3 ) Express f(t) in terms of the unit step function u(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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![### Educational Exercise on Piecewise Functions
**Problem Statement:**
(a) **Sketch the graph of the given function.**
The function \( f(t) \) is defined piecewise as follows:
\[
f(t) =
\begin{cases}
2, & 0 \leq t < 2 \\
2e^{-(t-2)}, & t \geq 2
\end{cases}
\]
**Graph Explanation:**
- The graph of \( f(t) \) consists of two parts:
- For \( 0 \leq t < 2 \), the function is constant at \( f(t) = 2 \). This is represented by a horizontal line segment from \( (0, 2) \) to \( (2, 2) \) with an open circle at \( t = 2 \).
- For \( t \geq 2 \), the function is \( f(t) = 2e^{-(t-2)} \). This is an exponentially decaying curve starting from \( (2, 2) \) with a closed circle, decreasing towards zero as \( t \) increases.
(b) **Express \( f(t) \) in terms of the unit step function \( u_c(t) \).**
A box is provided for the expression of \( f(t) \) using the unit step function, denoted as \( u_c(t) \), and currently, the expression is left blank.
**Note:** The unit step function \( u_c(t) \) is not explicitly defined in the problem but is generally used to represent piecewise functions with steps at specified points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83eb3156-1b4a-423a-812e-57012c069de9%2F607503d9-1138-4348-a699-7e56f87e89c3%2Fz516bfk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Exercise on Piecewise Functions
**Problem Statement:**
(a) **Sketch the graph of the given function.**
The function \( f(t) \) is defined piecewise as follows:
\[
f(t) =
\begin{cases}
2, & 0 \leq t < 2 \\
2e^{-(t-2)}, & t \geq 2
\end{cases}
\]
**Graph Explanation:**
- The graph of \( f(t) \) consists of two parts:
- For \( 0 \leq t < 2 \), the function is constant at \( f(t) = 2 \). This is represented by a horizontal line segment from \( (0, 2) \) to \( (2, 2) \) with an open circle at \( t = 2 \).
- For \( t \geq 2 \), the function is \( f(t) = 2e^{-(t-2)} \). This is an exponentially decaying curve starting from \( (2, 2) \) with a closed circle, decreasing towards zero as \( t \) increases.
(b) **Express \( f(t) \) in terms of the unit step function \( u_c(t) \).**
A box is provided for the expression of \( f(t) \) using the unit step function, denoted as \( u_c(t) \), and currently, the expression is left blank.
**Note:** The unit step function \( u_c(t) \) is not explicitly defined in the problem but is generally used to represent piecewise functions with steps at specified points.
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