Question 3.4.1: Calculate the Laplace transform of the derivative f(t)....

L[\dot{f}(t)]=\int_0^{\infty} \dot{f}(t) e^{-s t} d t=\int_0^{\infty} e^{-s t} \frac{d[f(t)]}{d t} d t

Integration by parts yields

L[\dot{f}(t)]=\left.e^{-s t} f(t)\right|_0 ^{\infty}+s \int_0^{\infty} e^{-s t} f(t) d t

Recognizing that the integral in the last term of the preceding equation is the definition of F(s) yields

L[\dot{f}(t)]=s F(s)-f(0)

where F(s) denotes the Laplace transform of f(t). Repeating this procedure on \ddot{f}(t) yields

L[\ddot{f}(t)]=s^2 F(s)-s f(0)-\dot{f}(0) .