(a) Sketch the graph of the given function. t, 0 7 2, f(t) = f(t) 8 7 6 5. 4 3 2 10 t (b) Express f(t) in terms of the unit step function uc(t). f(t) : tU (t) + (2 – t)U(t – 2) + (5 – t)U(t – 5) + (t – t)U (t – t)
(a) Sketch the graph of the given function. t, 0 7 2, f(t) = f(t) 8 7 6 5. 4 3 2 10 t (b) Express f(t) in terms of the unit step function uc(t). f(t) : tU (t) + (2 – t)U(t – 2) + (5 – t)U(t – 5) + (t – t)U (t – 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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Question
![(a) The problem involves sketching the graph of the given piecewise function:
\[
f(t) =
\begin{cases}
t, & 0 \leq t < 2 \\
2, & 2 \leq t < 5 \\
-t + 7, & 5 \leq t < 7 \\
0, & t \geq 7
\end{cases}
\]
The graph of the function is a piecewise linear graph with the following segments:
- From \( t = 0 \) to \( t = 2 \), the function increases linearly with a slope of 1, starting from \( f(0) = 0 \) and reaching \( f(2) = 2 \).
- From \( t = 2 \) to \( t = 5 \), the function is constant with \( f(t) = 2 \).
- From \( t = 5 \) to \( t = 7 \), the function decreases linearly with a slope of -1, starting from \( f(5) = 2 \) and reaching \( f(7) = 0 \).
- For \( t \geq 7 \), the function remains at \( f(t) = 0 \).
These behaviors create a graph consisting of a rising segment, a flat segment, a falling segment, and a flat line at zero.
(b) The task involves expressing \( f(t) \) in terms of the unit step function \( u_c(t) \).
Incorrect expression provided:
\[
f(t) = tU(t) + (2 - t)U(t - 2) + (5 - t)U(t - 5) + (t - t)U(t - t)
\]
The box indicates this expression is incorrect. Note: The correct use of unit step functions should activate and deactivate segments of the piecewise function correctly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83eb3156-1b4a-423a-812e-57012c069de9%2Ff4b3dd0e-ac3d-429e-acc6-5bfb24fea074%2Fi30gz5t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) The problem involves sketching the graph of the given piecewise function:
\[
f(t) =
\begin{cases}
t, & 0 \leq t < 2 \\
2, & 2 \leq t < 5 \\
-t + 7, & 5 \leq t < 7 \\
0, & t \geq 7
\end{cases}
\]
The graph of the function is a piecewise linear graph with the following segments:
- From \( t = 0 \) to \( t = 2 \), the function increases linearly with a slope of 1, starting from \( f(0) = 0 \) and reaching \( f(2) = 2 \).
- From \( t = 2 \) to \( t = 5 \), the function is constant with \( f(t) = 2 \).
- From \( t = 5 \) to \( t = 7 \), the function decreases linearly with a slope of -1, starting from \( f(5) = 2 \) and reaching \( f(7) = 0 \).
- For \( t \geq 7 \), the function remains at \( f(t) = 0 \).
These behaviors create a graph consisting of a rising segment, a flat segment, a falling segment, and a flat line at zero.
(b) The task involves expressing \( f(t) \) in terms of the unit step function \( u_c(t) \).
Incorrect expression provided:
\[
f(t) = tU(t) + (2 - t)U(t - 2) + (5 - t)U(t - 5) + (t - t)U(t - t)
\]
The box indicates this expression is incorrect. Note: The correct use of unit step functions should activate and deactivate segments of the piecewise function correctly.
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