What is the average rate of change of y = cos(2x) on the interval 0,? a. = b. -1 с. О d. 4 e.
What is the average rate of change of y = cos(2x) on the interval 0,? a. = b. -1 с. О d. 4 e.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Average rate of change?
![### Problem Statement
**Question:**
What is the average rate of change of \( y = \cos(2x) \) on the interval \([0, \frac{\pi}{2}]\)?
### Options:
a. \(- \frac{4}{\pi}\)
b. \(-1\)
c. \(0\)
d. \(\frac{\sqrt{2}}{2}\)
e. \(\frac{4}{\pi}\)
### Explanation:
To find the average rate of change of a function \( f(x) \) over an interval \([a, b]\), use the formula:
\[
\frac{f(b) - f(a)}{b - a}
\]
For the function \( y = \cos(2x) \) on \([0, \frac{\pi}{2}]\):
1. **Calculate \( f(a) \):**
\( f(0) = \cos(2 \times 0) = \cos(0) = 1 \)
2. **Calculate \( f(b) \):**
\( f\left(\frac{\pi}{2}\right) = \cos\left(2 \times \frac{\pi}{2}\right) = \cos(\pi) = -1 \)
3. **Apply the formula:**
\[
\text{Average Rate of Change} = \frac{-1 - 1}{\frac{\pi}{2} - 0} = \frac{-2}{\frac{\pi}{2}} = -\frac{4}{\pi}
\]
Thus, the correct answer is **a. \(- \frac{4}{\pi}\)**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7616f65-0250-4f00-8be8-7f3ea815d241%2F9d0805f2-fd6f-4990-a863-8267745ac049%2F3vrzf6_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Question:**
What is the average rate of change of \( y = \cos(2x) \) on the interval \([0, \frac{\pi}{2}]\)?
### Options:
a. \(- \frac{4}{\pi}\)
b. \(-1\)
c. \(0\)
d. \(\frac{\sqrt{2}}{2}\)
e. \(\frac{4}{\pi}\)
### Explanation:
To find the average rate of change of a function \( f(x) \) over an interval \([a, b]\), use the formula:
\[
\frac{f(b) - f(a)}{b - a}
\]
For the function \( y = \cos(2x) \) on \([0, \frac{\pi}{2}]\):
1. **Calculate \( f(a) \):**
\( f(0) = \cos(2 \times 0) = \cos(0) = 1 \)
2. **Calculate \( f(b) \):**
\( f\left(\frac{\pi}{2}\right) = \cos\left(2 \times \frac{\pi}{2}\right) = \cos(\pi) = -1 \)
3. **Apply the formula:**
\[
\text{Average Rate of Change} = \frac{-1 - 1}{\frac{\pi}{2} - 0} = \frac{-2}{\frac{\pi}{2}} = -\frac{4}{\pi}
\]
Thus, the correct answer is **a. \(- \frac{4}{\pi}\)**.
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