7. Given that {{cos bi} (s) = - S 2 + 12 : use the translation property to compute f (e at cos bt} Click here to view the table of properties of Laplace transforms. 13 the at cos bt} (s) = 13: Properties of Laplace Transforms { {f + g) = {{f) + {{g) {{cf) = cf {f) for any constant c fe atf(t)} (s) = Lif)(s - a f (f } (s) = siff}(s) - f(0) e (f'') ( 5 ) = 52 elf} (s ) - sf ( 0 ) - f' ( 0 ) effin/}(s) = s" eff)(s)- s" -1f(0) - sn - 2f'(0) - ... - f(n -1)(0) * (in f(1 ) (s ) = ( - 1 jn d" dan ,(s)) 8. Use the formula 1 {t" f(t)} (s) = (-1)n _do den ((f,(s) to help determine the following the expressions. (a) {{t cos bt) (b) 2 {th cos bt} Click here to view the table of Laplace transforms. 14 (a) {{t cos bt} (s) = (b) 2 {12 cos bt} (s) = 14: Table of Laplace Transforms f(t) F(s) = 2{f)(s) 1 at 1 s- a S >0 n! th, n = 1,2,... n+ 1, $ >0 b sin bt 52 + 1 2 5>0 S cos bt ni e ath , n = 1,2,... (S - a)n + 1 : $ >a b e at sin bt (s - a) + 2 : s>a s - a e at cos bt (s- a)2 + 2 : 5 - a