Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Example Problem: Laplace Transform Using Definition
**Problem Statement:**
Use the definition to find the Laplace Transform of the given function
\[f(t) =
\begin{cases}
t, & 0 \le t < 1 \\
1, & t \ge 1
\end{cases}
\]
**Solution Outline:**
To solve this problem, you will:
1. Apply the definition of the Laplace Transform.
2. Integrate the function piecewise according to its definition over the different intervals.
**Step-by-Step Solution:**
1. **Definition of Laplace Transform:**
The Laplace Transform of a function \( f(t) \), denoted as \( \mathcal{L}\{f(t)\} \) or \( F(s) \), is given by:
\[
\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt
\]
2. **Piecewise Integration:**
For the given function \( f(t) \), we break the integral into two parts corresponding to the intervals of the piecewise function.
\[
\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt = \int_{0}^{1} e^{-st} t \, dt + \int_{1}^{\infty} e^{-st} \cdot 1 \, dt
\]
3. **Evaluate Each Integral:**
- **First Integral:**
\[
\int_{0}^{1} e^{-st} t \, dt
\]
Use integration by parts:
Let \( u = t \) and \( dv = e^{-st} dt \).
Then, \( du = dt \) and \( v = -\frac{1}{s} e^{-st} \).
Thus,
\[
\int_{0}^{1} t e^{-st} \, dt = \left. -\frac{t}{s} e^{-st} \right|_{0}^{1} + \int_{0}^{1} \frac{1}{s} e^{-st} \, dt
\]
\[
= \left( -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab6a44fb-472d-4115-ae99-90593ad92ba0%2F982fbdfd-7399-45d5-8566-da8ab9c3bf4a%2Fj9fi27_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem: Laplace Transform Using Definition
**Problem Statement:**
Use the definition to find the Laplace Transform of the given function
\[f(t) =
\begin{cases}
t, & 0 \le t < 1 \\
1, & t \ge 1
\end{cases}
\]
**Solution Outline:**
To solve this problem, you will:
1. Apply the definition of the Laplace Transform.
2. Integrate the function piecewise according to its definition over the different intervals.
**Step-by-Step Solution:**
1. **Definition of Laplace Transform:**
The Laplace Transform of a function \( f(t) \), denoted as \( \mathcal{L}\{f(t)\} \) or \( F(s) \), is given by:
\[
\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt
\]
2. **Piecewise Integration:**
For the given function \( f(t) \), we break the integral into two parts corresponding to the intervals of the piecewise function.
\[
\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt = \int_{0}^{1} e^{-st} t \, dt + \int_{1}^{\infty} e^{-st} \cdot 1 \, dt
\]
3. **Evaluate Each Integral:**
- **First Integral:**
\[
\int_{0}^{1} e^{-st} t \, dt
\]
Use integration by parts:
Let \( u = t \) and \( dv = e^{-st} dt \).
Then, \( du = dt \) and \( v = -\frac{1}{s} e^{-st} \).
Thus,
\[
\int_{0}^{1} t e^{-st} \, dt = \left. -\frac{t}{s} e^{-st} \right|_{0}^{1} + \int_{0}^{1} \frac{1}{s} e^{-st} \, dt
\]
\[
= \left( -
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