Let $g(t)=df(t)/dt$ and $F(s)=(1+s)/(s^2+12s+32)$ be the Laplace transform of f(t). Find g(0), interpreted as the right-hand derivative of the ordinary time function.
The initial value f(0⁺)=lim sF(s)=1. For the right derivative, apply the initial-value theorem to sF(s)−f(0⁺): g(0⁺)=lim s[sF(s)−1]. Expanding the numerator gives s[(-11s−32)/(s²+12s+32)], tending to −11. Thus A. This excludes a causal extension’s impulse at the origin.
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