For a sanity check, can someone prove 1b, where we have to find f(t) from F(s) using 1a? There is an answer shown, but no work shown...
The Laplace Transform of the convolution in 1a would be the product of
e^(-pi s)/s and (1/s plus s/(s^2 + 1))
and that will be highly related to the question in 1b.
Conversation between a student and Steve Bell:
> 1. for the sample midterm found here: > http://www.math.purdue.edu/~bell/MA527/prac2a.pdf > > For #3, what i tried to do is just do the > Laplace transform on each function and then > multiply them together.
Yes, that will give you the Laplace transform of the convolution. But they also want you to compute the (Laplace) convolution of those two functions. I did a problem exactly like that near the end of my lecture today. Note that, from the point of view of Laplace transforms u(t) times e^t is the same as e^t.
> For #6 I looked up Laplace for periodic > function and came across something totally > different.
The relevant formula is the last one on the cover page of the exam found at
http://www.math.purdue.edu/~bell/MA527/laplace.pdf
> For #10 I haven't completed it yet but my plan > was to plug in f(x) in Parseval's identity and > see if the summation of the coefficient and the > integral of the function will be the same.
That is the right idea. The left hand side will be bigger than something and the right hand side will be smaller than something, and you get an impossible situation.
> 2. for the mid term found here: > http://www.math.purdue.edu/~bell/MA527/mid2.pdf > > For #4-b we are asked to find the real Fourier > series from the complex one, is there a way to > do this without doing some of the calculation again.
I mentioned today that there will be no problems about the complex Fourier SERIES on Exam 2. The point of the question you refer to is that the integrals for the complex Fourier series are very easy to compute. To get the real Fourier series from it, you just expand the e^(inx) and e^(-inx) in terms of sine and cosine and then take the real part of what you get.
Another conversation between a student and Steve Bell:
> 1.) Can I expect the test to be similar to > the homework like the first test?
Yes. I like to make the test rather straightforward. It measures everything I want to measure that way.
> 2.) I hope it is not too late to ask, but > could you explain the partial differential > problem tomorrow? Problem 5 from Lesson 24.
I wrote out a careful solution of that problem at
http://www.math.purdue.edu/~bell/MA527/Solutions/grd8.pdf
That problem was a good way to show you how useful these things are, but it is a little too involved to be an exam question, so I don't want to take time on Friday to talk too much about that.
> 3.) Will I need trig identities for the exam?
No. No trig identities. (Except basic things like how to integrate sine and cosine.)
> 4.) I am a bit confused with the relevance of > problem 2 on Lesson 27, where we explained the > effect of damping and spring rates on Cn. We > didn't talk about it much, and I don't see it > mentioned in the review material, so is this important?
It is important, but I'd have a hard time writing an exam question about it. The point is that resonance can happen from terms farther down in the fourier series than the principal frequency.
> 5.) Since you said that we won't be solving complex > integrals, will we not need the table on page 534?
I'll give you the Laplace Transform table. You won't need tables of anything else as long as you know how to integrate the basic functions like sin, cos, e^x, etc. You might want to write down the basic rules of Fourier transforms on your crib sheet though.
More conversations:
> Prob. 6: > Find the Laplace transform of the pi-periodic extension > of the function f(x) = x defined on the interval [0, pi]. > > My approach: > > For this problem, I made use of the formula of Laplace > Transform for periodic functions. Is my approach correct? > or should I compute the even or odd periodic extension of > this function and then compute the Laplace Transform for > the resultant function.
That formula says it all. It includes all the work that doing a periodic extension and summing a geometric series would be.
> Prob. 10: > Suppose the Fourier series of some 2-periodic function > f(x) starts from 1 + sin x + cos x (i.e. a0 = a1 = b1 = 1). > Could it happen that |f(x)| <= 1 for all x? > (Hint: use Parseval's equality/Bessel's inequality). > > My approach: > > since a0=a1=b1=1, I think the function f(x) is neither > an even nor an odd function. To make use of the Parseval's > equality/Bessel's inequality, I think we need the value of > the remaining co-efficients a2,b2 etc.. > I am not able to proceed from that point. It would be nice, > if you can provide some guidance on solving these problems.
If you write out Parseval's, you'll see
2+1+1+ a sum of squares = (1/pi) integral of |f|^2
The sum of squares is greater than or equal to zero. So the left hand side is greater than or equal to 4. If |f| where too small, the integral would be too small to be greater than or equal to 4...
Question from Mnestero:
Dr. Bell in your response above how did you get 2+1+1? I thought it should be 1+1+1 + a sum of squares. Am I missing something?
---
Question from djkees:
In the first set of practice problems #8, where you find the cosine, sine, and complex transforms, is the main point of that problem just to compute what appears to be a difficult integral, or am I going about it incorrectly?
My integral is x*exp(-x)*sin(w*x)dx for the fourier sine transform. I'm assuming I'm just doing something wrong, can someone point me in the right direction?
Hmmm. That exp(-x) makes me think of Laplace Transform. In fact, it makes me think of x=t and s=1.
--- From Mnestero:
Using the information above in prob 8, for the cosine transform I used L{tf(t)}=-F'(s) to get the Laplace of 1/(s^2+w^2)-2s^2/(s^2+w^2)^2. I substitute S=1 to get the answer. Is this correct?
Question from Chris:
Is there going to be a circuit problem on this exam? I want to know if I should review that Four-Terminal RLC-Network example in the book.
From Steve Bell:
Nope.
Professor,
I noted from above in regards to the PDE of section 12.12 with respect to LaPlace that you mentioned the problem too involved. Is it then safe to say that a Laplace PDE question will not show up on the midterm?
Best regards
Another conversation:
> 1) I was wondering if we're allowed to > take (rip) out the Laplace Transform Table > cover page to use during the exam and then > staple it back afterwards?
I suppose you could do that. In fact, you could rip it off and keep it as long as you write your name on the first page of what you turn in.
> 2) Do I need to include trig identities > in my cribsheet (for example to integrate sinAsinB)?
No, you won't need things like that.
> 3) In the Laplace Transform Table that > we will be given, the Laplace transform of > cosh at for example is not given, is it safe > to assume that whatever Laplace Transform > is not in there we will not require during > the exam? I ask this because I am wondering > if maybe you expect us to see that (for example) > sinh at=(e^(at)-e^(-at))/2 and then go on from there?
You won't need hyperbolic functions, but you will almost certainly need exponential functions. I won't expect you to convert exponential functions into hyperbolic functions or vice versa. They really are the same animals.
> 4) We are allowed one cribsheet (back and > front) right? Just reconfirming..
Right.
> 5) Will I be at a loss if I just do the past > 2 old exams posted, the Sample for the Midterm, > and instead of doing > (http://www.math.purdue.edu/~bell/MA527/prac2.pdf) > just go over the HW questions for review?
Anything you can do to convince yourself that you understand the various problems that have come up in the homework is a good thing to do. Going over the homwork problems is a less stimulating way of doing that than taking a stab at the practice problems. The practice problems give you a chance to test your skills on new problems that are slightly different than homework problems -- that's more like an expereince on an exam.
Has anyone looked at #5 from practice exam two (http://www.math.purdue.edu/~bell/MA527/prac2.pdf) I was wondering how to represent r(t) as an integral, is it to find the Laplace and inverse Laplace we stick what we find Y(s) to be (that includes r(t)) into an integral to find y(t) ?
From Jake Eppehimer:
Using convolution, I got r(t) convolved with (1/2)sin2t. Putting it into integral form, I got (1/2) integral from 0 to t of r(Tau)sin2(t-Tau)dTau
I apologize that I don't know how to type those in a better format.
The key here is convolution. If you think that way, you'll get the answer in the form of an integral involving your numerical data r(t).
From Jake Eppehimer:
Referring to problem 4 in "More Practice Problems," I got x1 = e^t - 6e^t and x2 = -2e^t by using laplace and cramers rule. However, when I plugged it in to check my answers, I realized that x1 needs a -7 and x2 needs a +2 to make it work. I don't understand the issue here...
From Mnestero:
In Problem 9 of the first practice set, is (1) asking to find the odd and even extensions, or is it asking to calculate the fourier series for the sawtooth equation?
From Christine: http://www.math.purdue.edu/~bell/MA527/Sample_Mid2.pdf I have a sign error on 1a. I got [-1 + cos(t-pi)], from solving -cos(t-tau) from pi to t...? Thanks!
I think the issue may be with the integral of [sin(t-tau)dtau], for that I get cos(t-tau) evaluated from pi to t. If you took the integral with respect to t, or [sin(t-tau)dt] you would get -cos(t-tau). -MBoersma
From Subrina:
Does anyone get the solution for #5b on mid2 (http://www.math.purdue.edu/~bell/MA527/mid2.pdf)?
Part 5b is taking the Fourier cosine transform of the answer to part a (to get the original function back) at x=2Pi. In the problem they already did some simplification so its not as obvious. This converges to the mid-point at the jump. The work is shown next to the answer box for part a. -MBoersma
Is this as simple as indicating how 5b can be written in the form of the inverse transform? Therefore, f(x) when x = 2pi, as in 5b, is the average of "1" and "0" where x = 2pi? No solving of integrals here, right? --christine
