\magnification 2200 \parindent 0pt \parskip 10pt \pageno=1 \hsize 7truein \vsize 9.2truein \def\u{\vskip -10pt} \def\v{\vskip -6pt} 6.3: Step functions. $u_c(t) = \cases{0 & $t < c$ \cr 1 & $t \geq c$}$ 1.) Graph $u_c(t)$: \vskip 0.99in 2.) Given $f$, graph $u_c(t) f(t-c)$: \vskip 0.8in \eject 3.) Calculate ${\cal L}(u_c(t) f(t-c))$ in terms of ${\cal L}(f(t))$: \vfill \vfill Example: Find the LaPlace transform of 4.) \centerline{$g(t) = \cases{0 & $t < 3$ \cr e^{t-3} & $t \geq 3$}$} 5.) \centerline{$f(t) = \cases{0 & $t < 3$ \cr 5 & $3 \leq t < 4$ \cr t - 5 & $t \geq 4$}$} 6.) Example: Find the inverse Laplace transform of ${e^{-8s} \over s^3 } 7.) Calculate ${\cal L}(e^{ct} f(t))$ in terms of $F(s) = {\cal L}(f(t))$ \vfill 8.) Example: Use formula 6 (p. 304) to find the inverse LaPlace transform of ${s - c \over (s-c)^2 + a^2}$. \eject \end