7.8. Applications

Try these first

1. 1.

   What is the Laplace Transform of sin(at)?

   Hint

   Recall the standard transformation formulas.

2. 2.

   If L{f(t)} = F(s), what does L{tf(t)} equal?

   Hint

   Think about how differentiation applies in the s-domain.

3. 3.

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

   • (-1)^n (d^n F(s)/(ds^n))
   • L{f(t)}
   • F(s)*tn

   Hint

   Remember how multiplication by powers affects differentiation.

4. 4.

   True or False: The function must be continuous for applying Laplace Transforms.

   • True
   • False

   Hint

   Consider the implications of function behavior in integration.

5. 5.

   How would you apply the multiplication by tn property to model a system where input changes in time with a polynomial ramp function?

   Hint

   Focus on understanding how input affects output in the transformed space.

6. 6.

   Given a complex function, use both the Laplace Transform and multiplication by tn to produce a comprehensive system response.

   Hint

   Trace through the functions for seamless integration into systems.