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Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function. -9s 1-e F(s) = s(1+e=9s) f(t) 4 1 | a 2a 3a 4aj meander function t

Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function. -9s 1-e F(s) = s(1+e=9s) f(t) 4 1 | a 2a 3a 4aj meander function t

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.3: The Chain Rule
Problem 65E
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### Periodic Function and Its Laplace Transform **Objective:** Use Theorem 7.4.3 to find the Laplace transform \( F(s) \) of the given periodic function. #### Given Function and Incorrect Transform The given function is a meander function, a type of periodic function that alternates between 1 and -1 at regular intervals. #### Provided (Incorrect) Laplace Transform \[ F(s) = \frac{1 - e^{-9s}}{s \left(1 + e^{-9s}\right)} \] The provided transform is indicated to be incorrect with a red cross. #### Graphical Representation - **x-axis (t):** Represents time. - **y-axis (f(t)):** Represents the function values which are 1 and -1. - Intervals: - At \( t = a \) the function jumps from 1 to -1. - At \( t = 2a \) the function rises from -1 to 1. - At \( t = 3a \) it drops from 1 to -1. - The cycle repeats periodically. This visual description illustrates the periodic nature of the function: 1. The function \( f(t) \) starts at 1 for time t=0. 2. When \( t \) reaches \( a \), \( f(t) \) shifts to -1 and remains there until \( t \) reaches \( 2a \). 3. This pattern continues, creating a rectangular waveform with period \( 2a \). Understanding and applying Theorem 7.4.3 correctly can help derive the accurate Laplace transform for such periodic functions.
Transcribed Image Text:### Periodic Function and Its Laplace Transform **Objective:** Use Theorem 7.4.3 to find the Laplace transform \( F(s) \) of the given periodic function. #### Given Function and Incorrect Transform The given function is a meander function, a type of periodic function that alternates between 1 and -1 at regular intervals. #### Provided (Incorrect) Laplace Transform \[ F(s) = \frac{1 - e^{-9s}}{s \left(1 + e^{-9s}\right)} \] The provided transform is indicated to be incorrect with a red cross. #### Graphical Representation - **x-axis (t):** Represents time. - **y-axis (f(t)):** Represents the function values which are 1 and -1. - Intervals: - At \( t = a \) the function jumps from 1 to -1. - At \( t = 2a \) the function rises from -1 to 1. - At \( t = 3a \) it drops from 1 to -1. - The cycle repeats periodically. This visual description illustrates the periodic nature of the function: 1. The function \( f(t) \) starts at 1 for time t=0. 2. When \( t \) reaches \( a \), \( f(t) \) shifts to -1 and remains there until \( t \) reaches \( 2a \). 3. This pattern continues, creating a rectangular waveform with period \( 2a \). Understanding and applying Theorem 7.4.3 correctly can help derive the accurate Laplace transform for such periodic functions.
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