1.8. Examples

Try these first

1. 1.

   Define the Heaviside step function.

   Hint

   Consider how the function behaves over different time intervals.

2. 2.

   What does the Second Shifting Theorem state?

   Hint

   Recall the notation and the meaning of each variable.

3. 3.

   What is the equation for the Second Shifting Theorem?

   • ℒ{f(t-a)u_a(t)} = e^{-as}F(s)
   • ℒ{f(t+a)u_a(t)} = e^{as}F(s)
   • ℒ{f(t)u_a(t)} = F(s)

   Hint

   Focus on the role of the negative exponent.

4. 4.

   True or False: The Heaviside function is zero for all time.

   • True
   • False

   Hint

   Consider when it activates.

5. 5.

   What is the Laplace transform of (t-3)^3u_3(t)? Provide a detailed solution.

   Hint

   Remember to find the initial Laplace transform of t^3.

6. 6.

   Consider a system where the response to an input signal starts after a delay of 5 seconds. How would you mathematically represent the output in terms of the Second Shifting Theorem?

   Hint

   Think about how time delays linearly translate in the Laplace domain.