Which of the following statements regarding Laplace transforms are CORRECT?
Options:
(A) The Laplace transform is a linear operator: $\mathcal{L}\{a f(t) + b g(t)\} = a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\}$.
(B) $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$ assuming $f(t)$ is continuous.
(C) According to the Final Value Theorem, $\lim_{t \to \infty} f(t) = \lim_{s \to 0} s F(s)$ if all poles of $s F(s)$ lie in the left half of the $s$-plane.
(D) $\mathcal{L}\{1\} = \frac{1}{s^2}$ for all $s > 0$.
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
Statements A, B, and C are standard operational theorems in Laplace transform theory. Statement D is false because $\mathcal{L}\{1\} = \frac{1}{s}$ (whereas $\mathcal{L}\{t\} = \frac{1}{s^2}$).