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Apply the final-value theorem to F(s)= 40(s+3)/s(s+4)² to find the final value of f(t). Express your answer using three significant figures.

F([infinity])=_______

User Enigmadan
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Final answer:

The final value of f(t) is 7.50 when evaluated using the final-value theorem on the function F(s) = 40(s+3)/s(s+4)².

Step-by-step explanation:

To apply the final-value theorem to the Laplace-transformed function F(s) = 40(s+3)/s(s+4)² we assume that the system is stable meaning that all poles of F(s) are in the left half of the complex plane (i.e., they all have negative real parts).

The final-value theorem states that the final value of f(t) as t approaches infinity is equal to the limit of s multiplied by F(s) as s approaches zero:

f(∞) = lims→0(s · F(s))

Plugging in our function, we get:

f(∞) = lims→0(s · 40(s+3) / s(s+4)²) = lims→0(40(s+3) / (s+4)²)

Since the s term in the numerator and the denominator cancel out, we are left with the limit of (s+3) as s approaches zero:

f(∞) = lims→0(40(s+3) / (4)²) = 40(3) / 16 = 120 / 16 = 7.5

Therefore the final value of f(t) expressed using three significant figures, is 7.50.

User Cartant
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