In Exercises 1-6 find the Laplace transform by the method of Example 8.4.1. Then express the given function f in terms of unit step functions as in Eqn. (8.4.6), and use Theorem 8.4.1 to find C(f). Where indicated by C/G graph f.\fIn Exercises 7-18 express the given function f in terms of unit step functions and use Theorem 8.4.1 to find L(f). Where indicated by C/G , graph f.0, 0 2. tet, 0 so. Then C (u(t - T)g(t)) exists for s > so, and Clu(t - T)g(t)) = e-TC (g(t + T)). Proof By definition, -DO Clu(t - T)g(t)) = estu(t - T)g(t) dt. From this and the definition of u(t - T), Clu(t - T)g(t)) = / e-st (0) di + / estg(t) dt. The first integral on the right equals zero. Introducing the new variable of integration r = t - 7 in the second integral yields Clu(t - T)g(t)) = es(+7)g(x + +) do = e-*T e-sz g(* + 7) dx. Changing the name of the variable of integration in the last integral from r to t yields Clu(t - T)g(t) ) = e-ST e g(t + T) dt = e- C(g(t + +)). _