1.5. Proof of the Second Shifting Theorem

Try these first

1. 1.

   What does the Heaviside step function represent?

   Hint

   Think about when signals start.

2. 2.

   State the Second Shifting Theorem.

   Hint

   Focus on the relationship between original and shifted functions.

3. 3.

   What is the Laplace transform of a delayed function?

   • It does not exist
   • It is e^(-as)F(s)
   • It is F(s)/e^(-as)

   Hint

   Recall the theorem's statement.

4. 4.

   True or False: The Heaviside function is used for functions that start at a given time.

   • True
   • False

   Hint

   Think about its definition and purpose.

5. 5.

   Determine the Laplace transform of the function u(t - 4)(t - 4)(t - 5).

   Hint

   Start by identifying the base function you'll transform.

6. 6.

   Prove how the Heaviside function alters the transform for f(t) = e^(-t) and if it affects its existing properties.

   Hint

   Consider the definition and behavior of the Heaviside function in your proof.