steps:**
1. **Determine the Range and Domain:**
- The domain includes all real numbers for both \( x \) and \( y \).
- The range of \( \sin(y) \) is \([-1, 1]\), so \( f(x,y) \) can take any real number values depending on \( x \).
2. **Analyze the Function:**
- The function is periodic in \( y \) with a period of \( 2\pi \).
- As \( y \) changes, the sine function will oscillate between -1 and 1.
- For a fixed \( x \), \( f(x, y) \) will oscillate between \(-x\) and \( x \).
3. **Sketching Key Points:**
- When \( y = n\pi \) (where \( n \) is an integer), \( \sin(y) = 0 \), so \( f(x, y) = 0 \).
- When \( y = (2n+1)\frac{\pi}{2} \), \( \sin(y) = \pm 1 \), so \( f(x, y) = \pm x \).
4. **Draw the Graph:**
- Plot the sine wave for \( y \) on the y-axis, showing oscillation.
- For each point on the \( x \)-axis, multiply the sine wave's value by \( x \).
**Example Graph:**
You should produce a three-dimensional graph where the \( z \)-axis represents \( f(x, y) \). The \( x \)-axis will represent \( x \), and the \( y \)-axis will represent \( y \). The resulting plot will look like a series of sine waves stretched and compressed along the \( z \)-axis by the multiplication with \( x \).
**Conclusion:**
This graph helps visualize the combined effect of a linear scalar \( x \) and a periodic function \( \sin(y) \), providing insights into oscillatory behaviors modified by a linear factor.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F416064f9-c57b-46ee-9b50-14b257515654%2Fbff3b4d7-5090-40cc-9b71-3684fb39fdcc%2F10iccp9_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise: Graphing a Function**
**Objective:** Sketch a graph of the function \( f(x, y) = x \sin(y) \).
**Instructions:** To understand the behavior of the function \( f(x, y) = x \sin(y) \) and visualize it, you can follow these steps:
**Steps:**
1. **Determine the Range and Domain:**
- The domain includes all real numbers for both \( x \) and \( y \).
- The range of \( \sin(y) \) is \([-1, 1]\), so \( f(x,y) \) can take any real number values depending on \( x \).
2. **Analyze the Function:**
- The function is periodic in \( y \) with a period of \( 2\pi \).
- As \( y \) changes, the sine function will oscillate between -1 and 1.
- For a fixed \( x \), \( f(x, y) \) will oscillate between \(-x\) and \( x \).
3. **Sketching Key Points:**
- When \( y = n\pi \) (where \( n \) is an integer), \( \sin(y) = 0 \), so \( f(x, y) = 0 \).
- When \( y = (2n+1)\frac{\pi}{2} \), \( \sin(y) = \pm 1 \), so \( f(x, y) = \pm x \).
4. **Draw the Graph:**
- Plot the sine wave for \( y \) on the y-axis, showing oscillation.
- For each point on the \( x \)-axis, multiply the sine wave's value by \( x \).
**Example Graph:**
You should produce a three-dimensional graph where the \( z \)-axis represents \( f(x, y) \). The \( x \)-axis will represent \( x \), and the \( y \)-axis will represent \( y \). The resulting plot will look like a series of sine waves stretched and compressed along the \( z \)-axis by the multiplication with \( x \).
**Conclusion:**
This graph helps visualize the combined effect of a linear scalar \( x \) and a periodic function \( \sin(y) \), providing insights into oscillatory behaviors modified by a linear factor.
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