15.9. Laplace Transform of Standard Functions

Try these first

1. 1.

   What is the Laplace transform of a constant function f(t) = 1?

   Hint

   Think about how constants behave in the s-domain.

2. 2.

   What is the Laplace transform for f(t) = t?

   Hint

   Consider how polynomial transforms relate to factorials.

3. 3.

   What is the Laplace transform of f(t) = cos(at)?

   • a/(s^2 + a^2)
   • s/(s^2 + a^2)
   • 1/s

   Hint

   Focus on the placement of 's' and 'a' in the formula.

4. 4.

   True or False: The Laplace transform of f(t) = t^n is given by n!/s^(n+1).

   • True
   • False

   Hint

   Try recalling the form of polynomial transformations.

5. 5.

   Find the Laplace transform of the function f(t) = t^3 + e^(2t) + sin(5t).

   Hint

   Break it down into pieces using known transforms for each part.

6. 6.

   A function is defined as f(t)= { 0, t<0; t^2, t≥0; e^(2t)sin(t), t≥0 }. Compute its Laplace transform.

   Hint

   You can handle each piece separately and apply known transforms!