mid1sol2 - Math 421 Midterm #1 Solutions 1. Function f ( t...

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Unformatted text preview: Math 421 Midterm #1 Solutions 1. Function f ( t ) is defined as follows: f ( t ) = , t < 1 , t + 1 , 1 ≤ t ≤ 2 , e- t , t > 2 . (a) Sketch the function f ( t ). (b) Express f ( t ) in terms of the Heaviside functions. f ( t ) = [ H ( t- 1)- H ( t- 2)]( t + 1) + H ( t- 2) e- t . (c) Calculate the Laplace transform of f ( t ). L [ f ] = L [ H ( t- 1)( t + 1)]-L [ H ( t- 2)( t + 1)] + L [ H ( t- 2) e- t ] . L [ H ( t- 1)( t +1)] = L [ H ( t- 1)( t- 1+2)] = L [ H ( t- 1)( t- 1)]+2 L [ H ( t- 1)] = e- s 1 s 2 +2 e- s 1 s . Similarly, L [ H ( t- 2)( t + 1)] = e- 2 s 1 s 2 + 3 e- 2 s 1 s . Finally, L [ H ( t- 2) e- t ] = e- 2( s +1) s + 1 , by 1st shifting theorem. All together, L [ f ] = e- s 1 s 2 + 2 e- s 1 s- e- 2 s 1 s 2- 3 e- 2 s 1 s + e- 2( s +1) s + 1 . 1 2. Find the inverse Laplace transform of the functions F ( s ) = 1 1- s 2 , and G ( s ) s 1- s 2 . By partial fractions, L- 1 " 1 (1- s )(1 + s ) # = 1 2 L- 1 • 1 1- s + 1 1 + s ‚ = 1 2 ‡- e t + e- t · = f ( t ) . Similarly, L- 1 " 2 (1- s )(1 + s ) # = 1 2 L- 1 • 1 1- s- 1 1 + s ‚ = 1 2 ‡- e t- e- t · = g ( t ) . (Note that f ( t ) =- sinh t and g ( t ) =- cosh t .) 2 3.Calculate Z ∞ µ δ ( t- π ) sin t 2 + 4 H ( t- 3) δ ( t- 7) ¶ dt....
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This note was uploaded on 01/12/2010 for the course MATH 421 taught by Professor Nataliakomarova during the Winter '07 term at San Jose State University .

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mid1sol2 - Math 421 Midterm #1 Solutions 1. Function f ( t...

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