12.4.4. Multiplication by a Function

Try these first

1. 1.

   What is the result of multiplying $f(x)$ by $\delta(x - a)$?

   Hint

   Think about what the delta function does at point a.

2. 2.

   Where does the Dirac delta function have non-zero value?

   Hint

   Recall the definition of the Dirac delta function.

3. 3.

   What does the multiplication of a function $f(x)$ by $\delta(x-a)$ yield?

   • Null function
   • f(a) \\delta(x-a)
   • f(a)

   Hint

   Think about the sifting property.

4. 4.

   The Dirac delta function has non-zero values at:

   • True
   • False

   Hint

   Recall how the delta function behaves.

5. 5.

   You are given a function $f(x) = 2x$ and tasked to evaluate the integral $I = \int_{-5}^{5} f(x) \delta(x - 3) dx$. Find the value of $I$.

   Hint

   Recall how the delta function selects the value from the function.

6. 6.

   Consider an impulse function $F(t) = 5 \delta(t - 1)$. If this force is applied to a mass of 10 kg, what is the instantaneous acceleration?

   Hint

   Apply Newton’s law of motion considering instantaneous force.