0 7 f(t) 3 2 10 -1 -3 (b) Express f (t) in terms of the unit step function uc(t). f(t) = 1– 3uz (t – 3) – u5(t) + 5 u7(t)
0 7 f(t) 3 2 10 -1 -3 (b) Express f (t) in terms of the unit step function uc(t). f(t) = 1– 3uz (t – 3) – u5(t) + 5 u7(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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![### Piecewise Function Representation
The function \( f(t) \) is defined as follows:
\[
f(t) =
\begin{cases}
1, & 0 \leq t < 3 \\
-2, & 3 \leq t < 5 \\
-3, & 5 \leq t < 7 \\
2, & t \geq 7
\end{cases}
\]
### Graph Explanation
The graph of the piecewise function \( f(t) \) is illustrated on a coordinate plane.
- The x-axis represents the variable \( t \), ranging from 0 to 10.
- The y-axis represents the function values, \( f(t) \), ranging from -3 to 3.
Key points in the graph:
- From \( t = 0 \) to \( t < 3 \), the function value is 1. This is shown by a horizontal line at \( f(t) = 1 \), ending with a closed circle at \( t = 0 \) and an open circle at \( t = 3 \).
- From \( t = 3 \) to \( t < 5 \), the function value is -2, indicated by a horizontal line at \( f(t) = -2 \), with closed at \( t = 3 \) and open at \( t = 5 \).
- From \( t = 5 \) to \( t < 7 \), the function value is -3, depicted by a horizontal line at \( f(t) = -3 \), closed at \( t = 5 \) and open at \( t = 7 \).
- For \( t \geq 7 \), the function value is 2, represented by a horizontal line at \( f(t) = 2 \), with a closed circle at \( t = 7 \).
### Expression with Unit Step Function
The task is to express \( f(t) \) in terms of the unit step function \( u_c(t) \).
Incorrect Expression:
\[
f(t) = 1 - 3u_3(t-3) - u_5(t) + 5u_7(t)
\]
This expression is marked with an "X" indicating it is incorrect.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F426fc7fa-6ee2-46f8-8cce-a78cc8035a57%2Ffeedf68c-ecf2-41d8-94b2-cbca01d1d915%2Fdgkd6ma_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise Function Representation
The function \( f(t) \) is defined as follows:
\[
f(t) =
\begin{cases}
1, & 0 \leq t < 3 \\
-2, & 3 \leq t < 5 \\
-3, & 5 \leq t < 7 \\
2, & t \geq 7
\end{cases}
\]
### Graph Explanation
The graph of the piecewise function \( f(t) \) is illustrated on a coordinate plane.
- The x-axis represents the variable \( t \), ranging from 0 to 10.
- The y-axis represents the function values, \( f(t) \), ranging from -3 to 3.
Key points in the graph:
- From \( t = 0 \) to \( t < 3 \), the function value is 1. This is shown by a horizontal line at \( f(t) = 1 \), ending with a closed circle at \( t = 0 \) and an open circle at \( t = 3 \).
- From \( t = 3 \) to \( t < 5 \), the function value is -2, indicated by a horizontal line at \( f(t) = -2 \), with closed at \( t = 3 \) and open at \( t = 5 \).
- From \( t = 5 \) to \( t < 7 \), the function value is -3, depicted by a horizontal line at \( f(t) = -3 \), closed at \( t = 5 \) and open at \( t = 7 \).
- For \( t \geq 7 \), the function value is 2, represented by a horizontal line at \( f(t) = 2 \), with a closed circle at \( t = 7 \).
### Expression with Unit Step Function
The task is to express \( f(t) \) in terms of the unit step function \( u_c(t) \).
Incorrect Expression:
\[
f(t) = 1 - 3u_3(t-3) - u_5(t) + 5u_7(t)
\]
This expression is marked with an "X" indicating it is incorrect.
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