9.1.3. Proof

Try these first

1. 1.

   State the definition of the unit step function.

   Hint

   Think about what happens before and after time a.

2. 2.

   What is the Laplace Transform of u(t)?

   Hint

   Recall how the unit step function behaves from a time perspective.

3. 3.

   What is the Laplace Transform of u(t-a)?

   • 0
   • 1
   • e^{-as}/s

   Hint

   Recall the definition of the transform.

4. 4.

   True or False: The unit step function can only take values 0 or 1.

   • True
   • False

   Hint

   Think about the properties of the unit step function.

5. 5.

   For a function f(t) = sin(t) and a = 2, derive the Laplace Transform of f(t)u(t-2).

   Hint

   Think about the second shifting theorem and how it applies here.

6. 6.

   Design a control system using u(t-1) that incorporates a delayed response. What does your Laplace Transform yield?

   Hint

   Consider the implications of your delay in a real system.