9. Laplace Transforms & Applications

Try these first

1. 1.

   Define the unit step function.

   Hint

   Think about its representation in terms of 'a'.

2. 2.

   What is the Laplace Transform of u(t)?

   Hint

   Consider what happens in the frequency domain.

3. 3.

   What is the Laplace Transform of the unit step function u(t-a)?

   • e^{-as}/s
   • as/e^{-s}
   • 1/s

   Hint

   Recall the general formula for shifts.

4. 4.

   True or False: The unit step function is continuous.

   • True
   • False

   Hint

   Think about how the function behaves at the switching point.

5. 5.

   For a function g(t) = 3t * u(t-4), find the Laplace Transform.

   Hint

   Remember to shift 't' by the delay 'a'.

6. 6.

   Model a physical system using the unit step function and solve its corresponding differential equation.

   Hint

   Translate the forces into the terms of the differential equation first.