Question

Help, please.

1. In each of the following piece-wise functions: (i) Sketch the graph of the given function, (ii) Express f(t) in terms of the unit step function ue(t)=uſt - c) where ul 0 t<0 t>O and (iii) find the Laplace transform of f(t). (a) 1, 0<t<2 e-(-2), t> 2 s(t)= { (b) f(t) = t, 2, 7-t, 0, 0<t<2 2<t<5 5<t<7 t>7

Answer 1

- >ه Note : (1) a {ucty = (II) a {uct-01 (III) IF G (3) = L { get ak get-es) hen G() olt-e). é es

oct<2 (a) f(t) = -(1-2) e t.7, 2 (i) then the groaph of fot) is given by

24 0 5 10 15 -2

Νσω + e-6-2) &{{e (t-2 _ Just-235 (W) f(t) = 1. 1. Zuct ult-0) - uct-2) ult-2) = u(t) + [4-(4-2) 1] uct-2) <{f(t)} =a1 <{ucts } + 98179 So UCH + Cé(4-2) ult + 11 ė 29 ( 1) +1 -22 e 1 2 3 (4+1) -25 5+1 e 22 ta

(b) t oct<2 f(t) = 2 9 7-t 2 <tL5 54 tl t77 0 (i) Then the is groach of the function f ct

5. 0 5 10 15 - 5

Νσω f(t) = t[ult-o) - ult-2)] + 2[uct-2) - u(t-1)] + (7-D) [ ult-5) - ult -7)] + 0. uct-7) tult) & (2-t) ult-2) + (5-t) ult-5) & (7-Hult-7) = t - 0) ult-o) - (t - 2)uct - 2) - (t - 5) uct-5) + (t-tuct-7) SO cov) { fit)} =a717-9) u (t-o) – (t-2) ult-2) - (4-5) 4(7-6) + (7-7) uC+ -+5) = < 7(4-0) oct-0)} -<{(7-2)uct-2) <<<t-6)u(t-1)} +& {{t-3) uct ->

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