c) f(t) = { t², 0≤t<3 9, 3≤t a
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
1C
![**Title: Expressing Functions in Terms of Unit Step Functions**
**Objective:**
To express the following piecewise functions using unit step functions and sketch their graphs.
---
**Problems:**
**1. Express the given function in terms of unit step function(s):**
---
**a) Function:**
\[
f(t) =
\begin{cases}
2, & 0 \leq t < 4 \\
5, & 4 \leq t
\end{cases}
\]
---
**b) Function:**
\[
f(t) =
\begin{cases}
-1, & 0 \leq t < 2 \\
3, & 2 \leq t
\end{cases}
\]
---
**c) Function:**
\[
f(t) =
\begin{cases}
t^2, & 0 \leq t < 3 \\
9, & 3 \leq t
\end{cases}
\]
---
**d) Function:**
\[
f(t) =
\begin{cases}
0, & 0 \leq t < \pi \\
\sin(t - \pi), & \pi \leq t
\end{cases}
\]
---
**e) Function:**
\[
f(t) =
\begin{cases}
5, & 0 \leq t < 2 \\
2, & 2 \leq t < 4 \\
4, & 4 \leq t
\end{cases}
\]
---
**f) Function:**
\[
f(t) =
\begin{cases}
2t, & 0 \leq t < \pi/2 \\
\pi, & \pi/2 \leq t < 3\pi/2 \\
\cos(t - 3\pi/2), & 3\pi/2 \leq t
\end{cases}
\]
---
**Instructions:**
1. Express each function in terms of unit step functions.
2. Sketch the graph of each function accurately to visualize the transitions defined in each piece.
**Note:**
Unit step function can be defined as:
\[ u(t-a) =
\begin{cases}
0, & t < a \\
1, & t \geq a
\end{cases](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb33255d3-bc66-4cf8-b4b6-36dba3c0ca10%2Ff88d18c9-5295-4a4a-b4f6-5886325c272a%2Fnr503ah_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Expressing Functions in Terms of Unit Step Functions**
**Objective:**
To express the following piecewise functions using unit step functions and sketch their graphs.
---
**Problems:**
**1. Express the given function in terms of unit step function(s):**
---
**a) Function:**
\[
f(t) =
\begin{cases}
2, & 0 \leq t < 4 \\
5, & 4 \leq t
\end{cases}
\]
---
**b) Function:**
\[
f(t) =
\begin{cases}
-1, & 0 \leq t < 2 \\
3, & 2 \leq t
\end{cases}
\]
---
**c) Function:**
\[
f(t) =
\begin{cases}
t^2, & 0 \leq t < 3 \\
9, & 3 \leq t
\end{cases}
\]
---
**d) Function:**
\[
f(t) =
\begin{cases}
0, & 0 \leq t < \pi \\
\sin(t - \pi), & \pi \leq t
\end{cases}
\]
---
**e) Function:**
\[
f(t) =
\begin{cases}
5, & 0 \leq t < 2 \\
2, & 2 \leq t < 4 \\
4, & 4 \leq t
\end{cases}
\]
---
**f) Function:**
\[
f(t) =
\begin{cases}
2t, & 0 \leq t < \pi/2 \\
\pi, & \pi/2 \leq t < 3\pi/2 \\
\cos(t - 3\pi/2), & 3\pi/2 \leq t
\end{cases}
\]
---
**Instructions:**
1. Express each function in terms of unit step functions.
2. Sketch the graph of each function accurately to visualize the transitions defined in each piece.
**Note:**
Unit step function can be defined as:
\[ u(t-a) =
\begin{cases}
0, & t < a \\
1, & t \geq a
\end{cases
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