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You have provided two functions to analyze:

27. \( f(x) = \frac{1}{x^2 + 1} \) on the interval \([-1, 1]\)
28. \( f(x) = \frac{1}{x} \) on the interval \([1, e]\)

Let me know if you would like me to solve a specific problem related to these functions, such as integration, finding derivatives, analyzing their behavior, or any other operation!

For now, here are five related questions:

1. What is the derivative of \( f(x) = \frac{1}{x^2 + 1} \) on \([-1, 1]\)?
2. Can you calculate the integral of \( f(x) = \frac{1}{x^2 + 1} \) on \([-1, 1]\)?
3. What is the limit of \( f(x) = \frac{1}{x} \) as \( x \) approaches 1 from the right on \([1, e]\)?
4. How does the function \( f(x) = \frac{1}{x^2 + 1} \) behave in terms of concavity and critical points?
5. Can we find the area under the curve for \( f(x) = \frac{1}{x} \) on \([1, e]\)?

### Tip: For rational functions, it's useful to identify key points such as limits, vertical asymptotes, and intervals of increasing/decreasing behavior to understand their overall structure.

27. f(x) = 1/(x^2 + 1) on [-1, 1]
28. f(x) = 1/x on [1, e]

This problem explores the rational functions f(x) = 1/(x^2 + 1) on the interval [-1, 1] and f(x) = 1/x on [1, e]. Learn how to find derivatives, integrals, and limits for these functions. The problem is suitable for students in Grades 11-12 or early college-level calculus.

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