1.1. Topic 3: First Shifting Theorem

Try these first

1. 1.

   What is the First Shifting Theorem?

   Hint

   Think about how exponential functions influence the transforms.

2. 2.

   Identify a condition for applying the theorem.

   Hint

   Check the relationship between s and a.

3. 3.

   What is the result of applying the First Shifting Theorem to $f(t) = \sin(b t)$?

   • A. \\( \\mathcal{L}\\{e^{at} sin(bt)\\} = \\frac{b}{(s-a)^2 + b^2} \\)
   • B. \\( \\mathcal{L}\\{e^{at} sin(bt)\\} = \\frac{a}{(s+a)^2 + b^2} \\)
   • C. \\( \\mathcal{L}\\{e^{at} sin(bt)\\} = \\frac{b}{(s + a)^2 + b^2} \\)

   Hint

   Recall the basic formula for sin under the shift.

4. 4.

   True or False: The First Shifting Theorem can only be applied when s > a.

   • True
   • False

   Hint

   Consider the implications of the theorem's requirements.

5. 5.

   Derive the Laplace Transform of $e^{0.5t} * t^2$ using the First Shifting Theorem.

   Hint

   Start with the basic transform and replace \\( s \\) in the final result.

6. 6.

   Find a Laplace Transform for $e^{4t} * an(t)$ using theoretical explanations.

   Hint

   Consider the behavior of tan in Laplace transforms and apply the theorem correctly.