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**Unformatted text preview: **—————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 254 Chapter Six Section 6.1 3. The function is 0 > a b continuous . 4. The function has a at . 0 > > œ " a b jump discontinuity 7. Integration is a linear operation. It follows that ( ( ( ( ( ! ! ! E E E => ,> => ,> => ! ! E E ,= > ,= >-9=2 ,> † / .> œ / † / .> / † / .> " " # # œ / .> / .> Þ " " # # a b a b Hence ( – — – — ! E => ,= E ,= E-9=2 ,> † / .> œ Þ " " / " " / # = , # = , a b a b Taking a , as , limit E p _ —————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 255 ( ” • ” • ! _ => # #-9=2 ,> † / .> œ " " " " # = , # = , œ Þ = = , Note that the above is valid for . = , k k 8. Proceeding as in Prob. , ( ( – — – — ! E => ,= E ,= E =382 ,> † / .> œ Þ " " / " " / # = , # = , a b a b Taking a , as , limit E p _ ( ” • ” • ! _ => # # =382 ,> † / .> œ " " " " # = , # = , œ Þ , = , The limit exists as long as . = , k k 10. Observe that It follows that / =382 ,> œ / / Î# Þ +> Ð+,Ñ> Ð+,Ñ> ˆ ‰ ( – — – — ! E +> => +,= E ,+= E / =382 ,> † / .> œ Þ " " / " " / # = + , # = , + a b a b Taking a , as , limit E p _ ( ” • ” • a b ! _ +> => # # / =382 ,> † / .> œ " " " " # = + , # = , + œ Þ , = + , The limit exists as long as . = + , k k 11. Using the of the Laplace transform, linearity _ _ _ c d ‘ ‘ =38 ,> œ / / Þ " " #3 #3 3,> 3,> Since ( ! _ +3, > => / / . > œ " = + 3, a b , we have —————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 256 ( ! _ „ 3,> => / / . > œ " = … 3, . Therefore _ c d ” • =38 ,> œ " " " #3 = 3, = 3, œ , = , # # . 12. Using the of the Laplace transform, linearity _ _ _ c d ‘ ‘-9= ,> œ / / Þ " " # # 3,> 3,> From Prob. , we have "" ( ! _ „ 3,> => / / . > œ " = … 3, . Therefore _ c d ” •-9= ,> œ " " " # = 3, = 3, œ = = , # # . 14. Using the of the Laplace transform, linearity _ _ _ ‘ ‘ ‘ / -9= ,> œ / / Þ " " # # +> +3, > +3, > a b a b Based on the integration in Prob. , "" ( !...

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