17Calculus - Laplace Transform First Shifting Theorem
17Calculus
On this page, we discuss the first of two important theorems related to Laplace Transforms. They are rather cleverly named the First Shifting Theorem and the Second Shifting Theorem.
First Shifting Theorem |
\(\displaystyle{ \mathcal{L}\{ e^{-at} f(t) \} = F(s+a) }\) |
\(\displaystyle{ \mathcal{L}\{ u(t-c)g(t-c) \} = e^{-cs}G(s) }\) |
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First Shifting Theorem
The first shifting theorem says that in the \(t\)-domain, if we multiply a function by \(e^{-at}\), this results in a shift in the \(s\)-domain of \(a\) units. In your Laplace Transforms table you probably see the line that looks like \[\displaystyle{ \mathcal{L}\{ e^{-at} f(t) \} = F(s+a) }\] This idea looks easy and watching other people using it may look easy but when you have to do it, it is not so easy. So it is important to get your hands dirty by working some problems on your own. After you watch this first video, you can work some practice problems to help you understand this better.
Let's watch a video that explains how this works. This video contains a very good explanation, a couple of examples and the proof of this theorem.
Dr Chris Tisdell - First shifting theorem of Laplace transforms: a how to [12min-6secs]
video by Dr Chris Tisdell |
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Practice
Unless otherwise instructed,
- if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the first shifting theorem
- if \(F(s)\) is given, find the inverse Laplace transform \( f(t) = \mathcal{L}^{-1} \{ F(s) \} \)
Give your answers in exact, completely factored form.
\( f(t) = u(t-4)[t-4] \)
Problem Statement
For \( f(t) = u(t-4)[t-4] \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution
Dr Chris Tisdell - 665 video solution
video by Dr Chris Tisdell |
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\( f(t) = u(t-4)[t-4]^2 \)
Problem Statement
For \( f(t) = u(t-4)[t-4]^2 \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution
Dr Chris Tisdell - 667 video solution
video by Dr Chris Tisdell |
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\( f(t) = u(t-2\pi) \sin(t-2\pi) \)
Problem Statement
For \( f(t) = u(t-2\pi) \sin(t-2\pi) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution
Dr Chris Tisdell - 668 video solution
video by Dr Chris Tisdell |
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\(\displaystyle{ F(s) = \frac{1-e^{-2s}}{s^2} }\)
Problem Statement
For \(\displaystyle{ F(s) = \frac{1-e^{-2s}}{s^2} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{-1} \{ F(s) \} \) using a shifting theorem.
Solution
Dr Chris Tisdell - 666 video solution
video by Dr Chris Tisdell |
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\( f(t) = \cos[3(t-1)] u(t-1) \)
Problem Statement
For \( f(t) = \cos[3(t-1)] u(t-1) \) find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution
Dr Chris Tisdell - 670 video solution
video by Dr Chris Tisdell |
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\( f(t) = \left\{\begin{array}{lr} t & 0 \leq 4 \\ 2 & t = 4 \\ 0 & t > 4 \end{array} \right. \)
Problem Statement
For \( f(t) = \left\{\begin{array}{lr} t & 0 \leq 4 \\ 2 & t = 4 \\ 0 & t > 4 \end{array} \right. \) find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution
Dr Chris Tisdell - 669 video solution
video by Dr Chris Tisdell |
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Practice Instructions
Unless otherwise instructed,
- if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the first shifting theorem
- if \(F(s)\) is given, find the inverse Laplace transform \( f(t) = \mathcal{L}^{-1} \{ F(s) \} \)
Give your answers in exact, completely factored form.
