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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Question
Find the intervals where the function is increasing or decreasing . Use a sign chart to organize your analysis.
#4
![### Mathematical Function Analysis
**Function:** \( f(x) = \sin x + \cos x \)
**Domain:** \( (0, 2\pi) \)
---
This content appears to focus on the trigonometric function \( f(x) = \sin x + \cos x \) over the interval \( (0, 2\pi) \).
### Exploration of the Function
1. **Understanding the Graph and Behavior:**
- The function is composed of the sine and cosine functions, which are periodic with a period of \( 2\pi \).
- Analyzing this on the interval \( (0, 2\pi) \), we expect the function to exhibit wave-like characteristics due to the nature of the sine and cosine components.
2. **Property Analysis:**
- **Amplitude and Phase Shift:** Both sine and cosine have an amplitude of 1. The resulting function \( \sin x + \cos x \) can be rewritten using the amplitude and phase shift formula:
\[
f(x) = \sqrt{2}\sin(x + \frac{\pi}{4})
\]
Thus, the amplitude is \(\sqrt{2}\), and there is a phase shift of \(\frac{\pi}{4}\).
- **Zeros and Critical Points:** Solving \( \sin x + \cos x = 0 \) within this interval will provide the zeros, and taking the derivative to find critical points will show where the function reaches maximum or minimum values.
3. **Practical Applications:**
- Understanding oscillations in wave motion.
- Analyzing alternating current electricity outcomes.
---
This content can be used on an educational website to teach students about trigonometric functions, their transformations, and their implications in real-world scenarios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66d30f65-22c6-47d5-b371-c8d8c2d2b3f2%2F7ac6780c-7862-4e48-b823-661fa5c9d1a6%2Faxvkrd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Function Analysis
**Function:** \( f(x) = \sin x + \cos x \)
**Domain:** \( (0, 2\pi) \)
---
This content appears to focus on the trigonometric function \( f(x) = \sin x + \cos x \) over the interval \( (0, 2\pi) \).
### Exploration of the Function
1. **Understanding the Graph and Behavior:**
- The function is composed of the sine and cosine functions, which are periodic with a period of \( 2\pi \).
- Analyzing this on the interval \( (0, 2\pi) \), we expect the function to exhibit wave-like characteristics due to the nature of the sine and cosine components.
2. **Property Analysis:**
- **Amplitude and Phase Shift:** Both sine and cosine have an amplitude of 1. The resulting function \( \sin x + \cos x \) can be rewritten using the amplitude and phase shift formula:
\[
f(x) = \sqrt{2}\sin(x + \frac{\pi}{4})
\]
Thus, the amplitude is \(\sqrt{2}\), and there is a phase shift of \(\frac{\pi}{4}\).
- **Zeros and Critical Points:** Solving \( \sin x + \cos x = 0 \) within this interval will provide the zeros, and taking the derivative to find critical points will show where the function reaches maximum or minimum values.
3. **Practical Applications:**
- Understanding oscillations in wave motion.
- Analyzing alternating current electricity outcomes.
---
This content can be used on an educational website to teach students about trigonometric functions, their transformations, and their implications in real-world scenarios.
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