7.5. Understanding the Formula

Try these first

1. 1.

   What does L{tf(t)} equal if L{f(t)} = F(s)?

   Hint

   Use the differentiation property.

2. 2.

   What is the basic definition of Laplace Transform?

   Hint

   Consider the integral representation.

3. 3.

   What is the formula for L{tnf(t)}?

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

   Hint

   Recall the alternating sign involved in the formula.

4. 4.

   True or False: The multiplication by tn property simplifies solving polynomial functions.

   • True
   • False

   Hint

   Think about its applications in differential equations.

5. 5.

   Given f(t) = cos(bt), calculate L{t^3*cos(bt)}. Show all steps.

   Hint

   You’ll need to differentiate L{cos(bt)} carefully.

6. 6.

   Discuss how the multiplication by tn property impacts the response of a control system under polynomial inputs.

   Hint

   Consider the control theory principles.