Question

12.53 Apply the initial- and final-value theorems to each transform pair in Problem 12.43. 12.43 Find f(t) for each of the following functions.

a) F(s) 135 s(s + 3)3•
b) F(s) = 10(s + 2)2 (52 + 25 + 2)2 ·
c) F(s) = 2582 + 395s + 1494 $2 + 15s + 54 108
d) F(s) 583 + 2052 49s 92 + 7s + 10

Answer 1

The initial- and final-value theorems predict the behavior of a function at the start and as time approaches infinity. They require finding the function in the time domain before applying the theorems to each transform pair provided.

Step-by-step explanation:

Applying the initial- and final-value theorems to each transform pair helps to find the limiting behavior of a function as time approaches zero and infinity. These theorems are particularly useful in control system analysis and signal processing, which falls under the domain of electrical engineering. The final value of the function, f(infinity), is obtained by evaluating sF(s) at s=infinity. Therefore, f(infinity) = 0.

The initial value theorem reveals the starting value of the function in the time domain, while the final-value theorem predicts the steady-state value as time approaches infinity. In each given case of a transform pair, one would use partial fraction decomposition or some other method to find the original time-domain function f(t) and then apply these theorems.