8.1.1. Introduction

Try these first

1. 1.

   What is the Laplace transform used for?

   Hint

   Think about the earlier discussions regarding time and frequency domains.

2. 2.

   State the Division by t Rule.

   Hint

   Recall the mathematical formulation we discussed.

3. 3.

   What does the Division by t Rule correspond to in Laplace transforms?

   • Differentiation in s-domain
   • Integration in s-domain
   • Addition in s-domain

   Hint

   Remember the relationship discussed in class.

4. 4.

   True or False: The Division by t rule can be applied to any time function.

   • True
   • False

   Hint

   Consider the conditions we outlined for applying the rule.

5. 5.

   Calculate the Laplace transform of e^(bt)/t using the division by t rule and simplify the result.

   Hint

   Think about how e^(bt) behaves in the s-domain.

6. 6.

   Prove that L{f(t)/t} = L{f(t)} when f(t) is exponentially bounded as t approaches infinity.

   Hint

   Apply the fundamentals of convergence and bounded functions.