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Laplace Transforms involve a technique to change an expression into another form using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, they require only improper integration techniques to use. So you may run across them in first year calculus.

The Laplace Transform

\(\displaystyle{ \mathcal{L}\{ f(t) \} = F(s) = \int_{0}^{\infty}{e^{-st}f(t)~dt} }\)

In the integral, \(s\) is considered a constant and \(s > 0\)

After integration, we have a function \(F(s)\), which is a function of \(s\) (now considered a variable)

Overview

This takes some thinking and getting used to. So let's start off by watching some videos. There is some duplication coming up but you really need to have that repetition in order to get your head around this concept.

This first video is a quick overview with some calculations to get you started. Do not worry too much about understanding everything here since we will revisit these concepts very soon. Just get the big picture.

Dr Chris Tisdell - Intro to Laplace transform and how to calculate them [8min-48secs]

video by Dr Chris Tisdell

Now that you have the big picture, let's start working on understanding the details of Laplace Transforms. This is a great video clip explaining this notation and the improper integral definition. It is relatively short but is packed with good information including explaining notation and calculating the integral.

Dr Chris Tisdell - Introduction to Laplace transforms (part 1) [5min-35secs]

video by Dr Chris Tisdell

Okay, as explained in that video, the Laplace Transform can only be used when the improper integral converges. So the next question is, what kind of functions can the Laplace Transform be applied to?

Piecewise Continuous Functions

This video clip explains one very important type of function, a piecewise continuous function, where the Laplace Transform really helps us with.

Dr Chris Tisdell - Introduction to Laplace transforms (part 2) [2min-56secs]

video by Dr Chris Tisdell

Now, from that last video, you can tell from his graph that a piecewise continuous function is NOT a piecewise function that is continuous. It is a piecewise function that has sections that are continuous, like the plot on the right. Applying the Laplace Transform to these kind of functions is where the Laplace Transform really shines.

Also notice in the last video, that he calculated the Laplace Transform of \(\cos(x)\) and \(\sin(x)\) without using the improper integral directly, which would require integration by parts. That is one of the powerful things about Laplace Transforms.

Table of Laplace Transforms

Fortunately, you are probably not going to be asked to use the improper integral at the top of the page to calculate most Laplace Transforms. You will use a Laplace Transform that you know and derive what you are looking for. In order to do that, you will usually be given a table of Laplace Transforms. These will also be used to get the inverse Laplace Transform (since no direct method exists that we can understand at this level of mathematics).

This video clip finishes the video shown previously and demonstrates how to use tables of Laplace Transforms.

Dr Chris Tisdell - Introduction to Laplace transforms (part 3) [13min-47secs]

video by Dr Chris Tisdell

Okay, now that you see the importance of tables for Laplace Transforms, you are probably expecting to see a table, right? Well, here is where you can find one. I highly recommend this book. It is free for you to download.
Engineering Math Workbook by Dr Chris Tisdell

Examples

Okay, let's look at a couple of examples. Try them on your own first. The tools you need are
1. the definition of the Laplace Transform at the top of the page
2. improper integral techniques (Make sure you use correct notation including the limit.)
3. integration by substitution
4. integration by parts
You should know all these by now, so you know how to do these calculations. The solutions are shown in video clips.

Example 1 - - Calculate the Laplace Transform of \(\displaystyle{ f(t) = e^{-at} }\).

This video clip shows the step-by-step solution. It also demonstrates the incredible power of the Laplace Transform. It is a little longer than the previous two video clips but make sure and watch the entire clip. Your time will be well spent.

Dr Chris Tisdell - Introduction to Laplace transforms (part 4) [21min-49secs]

video by Dr Chris Tisdell

Example 2 - - Calculate the Laplace Transform of \(tf(t)\).

Before you go to the practice problems, watch this video showing how to calculate the Laplace transform of \(tf(t)\). Yes, this probably appears your table, but it will help you get a better feel for Laplace transforms if you work through this video with the instructor. (If you feel adventurous, try working it on your own before watching the video. There is nothing new here, just an improper integral that uses integration by parts.)

Dr Chris Tisdell - Laplace Transform of tf(t) [7min-24secs]

video by Dr Chris Tisdell

Laplace Transform of a Periodic Function

A useful concept of the Laplace Transform is how it works with periodic functions. Periodic functions are, of course, functions that repeat in a periodic fashion. You've seen periodic functions in trigonometry. Sine and cosine are both periodic with period \(T=2\pi\).

The equation to find the Laplace Transform of a periodic function \(f(t)\) with period \(T\) is \(\displaystyle{ \mathcal{L}\{ f(t) \} = \frac{1}{1-e^{-sT}} \int_{0}^{T}{ e^{-st}f(t)~dt } }\)

Here is a great video clip showing the derivation of the last equation. Don't skip this video. It has important information that will help you when working with Laplace Transforms.

Dr Chris Tisdell - Laplace transform: square wave [8min-51secs]

video by Dr Chris Tisdell

Before you go on, work these practice problems. We do not have many since these are just to get your feet wet.

Okay, so why do we need Laplace Transforms? Why are they useful? We use them to solve differential equations that cannot be solved otherwise, sometimes involving some special functions. These special functions also have a purpose, which we discuss later. For now, let's just look at those functions and how the Laplace Transform can be applied.

special functions →

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Laplace Transforms - Practice Problems Conversion

[A01-650] - [A02-651] - [A03-653] - [A04-654] - [B01-652] - [C01-655]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

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Instructions - - Unless otherwise instructed,
- if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition
- 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.

Basic Problems

\( f(t) = t \)

Problem Statement

For \( f(t) = t \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

Solution

650 solution video

video by PatrickJMT

close solution

\(\displaystyle{ F(s) = \frac{1}{s-3} - \frac{16}{s^2+9} }\)

Problem Statement

For \(\displaystyle{ F(s) = \frac{1}{s-3} - \frac{16}{s^2+9} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{-1} \{ F(s) \} \).

Solution

651 solution video

video by PatrickJMT

close solution

\( f(t) = e^{3t} + \cos(6t) - e^{3t}\cos(6t) \); use a table

Problem Statement

For \( f(t) = e^{3t} + \cos(6t) - e^{3t}\cos(6t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a table.

Solution

653 solution video

video by Krista King Math

close solution

\(\displaystyle{ F(s) = \frac{s+3}{s^2+4s+13} }\)

Problem Statement

For \(\displaystyle{ F(s) = \frac{s+3}{s^2+4s+13} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{-1} \{ F(s) \} \).

Solution

654 solution video

video by Krista King Math

close solution

Intermediate Problems

\( f(t) = \sin(3t) \)

Problem Statement

For \( f(t) = \sin(3t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

Solution

652 solution video

video by Krista King Math

close solution

Advanced Problems

\( f(t) = t \cosh(3t) \)

Problem Statement

For \( f(t) = t \cosh(3t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.

Solution

655 solution video

video by Krista King Math

close solution
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