Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Transcription for Educational Website:
---
### Consider the function
\[
f(t) =
\begin{cases}
0 & \text{if } 0 \leq t < 10\pi \\
\sin(t - 10\pi) & \text{if } 10\pi \leq t.
\end{cases}
\]
#### a. Use the graph of this function to write it in terms of the Heaviside function. Use \( h(t - a) \) for the Heaviside function shifted \( a \) units horizontally.
\[
f(t) = \boxed{} \quad \text{help (formulas)}
\]
#### b. Find the Laplace transform \( F(s) = \mathcal{L} \{ f(t) \} \).
\[
F(s) = \mathcal{L} \{ f(t) \} = \boxed{} \quad \text{help (formulas)}
\]
#### Explanation:
- **Graph Interpretation**: The function \( f(t) \) starts as 0 for \( 0 \leq t < 10\pi \). At \( t = 10\pi \), the function transitions to a sine wave given by \( \sin(t - 10\pi) \).
- **Heaviside Function**: The Heaviside step function \( h(t - a) \) is used to shift the function horizontally by \( a \) units. It is defined as:
\[
h(t - a) =
\begin{cases}
0 & \text{if } t < a \\
1 & \text{if } t \geq a
\end{cases}
\]
- **Laplace Transform**: The Laplace transform of a function \( f(t) \) is represented by \( \mathcal{L} \{ f(t) \} \) and transforms the function from the time domain into the s-domain.
For help with formulas, click on the provided links.
---
This transcription captures the mathematical content and instructions for solving the problem, suitable for an educational context.](https://content.bartleby.com/qna-images/question/b04829d0-4645-426e-bf1a-7ada40b0786f/86481fd3-7e16-47cd-bb7d-5439b0049214/rg04k6_processed.jpeg)
Transcribed Image Text:### Transcription for Educational Website:
---
### Consider the function
\[
f(t) =
\begin{cases}
0 & \text{if } 0 \leq t < 10\pi \\
\sin(t - 10\pi) & \text{if } 10\pi \leq t.
\end{cases}
\]
#### a. Use the graph of this function to write it in terms of the Heaviside function. Use \( h(t - a) \) for the Heaviside function shifted \( a \) units horizontally.
\[
f(t) = \boxed{} \quad \text{help (formulas)}
\]
#### b. Find the Laplace transform \( F(s) = \mathcal{L} \{ f(t) \} \).
\[
F(s) = \mathcal{L} \{ f(t) \} = \boxed{} \quad \text{help (formulas)}
\]
#### Explanation:
- **Graph Interpretation**: The function \( f(t) \) starts as 0 for \( 0 \leq t < 10\pi \). At \( t = 10\pi \), the function transitions to a sine wave given by \( \sin(t - 10\pi) \).
- **Heaviside Function**: The Heaviside step function \( h(t - a) \) is used to shift the function horizontally by \( a \) units. It is defined as:
\[
h(t - a) =
\begin{cases}
0 & \text{if } t < a \\
1 & \text{if } t \geq a
\end{cases}
\]
- **Laplace Transform**: The Laplace transform of a function \( f(t) \) is represented by \( \mathcal{L} \{ f(t) \} \) and transforms the function from the time domain into the s-domain.
For help with formulas, click on the provided links.
---
This transcription captures the mathematical content and instructions for solving the problem, suitable for an educational context.
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