Let L{f(1)} be the Laplace Transform of a function f(t), defined as L{f(!)} = [ f(t)e*...

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Let L{f(1)} be the Laplace Transform of a function f(t), defined as L{f(!)} = [ f(t)e*" dt = F(s), say, in terms of a complex parameter s. (a) Show, from first principles (i.e., using the given integral definition of the Laplace Transform operator), that L{t} = 1/s', stating all necessary steps in the proof. (15 marks) (b) Hence, or otherwise, establish further that L{t } = 2/s'. Let L{f(1)} be the Laplace Transform of a function f(t), defined as L{f(!)} = [ f(t)e*" dt = F(s), say, in terms of a complex parameter s. (a) Show, from first principles (i.e., using the given integral definition of the Laplace Transform operator), that L{t} = 1/s', stating all necessary steps in the proof. (15 marks) (b) Hence, or otherwise, establish further that L{t } = 2/s'.