15.7.3. Derivative Theorem

Try these first

1. 1.

   What does the Derivative Theorem allow you to do with derivatives of a function in Laplace transforms?

   Hint

   Think about how we handle initial values.

2. 2.

   Use the Derivative Theorem to find the Laplace transform of the equation y' + 4y = 0 with y(0) = 3.

   Hint

   Apply the theorem then substitute the initial conditions.

3. 3.

   What does the Derivative Theorem express for the Laplace transform of a derivative?

   • It expresses it as F(s).
   • It expresses it as a function of s only.
   • It relates it to initial conditions and F(s).

   Hint

   Think about what initial conditions the theorem considers.

4. 4.

   True or False: The Derivative Theorem is not required for solving initial value problems.

   • True
   • False

   Hint

   Consider its application in ODEs.

5. 5.

   For the function f(t) = e^{-t}, calculate L{f''(t)} using the Derivative Theorem and given f(0) = 1 and f'(0) = -1.

   Hint

   Start by finding F(s) before computing the derivatives.