(a) Sketch the graph of the given function. t, 0 7 f(t) 7 6 5 4. 3 • 2 10 t (b) Express f(t) in terms of the unit step function uc(t). f(t)
(a) Sketch the graph of the given function. t, 0 7 f(t) 7 6 5 4. 3 • 2 10 t (b) Express f(t) in terms of the unit step function uc(t). f(t)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**(a) Sketch the Graph of the Given Function**
The function \( f(t) \) is defined piecewise as follows:
\[
f(t) =
\begin{cases}
t, & 0 \leq t < 2 \\
2, & 2 \leq t < 5 \\
-t + 7, & 5 \leq t < 7 \\
0, & t \geq 7
\end{cases}
\]
**Graph Explanation:**
- **Segment 1 \( (0 \leq t < 2): \)** The function \( f(t) = t \) is a linear function, depicted as a straight line starting from the origin (0, 0) and ending at (2, 2). The point at (2, 2) is an open circle, indicating that the value at \( t = 2 \) is not included in this segment.
- **Segment 2 \( (2 \leq t < 5): \)** In this interval, \( f(t) = 2 \), which is represented as a horizontal line from (2, 2) to (5, 2). This line includes the point at \( t = 2 \) and excludes the point at \( t = 5 \), indicated by an open circle at (5, 2).
- **Segment 3 \( (5 \leq t < 7): \)** Here, \( f(t) = -t + 7 \), a decreasing linear function that starts from (5, 2) to (7, 0). The point (7, 0) has an open circle, showing that this endpoint is not included.
- **Segment 4 \( (t \geq 7): \)** The function becomes 0 and is represented as a horizontal line starting from \( t = 7 \) onward on the t-axis.
**(b) Express \( f(t) \) in Terms of the Unit Step Function \( u_c(t) \).**
\[
f(t) =
\]
The box below the problem is intended for expressing \( f(t) \) using the unit step function, \( u_c(t) \). This method allows for the combination of the piecewise function segments into a single expression using step functions that activate at specific points \( t = c \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4bdfca1-c479-4089-8b74-947c8fb213c2%2F3301bcc6-41c9-4097-b833-f86e0017f208%2F2w5ip0m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**(a) Sketch the Graph of the Given Function**
The function \( f(t) \) is defined piecewise as follows:
\[
f(t) =
\begin{cases}
t, & 0 \leq t < 2 \\
2, & 2 \leq t < 5 \\
-t + 7, & 5 \leq t < 7 \\
0, & t \geq 7
\end{cases}
\]
**Graph Explanation:**
- **Segment 1 \( (0 \leq t < 2): \)** The function \( f(t) = t \) is a linear function, depicted as a straight line starting from the origin (0, 0) and ending at (2, 2). The point at (2, 2) is an open circle, indicating that the value at \( t = 2 \) is not included in this segment.
- **Segment 2 \( (2 \leq t < 5): \)** In this interval, \( f(t) = 2 \), which is represented as a horizontal line from (2, 2) to (5, 2). This line includes the point at \( t = 2 \) and excludes the point at \( t = 5 \), indicated by an open circle at (5, 2).
- **Segment 3 \( (5 \leq t < 7): \)** Here, \( f(t) = -t + 7 \), a decreasing linear function that starts from (5, 2) to (7, 0). The point (7, 0) has an open circle, showing that this endpoint is not included.
- **Segment 4 \( (t \geq 7): \)** The function becomes 0 and is represented as a horizontal line starting from \( t = 7 \) onward on the t-axis.
**(b) Express \( f(t) \) in Terms of the Unit Step Function \( u_c(t) \).**
\[
f(t) =
\]
The box below the problem is intended for expressing \( f(t) \) using the unit step function, \( u_c(t) \). This method allows for the combination of the piecewise function segments into a single expression using step functions that activate at specific points \( t = c \).
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