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### Problem 7b:
**Obtain the largest possible domain and corresponding range of the function**  
\[ f(x) = \frac{x - 2}{3 - x} \]

**Solution:**
- The function is defined as long as the denominator is not zero, i.e., \(3 - x \neq 0\). Therefore, \(x \neq 3\).
- Domain: All real numbers except \(x = 3\), i.e., \(\mathbb{R} \setminus \{3\}\).
- To find the range, rewrite \(f(x)\) in the form of \(y = \frac{x - 2}{3 - x}\). Solving for \(x\) in terms of \(y\), we get:  
  \[ y(3 - x) = x - 2 \implies x(y + 1) = 3y + 2 \implies x = \frac{3y + 2}{y + 1}. \]  
  The range is all real numbers except \(y = 1\).

### Problem 7c:
**Expand \( e^{2x} \) in powers of \((x - 1)\), up to four terms.**

**Solution:**
Using the Maclaurin series for \(e^{x}\):  
\[ e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]  
Substituting \(x = 2(x - 1) + 2\) into the series and expanding up to four terms:  
\[ e^{2x} = e^{2(x-1)} = 1 + 2(x - 1) + \frac{(2(x - 1))^2}{2!} + \frac{(2(x - 1))^3}{3!} + \cdots. \]

### Problem 7a:
**Verify Rolle’s theorem for the function \( f(x) = x(x - 2)e^{-x} \) on the interval \([0, 2]\).**

**Solution:**
1. Check continuity on \([0, 2]\) and differentiability on \((0, 2)\).
2. Verify \(f(0) = f(2)\).
3. Find \(f'(x)\) and solve \(f'(x) = 0\) within the interval.

### Problem 7b (Continued):
**Is the function \( f(x) = \frac{x^2 - 5x + 4}{x^2 - 16}, x \neq 4 \), continuous at \(x = 4\)?**

**Solution:**
1. Simplify \(f(x)\) to \(f(x) = \frac{(x - 4)(x - 1)}{(x - 4)(x + 4)}\). When \(x \to 4\), check continuity by direct substitution.
   
### Problem 7c (Continued):
**Evaluate \(\int_{0}^{1} x^2 e^{3x} \, dx\).**

**Solution:**
Use integration by parts with \(u = x^2\) and \(dv = e^{3x} dx\).

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Would you like me to continue solving each part in detail, or focus on a specific problem? Here's a list of further expansions I can work on:

1. Detailed evaluation of each integral.
2. Verification of continuity and Rolle’s theorem step-by-step.
3. Expansion and series solutions for exponential functions.
4. Techniques for finding ranges and solving domain restrictions.
5. Graphical interpretation and deeper insights into the behavior of each function.

**Tip:** Always check domain restrictions carefully when dealing with rational functions.

Explore a comprehensive set of calculus problems involving Rolle’s theorem, series expansions, integration techniques, and curve analysis. This guide includes step-by-step solutions to evaluating limits, verifying continuity, finding derivatives, and more. Suitable for undergraduate calculus students.

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