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429-2009-Q2key - t with x = for t we can write x s = H s s...

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580.429 SBE3 2009 Quiz 2 Name: Section meeting time: Section meeting location: (2 pts) 1. The Laplace transform of a function is defined as ! f ( s ) = exp( ! st ) f ( t ) dt 0 " # . Suppose that f ( t ) = 0 for t < a and f ( t ) = C for t ! a , with a ! 0 . What is ! f ( s ) ? ! f ( s ) = C a ! " e # st dt = # ( C / s ) e # st | a ! = ( C / s ) e # as (2 pts) 2. Suppose that ! f ( s ) = 1/ ( s + a ) . What is f ( t ) for t ! 0 ? f ( t ) = e ! at (2 pts) 3. Using ! f ( s ) = 1/ ( s + a ) and ! g ( s ) = 1/ ( s + b ) , what is the Laplace transform of C ( t ) = f ( t ! " t ) g ( " t ) d " t 0 t # ? ! C ( s ) = 1/[( s + a )( s + b )] (2 pts) 4. Given the ODE ! x ( t ) = ! ( t ) " # x ( t
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Unformatted text preview: t ) , with x = for t ! , we can write ! x ( s ) = ! H ( s ) ! ( s ) to define a transfer function ! H ( s ) . What is ! H ( s ) ? ! H ( s ) = 1 / ( s + ) (2 pts) 5. Suppose that ( t ) = cos( " t ) = Re e i t . What is ! ( s ) ? ! ( s ) = Re[1 / ( s " i )] = Re[( s + i ) / ( s 2 + 2 )] = s / ( s 2 + 2 )...
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