1.1.4. Proof of First Derivative

Try these first

1. 1.

   What is the Laplace Transform of f(t) = t?

   Hint

   Use the basic formula of Laplace Transform.

2. 2.

   Write the formula for L{f'(t)}.

   Hint

   Recall the transformation equation discussed.

3. 3.

   What is the Laplace Transform of f'(t)?

   • sF(s) + f(0)
   • sF(s) - f(0)
   • 0

   Hint

   Remind yourself of the transformation formulas.

4. 4.

   True or False: The Laplace Transform can only be used for first derivatives.

   • True
   • False

   Hint

   Think about the different orders of derivatives we've covered.

5. 5.

   Given f(t) = e^{at} sin(bt), find the Laplace Transform using properties of derivatives implicitly.

   Hint

   Use the formula for the Laplace Transform of functions multiplied by an exponential.

6. 6.

   Prove the formula for L{f(n)(t)} = s^nF(s) - Σ from k=0 to n-1 of s^{n-1-k} f^(k)(0).

   Hint

   Work step-by-step through the integrations as shown in the previous examples.