1.1. Second Shifting Theorem

Try these first

1. 1.

   What is the definition of the Heaviside step function?

   Hint

   Think about its piecewise nature.

2. 2.

   When does the Second Shifting Theorem apply?

   Hint

   Remember the conditions of the theorem.

3. 3.

   What is the result of applying the Second Shifting Theorem?

   • $e^{-as}F(s)$
   • $e^{as}F(s)$
   • $F(s)e^{as}$

   Hint

   Focus on how the delay affects the transform.

4. 4.

   True or False: The Heaviside function starts its value at $t=0$.

   • True
   • False

   Hint

   Remember the piecewise definition of the step function.

5. 5.

   Using the Second Shifting Theorem, find the Laplace transform of $e^{(t-4)}u(t-4)$ and describe its significance.

   Hint

   Focus on both parts of the function during transformation.

6. 6.

   Prove that the Laplace transform of $f(t-a)u(t-a)$ leads to $e^{-as}F(s)$, detailing all steps.

   Hint

   Recall the substitution and limit adjustments.