1.2. Conditions for Existence (Dirichlet Conditions)

Try these first

1. 1.

   What does piecewise continuity mean?

   Hint

   Think about the continuity of the function.

2. 2.

   State the Dirichlet Conditions.

   Hint

   Consider which two major conditions we discussed.

3. 3.

   What is a requirement for a function to possess a Laplace Transform?

   • It must be continuous everywhere
   • It must be piecewise continuous
   • It can have infinite discontinuities

   Hint

   Look back at the first condition we discussed.

4. 4.

   True or False: A function growing faster than e^(2t) can be transformed using Laplace.

   • True
   • False

   Hint

   Recall the defining rules about growth.

5. 5.

   Evaluate if the function f(t) = { 3t for 0 ≤ t < 5; t^2 - 10 for t ≥ 5 } meets the Dirichlet Conditions.

   Hint

   Analyze the function parts for continuity and check against the nature of growth.

6. 6.

   Given f(t) = sin(t) + e^(t^2), determine if it is suitable for Laplace Transform and explain.

   Hint

   Consider the rapid increase of e^(t^2) compared to its constant multiplicative limits.