3.4 In the following problems properties of the Laplace transform are used. (a) Show that the Laplace transform of x(t)e-a' u(t) is X (s +a), where X (s) = L[x(t)] and then use it to find the Laplace transform of y(t) = cos(t)e¬"u(t). (b) A signal x1(t) has as Laplace transform s +2 X1(s)= (s + 2)2 +1' find poles and zeros of X1(s) and find x1 (t) as t → o from the location of the poles. (c) The signal z(t) =de¯u(t)/dt. i. Compute the derivative z(t) and then find its Laplace transform Z(s). ii. Use the derivative property to find Z(s). Compare your result with the one obtained above. Answer: (a) Y (s) = (s +2)/((s+2)² + 1); (b) x1(t) → 0 as t → oo; (c) Z(s) = s/(s+1).
3.4 In the following problems properties of the Laplace transform are used. (a) Show that the Laplace transform of x(t)e-a' u(t) is X (s +a), where X (s) = L[x(t)] and then use it to find the Laplace transform of y(t) = cos(t)e¬"u(t). (b) A signal x1(t) has as Laplace transform s +2 X1(s)= (s + 2)2 +1' find poles and zeros of X1(s) and find x1 (t) as t → o from the location of the poles. (c) The signal z(t) =de¯u(t)/dt. i. Compute the derivative z(t) and then find its Laplace transform Z(s). ii. Use the derivative property to find Z(s). Compare your result with the one obtained above. Answer: (a) Y (s) = (s +2)/((s+2)² + 1); (b) x1(t) → 0 as t → oo; (c) Z(s) = s/(s+1).
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Question
![3.4 In the following problems properties of the Laplace transform are used.
(a) Show that the Laplace transform of x(t)e-al u (t) is X (s + a), where X (s) =L[x(t)] and
then use it to find the Laplace transform of y(t) = cos(t)e-2u (t).
(b) A signal x1(t) has as Laplace transform
s +2
X1(s) =
(s + 2)2 +1'
find poles and zeros of X1(s) and find x1(t) as t → ∞ from the location of the poles.
(c) The signal z(t) = de¯'u(t)/dt.
i. Compute the derivative z(t) and then find its Laplace transform Z(s).
ii. Use the derivative property to find Z(s). Compare your result with the one obtained
above.
Answer: (a) Y (s)= (s +2)/((s +2)² + 1); (b) x1(t) → 0 as t → ; (c) Z(s) = s/(s+1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F715ad422-c0c9-4bb1-9209-6734503c0d0c%2Fa90c6f3a-ad59-4597-8556-525196cd88ed%2Fi5qhss_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.4 In the following problems properties of the Laplace transform are used.
(a) Show that the Laplace transform of x(t)e-al u (t) is X (s + a), where X (s) =L[x(t)] and
then use it to find the Laplace transform of y(t) = cos(t)e-2u (t).
(b) A signal x1(t) has as Laplace transform
s +2
X1(s) =
(s + 2)2 +1'
find poles and zeros of X1(s) and find x1(t) as t → ∞ from the location of the poles.
(c) The signal z(t) = de¯'u(t)/dt.
i. Compute the derivative z(t) and then find its Laplace transform Z(s).
ii. Use the derivative property to find Z(s). Compare your result with the one obtained
above.
Answer: (a) Y (s)= (s +2)/((s +2)² + 1); (b) x1(t) → 0 as t → ; (c) Z(s) = s/(s+1).
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