#5: Determine the Laplace transform for the following signal (Hint: Draw it first) f(t) = 0, 2t4, 4, 0, t<2, 2≤t < 4, 4 < t < 6, 6 ≤t.
#5: Determine the Laplace transform for the following signal (Hint: Draw it first) f(t) = 0, 2t4, 4, 0, t<2, 2≤t < 4, 4 < t < 6, 6 ≤t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Exercise #5: Find the Laplace Transform of the Given Signal**
**Problem Statement:**
Determine the Laplace transform for the following signal. *(Hint: Draw it first)*
\[
f(t) =
\begin{cases}
0, & t < 2, \\
2t - 4, & 2 \leq t < 4, \\
4, & 4 \leq t < 6, \\
0, & 6 \leq t.
\end{cases}
\]
**Explanation:**
To solve this problem, we break the function \( f(t) \) into time intervals and analyze each part:
1. **Interval \( t < 2 \):** The function \( f(t) = 0 \).
2. **Interval \( 2 \leq t < 4 \):** The function is linear and given by \( f(t) = 2t - 4 \).
3. **Interval \( 4 \leq t < 6 \):** The function becomes constant with \( f(t) = 4 \).
4. **Interval \( 6 \leq t \):** The function returns to zero, \( f(t) = 0 \).
**Graph Explanation:**
A graph of \( f(t) \) would show a piecewise function:
- A horizontal line at \( f(t) = 0 \) for \( t < 2 \).
- A rising line from \( (2, 0) \) to \( (4, 4) \) for \( 2 \leq t < 4 \).
- A constant line at \( f(t) = 4 \) for \( 4 \leq t < 6 \).
- A horizontal line at \( f(t) = 0 \) for \( t \geq 6 \).
For each segment, you can calculate the Laplace transform separately and add them up using the properties of linearity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2Fa6ac0ae9-504d-48f5-85bc-1b88e9ec79af%2Fwcmljws_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise #5: Find the Laplace Transform of the Given Signal**
**Problem Statement:**
Determine the Laplace transform for the following signal. *(Hint: Draw it first)*
\[
f(t) =
\begin{cases}
0, & t < 2, \\
2t - 4, & 2 \leq t < 4, \\
4, & 4 \leq t < 6, \\
0, & 6 \leq t.
\end{cases}
\]
**Explanation:**
To solve this problem, we break the function \( f(t) \) into time intervals and analyze each part:
1. **Interval \( t < 2 \):** The function \( f(t) = 0 \).
2. **Interval \( 2 \leq t < 4 \):** The function is linear and given by \( f(t) = 2t - 4 \).
3. **Interval \( 4 \leq t < 6 \):** The function becomes constant with \( f(t) = 4 \).
4. **Interval \( 6 \leq t \):** The function returns to zero, \( f(t) = 0 \).
**Graph Explanation:**
A graph of \( f(t) \) would show a piecewise function:
- A horizontal line at \( f(t) = 0 \) for \( t < 2 \).
- A rising line from \( (2, 0) \) to \( (4, 4) \) for \( 2 \leq t < 4 \).
- A constant line at \( f(t) = 4 \) for \( 4 \leq t < 6 \).
- A horizontal line at \( f(t) = 0 \) for \( t \geq 6 \).
For each segment, you can calculate the Laplace transform separately and add them up using the properties of linearity.
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