15.7.2. First Shifting Theorem

Try these first

1. 1.

   What is the First Shifting Theorem?

   Hint

   Think about how the Laplace transform changes with exponential functions.

2. 2.

   If f(t) = cos(t), how would you express L{e^{5t}cos(t)}?

   Hint

   Remember to identify the original function before applying the theorem.

3. 3.

   What does the First Shifting Theorem state?

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

   Hint

   Focus on the relationship between exponential and Laplace transforms.

4. 4.

   True or False: The First Shifting Theorem is used solely in theoretical mathematics.

   • True
   • False

   Hint

   Think about where Laplace transforms are applied.

5. 5.

   Apply the First Shifting Theorem to the function f(t) = e^{4t} * t^3 and find its Laplace transform.

   Hint

   Calculate the basic Laplace transform first.

6. 6.

   Explain how the First Shifting Theorem applies when analyzing a damped harmonic oscillator with an exponential decay factor added.

   Hint

   Consider how damping impacts frequency responses.