1.3. Summary

Try these first

1. 1.

   What is the Laplace Transform of the first derivative?

   Hint

   Remember the general transformation rule.

2. 2.

   What transformation does L{f(t)} yield?

   Hint

   Think about the integral of the function.

3. 3.

   What is L{f'(t)}?

   • L{f'(t)} = sF(s) - f(0)
   • L{f'(t)} = s²F(s) - f(0)
   • L{f'(t)} = F(s) - tf(0)

   Hint

   Think about the basic formula for the Laplace transformation.

4. 4.

   True or False: The Laplace Transform can only be applied to first derivatives.

   • True
   • False

   Hint

   Review the formulas for different orders of derivatives.

5. 5.

   Derive the Laplace Transform of f'(t) using integration by parts from first principles.

   Hint

   Pay close attention to the boundary terms as t approaches infinity.

6. 6.

   Prove that L{f' + g' (t)} = L{f(t)} + L{g(t)} for two functions and their derivatives.

   Hint

   Consider how the Laplace Transform can be distributed across addition.