9.1.5.2. Time-Shift for General Function

Try these first

1. 1.

   What defines the unit step function?

   Hint

   Think about what happens at the jump point.

2. 2.

   Write the Laplace Transform of u(t-a).

   Hint

   Look for the exponential function in the formula.

3. 3.

   What is the unit step function?

   • A function that oscillates.
   • A function that stays constant.
   • A function that jumps at a specific point.

   Hint

   Focus on the defining feature of the function.

4. 4.

   True or False: The Laplace Transform is used to convert functions from the frequency domain to time domain.

   • True
   • False

   Hint

   Remember the direction of the transform.

5. 5.

   A control system has an input that is turned on at t=3 seconds and remains on indefinitely. Describe how you would model this using the unit step function and derive the Laplace Transform.

   Hint

   Start by writing down what happens at the time 3 seconds.

6. 6.

   You are given a mechanical system that each force is applied suddenly at intervals of 2 seconds. Explain how you would set up the differential equation using the Laplace Transform and the unit step function.

   Hint

   How does the forces’ sudden application affect the system's motion?