Final answer:
The expression e^{-t}δ(t-2) simplifies to e^{-2} at t = 2 due to the 'sifting' property of the Dirac delta function, which is zero everywhere except at the point t = 2.
Step-by-step explanation:
The question asks to calculate or simplify an expression involving the Dirac delta function, specifically e^{-t}δ(t-2). The Dirac delta function δ(t-a) is defined such that it is zero everywhere except at t = a, where it is infinite, and its integral over the entire real line is 1. This means that when you multiply the Dirac delta function by another function and integrate, you are left with the value of that function at the point where the Dirac delta function is centered.
For the expression e^{-t}δ(t-2), if you were to integrate this expression over t, you would simply get e^{-2} at t = 2, and zero everywhere else because of the properties of the Dirac delta function. The Dirac delta function 'picks out' the value of e^{-t} when t is equal to 2.