Practice problems for Exam 2 discussion area
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.
