1. Unit 1: Laplace Transforms & Applications

Try these first

1. 1.

   What is the First Shifting Theorem?

   Hint

   Think about how an exponential affects the transform.

2. 2.

   Provide the condition needed for the First Shifting Theorem to hold.

   Hint

   Consider what the Laplace Transform needs to work.

3. 3.

   What does the First Shifting Theorem indicate?

   • It results in a vertical shift
   • It results in a horizontal shift
   • It has no effect

   Hint

   Review what happens when functions are modified.

4. 4.

   True or False: The Laplace Transform of $e^{-at}f(t)$ results in $F(s + a)$.

   • True
   • False

   Hint

   Recollect the shifting directions.

5. 5.

   Find the Laplace Transform of $e^{2t} ext{cos}(3t)$ and explain your steps.

   Hint

   Start from the basic Laplace transform of cos.

6. 6.

   A function $f(t) = t^3$ is affected by an exponential decay $e^{-kt}$. Determine the Laplace transform of this entire function.

   Hint

   Recall the shifts and how the multipliers affect the base function.