The Laplace transform of a function of time f(t) is a function of a new variable s and is designated as (1) u(t — s) (2) o(s) (3) F(s) (4) C" 2. Which of the following is the definition of the Laplace...

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Answered Same DayDec 21, 2021

Answer To: The Laplace transform of a function of time f(t) is a function of a new variable s and is designated...

Robert answered on Dec 21 2021
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1. The Laplace transform of a function of time  f t is a function of a new variable s and is
designated as  F s .
2. The definition of the Laplace transform of  f t is

     
0
.stf t F s f t e dt

    L
3. The Laplace transform of the sum of several functions
is equal to the sum of the
individual transforms. The Laplace transform of the product of several functions is not
equal to the product of the individual transforms.
4. The Laplace transform of    20f t u t is given by

   
 
 
0
0
0
0
20
20
20
20
.
st
st
st
st
F s f t e dt
u t e dt
u t e dt
e dt
s

















5. The Laplace transform of   23 tf t e is given by

   
 
0
2
0
2
0
3
3
3
.
2
st
t st
s t
F s f t e dt
e e dt
e dt
s



 

 








6. The Laplace transform of    10sin 2 30f t t   is given by [1]

   
 
 
 
   
0
0
0
0
2 2
2
2
10sin 2 30
10 sin 2 cos30 cos 2 sin 30
5 3 sin 2 cos 2
5 3 sin 2 5 cos 2
2
5 3 5
4 4
5 10 3
4
5 17.32
.
4
st
st
st
st
F s f t e dt
t e dt
e t t dt
e t t dt
t t
s
s s
s
s
s
s









  
  
 
 
   
    
    
 


 






L L
7. The Laplace transform of

   
0
t
g t f u du 
is given by
   
   
    
   
 
 
0
0
0
0
0
1
1 1
1 1
0 0
0
.
st
st
st st
t
st
G s g t e dt
g t d e
s
g t e e d g t
s s
g f t e dt
s s
F s
s
F s
s






 




 
    
     
 





8. We are given the function

 
2
2
3
.
1 s
s s
G s
e



We wish to compute  
0
lim .
s
G s

Using L’Hopital’s rule, we obtain
  20 0
2 3 3
lim lim 1.5.
2 2ss s
s
G s
e 
 
   
 
9. We wish to find the Laplace transform of the periodic waveform shown below:



We have
       2sin 2 2 1 ,
n
f t t u t n u t n


      
whence
   
     
 
   
   
   
0
0 0
2 1
0 2
2 1
0 2
2 1
0
2
2 1 2 12 2
2 sin 2 2 1
2 sin
1
1
1
st
st
n
n
st
n n
n
s i t s i t
n n
n
s i t s i t
n
t n
n s n sns ns
F s f t e dt
t e u t n u t n dt
t e dt
e e dt
i
e e
i s i s i
e e e e
i s i s
 
 


 










   


   


    

      

  
 
 
  
    
   
 
   


 
 

0
0
2 2
0
2 2
0
1 1 1
1
2
.
n
ns
n
ns
n
ns
n
i
e
i s i s i
s i s i
e
i s
e
s

 
 














 
 
 
 
  
  
   
  
 






10. A voltage   2 32 tv t t e is passed through a delay line having a delay of 4 s. We wish to...
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