1.6. Important Notes

Try these first

1. 1.

   Define the Heaviside Step Function.

   Hint

   What happens before and after time c?

2. 2.

   State the main statement of the Second Shifting Theorem.

   Hint

   Think about what happens with the function when it’s delayed.

3. 3.

   What does the Second Shifting Theorem help in modeling?

   • Functions without delays
   • Delayed activation of functions
   • Nonlinear equations

   Hint

   Focus on the 'shifting' aspect of the theorem.

4. 4.

   True or False: The Heaviside step function must be included for the Second Shifting Theorem to apply.

   • True
   • False

   Hint

   Recall what the Heaviside function actually does in modeling.

5. 5.

   Prove the Second Shifting Theorem by calculating the Laplace transform of f(t) = e^(at)u(t-b).

   Hint

   Introduce a substitution of u(t-b) correctly before integrating.

6. 6.

   Determine the Laplace transform for f(t) = sin(ω(t-5))u(t-5).

   Hint

   Standardize to sin(ωt) first before incorporating the delay.