1.6. Proof of the First Shifting Theorem

Try these first

1. 1.

   What is the Laplace Transform of e^(2t)?

   Hint

   Use the basic exponential Laplace Transform.

2. 2.

   What happens to the Laplace Transform when multiplying by e^(at)?

   Hint

   Refer to the theorem's statement.

3. 3.

   What is the result of applying the First Shifting Theorem if ℒ{f(t)} = F(s)?

   • F(s + a)
   • F(s - a)
   • F(2s)

   Hint

   Think about how e^(at) affects the Laplace variable.

4. 4.

   True or False: Using the theorem, ℒ{e^(-2t)cos(t)} results in F(s + 2).

   • True
   • False

   Hint

   Recall the correct direction of the shift.

5. 5.

   Using the First Shifting Theorem, derive the transform for e^(3t)sin(2t).

   Hint

   First find the transform of sin(2t) then apply the theorem.

6. 6.

   Find the Laplace Transform of e^(-5t)(t^3).

   Hint

   Apply integration by parts, track using the theorem step-by-step.