Let's solve each problem step-by-step. Below, I provide detailed solutions for each problem from the uploaded image.

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 $dv = e^{3x} dx$.

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.