(a) Sketch the graph of the given function. 0
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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![**Task: Sketch the graph of the given function.**
The function \( f(t) \) is defined piecewise as follows:
\[
f(t) =
\begin{cases}
4t, & 0 \leq t < 2 \\
8, & 2 \leq t < 5 \\
-4t + 28, & 5 \leq t < 7 \\
0, & t \geq 7
\end{cases}
\]
**Graph Details:**
- **Interval \(0 \leq t < 2\):** The graph is a line with a slope of 4, starting at \( (0, 0) \) and ending at \( (2, 8) \), where the point at \( (2, 8) \) is hollow.
- **Interval \(2 \leq t < 5\):** The graph is a horizontal line at \( f(t) = 8 \). It starts with a filled dot at \( (2, 8) \) and ends with a hollow dot at \( (5, 8) \).
- **Interval \(5 \leq t < 7\):** This segment is a line with a negative slope of -4. It starts with a filled dot at \( (5, 8) \) and ends with a hollow dot at \( (7, 0) \).
- **Interval \(t \geq 7\):** The graph is a horizontal line at \( f(t) = 0 \), starting from a filled dot at \( (7, 0) \) and continuing indefinitely.
**Task: Express \( f(t) \) in terms of the unit step function \( u_c(t) \).**
The space is provided to write the expression for \( f(t) \) using unit step functions \( u_c(t) \), which are used to switch functions on or off at specific values of \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdddd00a4-d31c-4cdf-9a7a-3406e89ad378%2Fc24beb45-3bd3-43a9-9227-d1058bd1258f%2F54rscf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Sketch the graph of the given function.**
The function \( f(t) \) is defined piecewise as follows:
\[
f(t) =
\begin{cases}
4t, & 0 \leq t < 2 \\
8, & 2 \leq t < 5 \\
-4t + 28, & 5 \leq t < 7 \\
0, & t \geq 7
\end{cases}
\]
**Graph Details:**
- **Interval \(0 \leq t < 2\):** The graph is a line with a slope of 4, starting at \( (0, 0) \) and ending at \( (2, 8) \), where the point at \( (2, 8) \) is hollow.
- **Interval \(2 \leq t < 5\):** The graph is a horizontal line at \( f(t) = 8 \). It starts with a filled dot at \( (2, 8) \) and ends with a hollow dot at \( (5, 8) \).
- **Interval \(5 \leq t < 7\):** This segment is a line with a negative slope of -4. It starts with a filled dot at \( (5, 8) \) and ends with a hollow dot at \( (7, 0) \).
- **Interval \(t \geq 7\):** The graph is a horizontal line at \( f(t) = 0 \), starting from a filled dot at \( (7, 0) \) and continuing indefinitely.
**Task: Express \( f(t) \) in terms of the unit step function \( u_c(t) \).**
The space is provided to write the expression for \( f(t) \) using unit step functions \( u_c(t) \), which are used to switch functions on or off at specific values of \( t \).
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