14.1. Initial Value Theorem

Try these first

1. 1.

   What is the initial value of f(t) if F(s) = 3/(s + 1)?

   Hint

   Evaluate lim (s→∞)[3s/(s + 1)].

2. 2.

   List one condition for applying the Initial Value Theorem.

   Hint

   Think about the requirements needed for the theorem.

3. 3.

   What does the Initial Value Theorem state about f(t) and F(s)?

   • a) lim(t→0+) f(t) = 0
   • b) lim(t→0+) f(t) = lim(s→∞) sF(s)
   • c) lim(s→0) F(s) = f(0)

   Hint

   Focus on the relationship between limits involving f(t) and F(s).

4. 4.

   True or False: The Initial Value Theorem can be used with functions that contain impulse functions.

   • True
   • False

   Hint

   What happens to the function's continuity?

5. 5.

   Apply the Initial Value Theorem to find the initial value of f(t) for F(s) = 6/(s^2 + 3s + 2).

   Hint

   Simplify the expression before taking the limit.

6. 6.

   Determine why the IVT fails for F(s) = e^(-s)/(s + 1) in computing f(0).

   Hint

   Evaluate the limits carefully.