Question

Which of the following statements about the Dirac delta function Dirac(t) is FALSE?

Options:

Option 1: The Integral from 0 to [infinity] of Dirac(t - 2) t dt = 2
Option 2: The Laplace transform of Dirac(t - 4) t is s^2 / (e^(4s))
Option 3: Dirac(t - 3) = 0 for t = 5
Option 4: The convolution Dirac(t) * f(t) is equal to f(t)
Option 5: Dirac(t) may be considered as the derivative of the Heaviside function u(t)
Option 6: Dirac(t) may be defined as the limit of the function f(t) = 1/ϵ if 0 < t < ϵ, and 0 otherwise, as ϵ approaches 0

Answer 1

The FALSE statement is Option 3: Dirac(t - 3) = 0 for t = 5.

Step-by-step explanation:

The Dirac delta function, often denoted as δ(t), is a mathematical function widely used in engineering and physics. It is characterized by the property that it is zero everywhere except at t = 0, where it is infinite, and its integral over the entire real line is equal to 1.

Option 3 states that Dirac(t - 3) = 0 for t = 5. This statement is FALSE. The Dirac delta function evaluated at t = 5, with an argument of (t - 3), means that the function is zero everywhere except at t = 3. So, Dirac(5 - 3) = Dirac(2), not zero. The Dirac delta function is non-zero only at its argument, which in this case is 2, not 5. Therefore, Option 3 is incorrect.

It's crucial to understand the nature of the Dirac delta function and how it behaves under different conditions to accurately evaluate statements involving it. In this case, the misunderstanding likely stems from confusion about the argument of the delta function and its implications at the specified point in time.