Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Problem 2
The average value of \( f(x) = \frac{1}{1 + x^2} \) on the interval \([-1, 1]\) is equal to \(\frac{\pi}{4}\). Suppose \( g(x) = \frac{1}{(1 + x^2)^2} \).
#### Tasks:
(a) **Plot \( f(x) \) and \( g(x) \) on the same graph for \(-1 \leq x \leq 1\). Which function should have the smaller average value?**
(b) **Use trig substitution to find the average value of \( g(x) \) on \([-1, 1]\).**
- **Hint**: Since the function is even, you can use symmetry to make your life a bit easier.
#### Graph Explanation:
- There is no provided graph in the image, but you are instructed to plot the functions \( f(x) \) and \( g(x) \) on the same graph within the interval \([-1, 1]\).
- \( f(x) \) is given as \( \frac{1}{1 + x^2} \).
- \( g(x) \) is given as \( \frac{1}{(1 + x^2)^2} \).
By plotting these functions, visually compare their average values over the interval \([-1, 1]\).
#### Analytical Steps:
1. **Graph Plotting**:
- For \( f(x) = \frac{1}{1 + x^2} \), the function has a symmetric bell shape, peaking at \( x = 0 \).
- For \( g(x) = \frac{1}{(1 + x^2)^2} \), the function also has a symmetric bell shape, but it is narrower and taller at the peak compared to \( f(x) \).
2. **Function Comparison**:
- \( g(x) \) decays more quickly than \( f(x) \) as \( x \) moves away from zero, leading to a smaller integral value over a symmetric interval.
3. **Trig Substitution for Integration**:
- Use trigonometric substitution to integrate \( g(x) \) over \([-1, 1]\).
- The hint suggests using the symmetry property of even functions to simplify calculations (i.e.,](https://content.bartleby.com/qna-images/question/d877c616-2afd-46f9-89f1-75e994d7b3ab/8536b65a-138f-4f21-9fab-c8d9b18b945c/wg0d0s7_processed.png)
Transcribed Image Text:### Problem 2
The average value of \( f(x) = \frac{1}{1 + x^2} \) on the interval \([-1, 1]\) is equal to \(\frac{\pi}{4}\). Suppose \( g(x) = \frac{1}{(1 + x^2)^2} \).
#### Tasks:
(a) **Plot \( f(x) \) and \( g(x) \) on the same graph for \(-1 \leq x \leq 1\). Which function should have the smaller average value?**
(b) **Use trig substitution to find the average value of \( g(x) \) on \([-1, 1]\).**
- **Hint**: Since the function is even, you can use symmetry to make your life a bit easier.
#### Graph Explanation:
- There is no provided graph in the image, but you are instructed to plot the functions \( f(x) \) and \( g(x) \) on the same graph within the interval \([-1, 1]\).
- \( f(x) \) is given as \( \frac{1}{1 + x^2} \).
- \( g(x) \) is given as \( \frac{1}{(1 + x^2)^2} \).
By plotting these functions, visually compare their average values over the interval \([-1, 1]\).
#### Analytical Steps:
1. **Graph Plotting**:
- For \( f(x) = \frac{1}{1 + x^2} \), the function has a symmetric bell shape, peaking at \( x = 0 \).
- For \( g(x) = \frac{1}{(1 + x^2)^2} \), the function also has a symmetric bell shape, but it is narrower and taller at the peak compared to \( f(x) \).
2. **Function Comparison**:
- \( g(x) \) decays more quickly than \( f(x) \) as \( x \) moves away from zero, leading to a smaller integral value over a symmetric interval.
3. **Trig Substitution for Integration**:
- Use trigonometric substitution to integrate \( g(x) \) over \([-1, 1]\).
- The hint suggests using the symmetry property of even functions to simplify calculations (i.e.,
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