1.7. Application Scenarios

Try these first

1. 1.

   Find the Laplace Transform of e^3t * cos(5t).

   Hint

   Start with the transform of cos(bt).

2. 2.

   What shift do we apply for e^{-2t} f(t)?

   Hint

   Remember the sign when applying the shift.

3. 3.

   What is the result of applying the First Shifting Theorem to e^{2t} * sin(3t)?

   • s^2 + 9
   • (s-2)^2 + 9
   • (s+2)^2 + 9

   Hint

   Remember the shift rule defined by the theorem.

4. 4.

   True or False: The conditions s > Re(a) must be met when applying the First Shifting Theorem.

   • True
   • False

   Hint

   Think about the nature of the exponentials involved.

5. 5.

   Using the First Shifting Theorem, compute the Laplace Transform of e^{-2t}cos(6t).

   Hint

   What is the standard Laplace Transform for cos(bt)?

6. 6.

   Explain how the presence of e^{4t} affects the stability of the system modeled by the equation.

   Hint

   Consider the implications of shifting the transform.