\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