Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
Find the Laplace transform \( F(s) \) of the function \( f(t) = -1u(t - 4) - 6u(t - 6) - 1u(t - 8) \).
**Solution:**
The Laplace transform \( F(s) \) is represented by the equation:
\[
F(s) = \boxed{\phantom{F(s) = }}
\]
**Explanation:**
In the given problem, the function \( f(t) \) is comprised of step functions \( u(t - a) \), where \( a \) represents the point at which the step occurs. These are common in control theory and signal processing to model changes at specific times. Each term can be processed individually using the properties of the Laplace transform. The final result encompasses the transforms of each of these shifted unit step functions.](https://content.bartleby.com/qna-images/question/16510717-aebe-4647-9d51-4dea6e47545d/f86b8a33-23ac-4f5c-83b7-f8ffea9d8303/1nsm8t4_processed.png)
Transcribed Image Text:**Problem Statement:**
Find the Laplace transform \( F(s) \) of the function \( f(t) = -1u(t - 4) - 6u(t - 6) - 1u(t - 8) \).
**Solution:**
The Laplace transform \( F(s) \) is represented by the equation:
\[
F(s) = \boxed{\phantom{F(s) = }}
\]
**Explanation:**
In the given problem, the function \( f(t) \) is comprised of step functions \( u(t - a) \), where \( a \) represents the point at which the step occurs. These are common in control theory and signal processing to model changes at specific times. Each term can be processed individually using the properties of the Laplace transform. The final result encompasses the transforms of each of these shifted unit step functions.
![**Laplace Transform Problem**
**Objective:** Find the Laplace transform \( F(s) = \mathcal{L}(f(t)) \) given the piecewise function \( f(t) \).
**Function Definition:**
\[
f(t) =
\begin{cases}
0, & t < 3 \\
(t - 3)^3, & t \geq 3
\end{cases}
\]
**Task:** Determine \( F(s) = \) (Provide the answer in the space provided).
**Explanation:**
The problem involves finding the Laplace transform of a piecewise function where:
- For \( t < 3 \), the function \( f(t) \) is zero.
- For \( t \geq 3 \), the function \( f(t) \) takes the form \( (t-3)^3 \).
This requires applying the definition of the Laplace transform to piecewise functions, particularly using methods that accommodate shifting and handling polynomials.](https://content.bartleby.com/qna-images/question/16510717-aebe-4647-9d51-4dea6e47545d/f86b8a33-23ac-4f5c-83b7-f8ffea9d8303/lruc2_processed.png)
Transcribed Image Text:**Laplace Transform Problem**
**Objective:** Find the Laplace transform \( F(s) = \mathcal{L}(f(t)) \) given the piecewise function \( f(t) \).
**Function Definition:**
\[
f(t) =
\begin{cases}
0, & t < 3 \\
(t - 3)^3, & t \geq 3
\end{cases}
\]
**Task:** Determine \( F(s) = \) (Provide the answer in the space provided).
**Explanation:**
The problem involves finding the Laplace transform of a piecewise function where:
- For \( t < 3 \), the function \( f(t) \) is zero.
- For \( t \geq 3 \), the function \( f(t) \) takes the form \( (t-3)^3 \).
This requires applying the definition of the Laplace transform to piecewise functions, particularly using methods that accommodate shifting and handling polynomials.
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