Section 7.6  Transforms of Discontinuous and Periodic Functions

• to be able to express discontinuous functions in terms of  the step function
• to be able to use the Translation in t Theorem to determine Laplacians and their inverses
• to be able to determine the Laplacians of periodic functions

• unit step function

 
• Laplace transform of u(t-a)

 
• translation in t

 
• periodic functions

 

solved problems - using key ideas

• prove L{u(t-a)}(s) = (1/s)e^-as

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• sketching graphs involving step functions, #1,5,10
• computing their Laplacians

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• using TI-89 when (x <  3,when(x< 2,when(x < 1 , 0, 2), 1, 3)

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• determining inverse Laplacians, #11,16

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• determining the Laplacian of periodic functions, #21,22

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• solving initial value problems, #29,33,37

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

• using the Taylor series, #47

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

QUESTION OF THE DAY

Find the inverse Laplacian transform of

SOLUTION

SOLUTION.