1.6. Properties of Laplace Transform (To Be Explored in Later Sections)

Try these first

1. 1.

   What does linearity in Laplace Transform imply?

   Hint

   Think about how you might combine functions in algebra.

2. 2.

   State the Initial Value Theorem.

   Hint

   Consider what happens to F(s) at 's' approaches infinity.

3. 3.

   What is the property that allows the transformation of linear combinations of functions?

   • Additivity
   • Linearity
   • Distributive

   Hint

   Consider the term that describes linear functions.

4. 4.

   True or False: The Final Value Theorem can help find the value of a function at t = ∞.

   • True
   • False

   Hint

   Think about what happens to functions at very large time values.

5. 5.

   For the functions f(t) = t^2 and g(t) = e^(-3t), apply the linearity property to compute ℒ{4f(t) - 2g(t)}.

   Hint

   Look up the transforms for t^2 and e^(-3t).

6. 6.

   Given F(s) = 4/(s^2 + 9), utilize both Initial and Final Value Theorems to find the initial and final values without deriving f(t).

   Hint

   Think about what happens as s approaches these two different infinite situations.