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This document is the midterm examination for the course *Differential and Integral Calculus (MAT141)* at Southeast University, Summer 2024. It contains eight calculus-based problems. Here's a summary of the questions:

1. **Function Properties**: The function \( y = \ln(e^{x} + 1) \)
    - Show the function is odd.
    - Find its inverse.
    - Determine its domain and range.
    - Differentiate \( y \) with respect to \( x \).

2. **Limit Problem**: Solve \( \lim_{x \to \infty} (a - x)(b - x) \).

3. **Continuity**: For a piecewise function \( f(x) \), determine if it's continuous at \( x = 3 \).

4. **Limit and Continuity**: Find the value of \( b \) such that \( \lim_{x \to 1} f(x) \) exists.

5. **Differentiation**: Prove a derivative involving trigonometric functions \( y = \sin(x + \sin y) \).

6. **Parametric Differentiation**: Show a second-order differential equation using parametric equations \( x = \sin t \) and \( y = \sin(kt) \).

7. **Inverse Trigonometry Derivative**: Differentiate \( y = \frac{1}{2} \tan^{-1} \left( \frac{1}{x} + x \right) \).

8. **Reflection**: Describe any problems faced and solutions during the exam.

Would you like to see detailed solutions to these problems? Here are five relative questions to expand your understanding:

1. How do you prove that a function is odd?
2. What is the process for finding the inverse of a logarithmic function?
3. What conditions ensure the continuity of a piecewise function?
4. How does one differentiate trigonometric functions involving both \( x \) and \( y \)?
5. What is parametric differentiation and how is it used in second-order differential equations?

**Tip**: For proving continuity, always check the function's value at the point and its limit from both sides.

This document is the midterm examination for the course Differential and Integral Calculus (MAT141) at Southeast University, Summer 2024. It contains eight calculus-based problems.

This midterm exam features calculus problems including logarithmic functions, limits, continuity, and parametric differentiation. It assesses undergraduate students' understanding of core concepts such as the odd function theorem, limit theorem, and chain rule.

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