If $f(t)$ is continuous for $t \ge 0,$ the {\it Laplace transform} of $f$ is the function $F$ defined by $$F(s)=\int_{0}^{\,\infty} {f(t)e^{-st}}\, dt$$ and the domain of $F$ is the set consisting of all number $s$ for which the integral converges.

(a) Find the Laplace transform of $f(t)=1.$ (Make sure you can state the domain of $F$ if we ask for it later!)

$F(s) =$

(b) Find the Laplace transform of $f(t)=e^t.$ (Make sure you can state the domain of $F$ if we ask for it later!)

$F(s) =$

(c) Find the Laplace transform of $f(t)=t.$ (Make sure you can state the domain of $F$ if we ask for it later!)

$F(s) =$