1.9. Common Mistakes to Avoid

Try these first

1. 1.

   What happens to the Laplace Transform when applying to e^{-2t}f(t)?

   Hint

   Think about how the exponential term affects the shift.

2. 2.

   True or False: The integral converges if s < Re(a).

   Hint

   What do we need for convergence?

3. 3.

   When applying the First Shifting Theorem with e^{-4t}, what happens to s?

   • s + 4
   • s - 4
   • s

   Hint

   Think about whether you are shifting left or right.

4. 4.

   True or False: It's always okay to apply the theorem without checking s > Re(a).

   • True
   • False

   Hint

   How does convergence affect your result?

5. 5.

   Given the function f(t) = e^{-2t} sin(4t), calculate the Laplace Transform using the First Shifting Theorem.

   Hint

   Start with finding the base transform then shift it.

6. 6.

   How would you apply the theorem to calculate the Laplace Transform of e^{3t} e^{-t}cos(5t)?

   Hint

   Ensure the shifts combine correctly.