7.7. Proof (for n=1)

Try these first

1. 1.

   What is the Laplace Transform of 1?

   Hint

   Think of the integral from 0 to infinity.

2. 2.

   What is L{t}?

   Hint

   Remember the basic derivatives.

3. 3.

   What is the result of L{t * f(t)}?

   • F(s)
   • - (d/ds)F(s)
   • 0

   Hint

   Consider the relationship from the formula.

4. 4.

   The alternating sign in the multiplication property comes from:

   • True
   • False

   Hint

   Think about differentiation rules.

5. 5.

   If L{f(t)} = sin(t), compute L{t * f(t)} and provide step-by-step justification.

   Hint

   Follow through the differentiation carefully.

6. 6.

   For the function e^{-t} f(t) = e^{-t} * t^2, find the Laplace Transform and express using multiplication property.

   Hint

   Carefully differentiate according to the rules.