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Interactive Audio Lesson
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Introduction to the First Shifting Theorem
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Today, weβre diving into the First Shifting Theorem for Laplace Transforms! Can anyone tell me what the theorem states?
Isnβt it that if we have β{f(t)} = F(s), then β{e^(at)f(t)} = F(s-a)?
Exactly! That horizontal shift in the Laplace domain is crucial when dealing with functions that include exponential factors. We can remember it using the acronym 'SHFT'βShift Helps Find Transform!
How does this apply in real-world problems, especially in engineering?
Great question! This theorem helps in solving differential equations in control systems, electrical circuits, and mechanical vibrations. It makes handling exponential terms much simpler.
Can you give an example?
Sure! If we take f(t) = sin(bt), then by the theorem, β{e^(at)sin(bt)} translates to F(s-a).
So we have to ensure that s > a for it to converge, right?
Yes, precisely! Understanding these conditions is vital. Letβs move on to some exercises to apply what weβve learned.
Application Exercises
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Let's tackle the exercises from the section. The first one is to find the Laplace Transform of e^(4t)cos(3t). Any thoughts?
I think we should start with β{cos(3t)} first, right? Itβs s/(s^2 + 9).
Correct! Now, applying the theorem, what do we get?
Right, so it becomes β{e^(4t)cos(3t)} = F(s-4), which means we replace s with s-4 in the original Laplace transform.
Excellent! You did that quickly and correctly. Now, let's try e^(-2t)t^2. Who wants to take a shot?
The Laplace transform of t^2 is 2/s^3! So replacing s with s+2 would give us 2/(s+2)^3.
Fantastic! Just a tiny detailβremember you are shifting to the left, whatβs important? What inequality do we need to check?
We must ensure s+2 is greater than zero for convergence!
Exactly right! Convergence conditions are crucial. Letβs finish off with the last exercise!
Introduction & Overview
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Quick Overview
Standard
Users are encouraged to practice finding Laplace Transforms using the First Shifting Theorem. This section provides specific exercises to reinforce understanding of how to apply the theorem in given cases.
Detailed
The First Shifting Theorem is a crucial tool in engineering and applied mathematics, allowing us to find Laplace Transforms of functions multiplied by exponential factors. This section emphasizes the importance of practicing the theorem through exercises where learners are required to find the Laplace Transforms of various functions, showcasing the theorem's applications in problem-solving scenarios. Each exercise requires careful application of the theorem, ensuring that learners grasp both the conceptual and computational aspects of using this theorem effectively.
Audio Book
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Exercise 1: Laplace Transform of Exponential and Cosine
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Find the Laplace Transform of the following using the First Shifting Theorem:
1. π4π‘ β
cos(3π‘)
Detailed Explanation
To find the Laplace Transform of the function π(π‘) = π^{4π‘} imes ext{cos}(3π‘), we can apply the First Shifting Theorem. First, we note that the Laplace Transform of cos(3t) is known:
$$
β{cos(3t)} = \frac{s}{s^2 + 9}
$$
Since we have an exponential term e^(4t), we can let 'a' be 4. According to the theorem, we will shift 's' to 's - 4'. Therefore:
$$
β{βe^{4t} \cdot cos(3t)} = β{cos(3t)} |{s=s-4} = β{cos(3t)}|{s-4} = \frac{s-4}{(s-4)^2 + 9}
$$
Examples & Analogies
Think of it like tuning a radio. When you're trying to catch a specific station (the frequency of cos(3t)), sometimes you need to adjust your position slightly to account for interference, like the 4 from e^(4t). This adjustment represents shifting your frequency to get a clearer signal.
Exercise 2: Laplace Transform of Exponential and Quadratic
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- πβ2π‘ β π‘^2
Detailed Explanation
To solve for the Laplace Transform of the function π(π‘) = π^{-2π‘} imes t^2, we first find the Laplace Transform of tΒ². The Laplace Transform of t^n is:
$$
β{t^n} = \frac{n!}{s^{n+1}} ext{ for } s > 0
$$
So, for n = 2:
$$
β{t^2} = \frac{2!}{s^3} = \frac{2}{s^3}
$$
Next, applying the First Shifting Theorem with a = -2, we shift 's' to 's + 2':
$$
β{βe^{-2t}t^2} = \frac{2}{(s + 2)^3}
$$
Examples & Analogies
Imagine you're pushing a toy car that slows down over time due to friction. The quadratic term represents the distance it covers, and the e^{-2t} is like the slowing force. We adjust our expectation of distance based on how quickly it slows down, which mirrors how we adjust 's' in the transform.
Exercise 3: Laplace Transform of Exponential and Sine
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- π5π‘ β sin(2π‘)
Detailed Explanation
To find the Laplace Transform of the function π(π‘) = π^{5π‘} imes ext{sin}(2π‘), we start by finding the Laplace Transform of sin(2t):
$$
β{sin(2t)} = \frac{2}{s^2 + 4}
$$
Now, since we have an exponential e^ ext{5t}, and setting a = 5, we apply the First Shifting Theorem, which leads to shifting 's' to 's - 5':
$$
β{e^{5t} \cdot sin(2t)} = \frac{2}{(s-5)^2 + 4}
$$
Examples & Analogies
Think of this process like adjusting the brightness on your smartphone screen. The sine function (sin(2t)) behaves predictively like the brightness levels, but when someone introduces a constant increase (e^{5t}), it's like someone adjusting the overall brightness. The transformation reflects this change in the 's' domain.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
First Shifting Theorem: A method used to find Laplace Transforms of functions multiplied by exponential terms.
-
Exponential Shift: The way the shifting property alters the parameter in the Laplace domain.
-
Conditions for Application: Ensuring s > a for proper convergence when applying the theorem.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1: β{e^(4t)cos(3t)} = F(s-4) with F(s) = s/(s^2 + 9) results in s/(s^2 + 9) | s -> s-4.
-
Example 2: β{e^(-2t)t^2} = 2/(s+2)^3 derived from β{t^2} = 2/s^3.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
-
If e is near, change the s with cheer, or else convergence will disappear!
π Fascinating Stories
-
Imagine a ship named 'Laplace' sailing from the point 's'. When it hits 'e^(2t)', it takes a joyful leap left, landing at 's-2' where the seas are calm and equations are solved!
π§ Other Memory Gems
-
Remember: S.H.F.T for Shift Helps Find Transform when using the theorem!
π― Super Acronyms
S.H.F.T
- S: = shift
- H: = helps
- F: = find
- T: = transform.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
-
Term: Laplace Transform
Definition:
An integral transform that converts a function of time into a function of a complex variable, simplifying the analysis of linear systems.
-
Term: First Shifting Theorem
Definition:
A theorem stating that β{e^(at)f(t)} = F(s - a), which shows how multiplying by an exponential function shifts the Laplace transform.
-
Term: Exponential Function
Definition:
A mathematical function of the form e^(at) where 'a' is a constant, often appearing in differential equations.