13.1.8. Graphical Interpretation

Try these first

1. 1.

   What is the mathematical representation of convolution?

   Hint

   Think about the integration process and shifting one function.

2. 2.

   State one property of convolution.

   Hint

   Recall the basic properties discussed.

3. 3.

   What is defined as $(f * g)(t)$?

   • The product of functions
   • The convolution of functions
   • The Laplace transform

   Hint

   Recall the definition of convolution.

4. 4.

   True or False: Convolution is only applicable to continuous functions.

   • True
   • False

   Hint

   Consider the types of functions that can be convolved.

5. 5.

   Given $f(t) = e^{-t}$ and $g(t) = t^2$, find $(f * g)(t)$.

   Hint

   Break down the integral into simpler parts to solve iteratively.

6. 6.

   Demonstrate that convolution is a linear operation, proving that $a(f * g)(t) = (af * g)(t) = (f * ag)(t)$ for any constant a.

   Hint

   Manipulate the integral based on properties of multiplication and addition.