
Laplace Transforms
In the last few chapters, we have looked at several ways to use
The Laplace transform is defined in terms of an integral as
7
e""
ft
Note that the input to a Laplace transform is a function of time, /(/), and the output is a function of frequency, F(j), Although many real-world examples require the use of
Let's stan with a simple example. Here we calculate the Laplace transform of /(f) = t. We have
This is an improper integral, so we express it in terms of a limit, which gives
Now we use integration by pans to evaluate the integral. Note that we are integrating with respect to t, so we treat the variable s as a constant. We have
u—t dv — dt
du=dt v
— — ye_ir.
Then we obtain
= + +
= ~K + °1 -
= JinL[[-i,-]-±[e--lj]
- c + c
= 0-0 + -L
s“
_x
2* s
1. Calculate the Laplace transform of /(/) = 1.
3.Calculate the Laplace transform of /(/) = : (Note, you will have to integrate by parts twice.)
Laplace transforms are often used to solve differential equations. Differential equations are not covered in detail until later in this book; but, for now, let’s look at the relationship between the Laplace transform of a function and the Laplace transform of its derivative.
Let’s start with the definition of the Laplace transform. We have
WW! = r™ r'
= lim / e~st fifth.
4.Use integration by parts to evaluate Jjm^ e~sl fifth. (Let « = /{/) and dv — e '!dt.) After integrating by parts and evaluating the limit, you should see that
Then,
Thus, differentiation in the time domain simplifies to multiplication by s in the frequency domain.
The final thing we look at in this project is how the Laplace transforms of fit] and its antiderivative are
related. Let g(r) — f(u}dii. Then,
¦'o
lim /
:-* caj" 5.Use integration by parts to evaluate hrn^y e ’ g(t)dl. (Let u = gif) and dv = e dt. Note, by the way,
that we have defined gif, du — fifth.)
As you might expect, you should see that
L|^(r)| = |-L[/(/)i.
Integration in the time domain simplifies to division by s in ±e frequency domain.

Want to see the full answer?
Check out a sample textbook solution
Chapter 3 Solutions
Calculus Volume 2
Additional Math Textbook Solutions
Elementary Statistics: Picturing the World (7th Edition)
Introductory Statistics
Elementary Statistics (13th Edition)
Calculus: Early Transcendentals (2nd Edition)
Calculus: Early Transcendentals (2nd Edition)
- For the system consisting of the three planes:plane 1: -4x + 4y - 2z = -8plane 2: 2x + 2y + 4z = 20plane 3: -2x - 3y + z = -1a) Are any of the planes parallel and/or coincident? Justify your answer.b) Determine if the normals are coplanar. What does this tell you about the system?c) Solve the system if possible. Show a complete solution (do not use matrix operations). Classify the system using the terms: consistent, inconsistent, dependent and/or independent.arrow_forwardOpen your tool box and find geometric methods, symmetries of even and odd functions and the evaluation theorem. Use these to calculate the following definite integrals. Note that you should not use Riemann sums for this problem. (a) (4 pts) (b) (2 pts) 3 S³ 0 3-x+9-dz x3 + sin(x) x4 + cos(x) dx (c) (4 pts) L 1-|x|dxarrow_forwardA movie studio wishes to determine the relationship between the revenue generated from the streaming of comedies and the revenue generated from the theatrical release of such movies. The studio has the following bivariate data from a sample of fifteen comedies released over the past five years. These data give the revenue x from theatrical release (in millions of dollars) and the revenue y from streaming (in millions of dollars) for each of the fifteen movies. The data are displayed in the Figure 1 scatter plot. Theater revenue, x Streaming revenue, y (in millions of (in millions of dollars) dollars) 13.2 10.3 62.6 10.4 20.8 5.1 36.7 13.3 44.6 7.2 65.9 10.3 49.4 15.7 31.5 4.5 14.6 2.5 26.0 8.8 28.1 11.5 26.1 7.7 28.2 2.8 60.7 16.4 6.7 1.9 Streaming revenue (in millions of dollars) 18+ 16+ 14 12+ xx 10+ 8+ 6+ 2- 0 10 20 30 40 50 60 70 Theater revenue (in millions of dollars) Figure 1 Send data to calculator Send data to Excel The least-squares regression line for these data has a slope…arrow_forward
- 2arrow_forwardAn engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides to do it in three sections. The first section runs from point P to point Q, and costs $48 per mile to lay, the second section runs from point Q to point R and costs $54 per mile, the third runs from point R to point S and costs $44 per mile. Looking at the diagram below, you see that if you know the lengths marked x and y, then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline. Please show your answers to 4 decimal places. 2 Miles x = 1 Mile R 10 miles miles y = milesarrow_forwardhelp on this, results givenarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage