9.1.5.1. Linearity

Try these first

1. 1.

   What is the definition of the unit step function?

   Hint

   Think about how it models sudden changes.

2. 2.

   What is the Laplace Transform of $u(t)$?

   Hint

   Remember it's the transform of the basic unit step function.

3. 3.

   What is the Laplace Transform of $u(t-a)$?

   • \\( \\frac{1}{s} \\)
   • \\( \\frac{e^{-as}}{s} \\)
   • \\( s \\)

   Hint

   Focus on how the step function impacts transformation.

4. 4.

   True or False: The unit step function is always equal to 1.

   • True
   • False

   Hint

   Recall the conditions for the unit step function.

5. 5.

   Consider a mechanical system where a force is applied at t=2 seconds. Model the system using the unit step function and derive the Laplace Transform of the applied force.

   Hint

   Think about the timing and impact of the function before and after the shift.

6. 6.

   For the piecewise function defined as $f(t) = t \cdot u(t-1) + 2 \cdot u(t-2)$, calculate its Laplace Transform.

   Hint

   Separate each part according to the shifting theorem.