Next time bring your books. It'll help the class move along more smoothly and allow for better group discussions.
Do you like working in partners? Do you like being forced to work with someone or would you rather partner with someone on your own?
What don't you like about the discussion section?
What do you like about discussion section?
People did a good job of stapling their papers, I would like to remind that failure to do so will result in a loss of a point on homework.
I liked the examples we did in class today because they were more thorough than the in-class examples. The example with the repeated irreducible quadratic factor was helpful. I like that we use the textbook's examples and terminology so that when I review the text on my own I am already familiar with the material.
ReplyDeleteThe examples that were used in class today gave me a better understand of the concepts that we learned in class. I also found that going over how to set up a partial fraction with an irreductable quadric to be very helpful
ReplyDeleteTim... When are your office hours and where?
ReplyDeleteThanks a lot,
Marcelo
The information is listed on the right-hand side of this website.
DeleteI also liked the examples we did in class because they were more involved than the easy straightforward ones we did in class. Also I think people tend to sit with the people they'd choose to work with anyway so I don't think it makes much difference since you paired us based on who we were sitting near.
ReplyDeleteAlso some math comic relief. My friend told me this joke:
A guy at a party sees e^x sitting alone in the corner and he goes up to it and says "why don't you try to integrate yourself into the party?" and e^x sighs and says "it wouldn't make a difference"
Hahaha. My friend once told me a joke related to $e^x$. I personally didn't remember how to tell it, but luckily I was able to find a version of it online. It's a bit educational too. :p
DeleteThe Deadly Differential Operator
(from http://www.dansmath.com/pages/jokes.html)
A constant (linear) function and an exponential function are out walking, when, off in the distance, they spot a differential operator. The constant function cries out, turns around, and runs away. The exponential function quickly follows. The exponential function asks, "Hey, come on, what's the matter? Don't you want to meet her?" The constant function replies, "Well... no. She's a differential operator. If we meet, she'll differentiate me, and there'll be nothing left of me!" The exponential function nods. "Okay, then; I'll go and talk to her. She doesn't scare me -- I'm e to the x!" With that, the exponential function walks, alone, to the oncoming differential operator. He introduces himself, "Hi! I'm e to the x." The differential operator replies, "Hi! I'm d/dy."
I felt like we weren't really given enough time to do the example questions that you put on the board during class. I might be alone in this but it seemed to me that I ran out of the time as soon as I looked down at the question.
ReplyDeleteThank you for your feedback. There are multiple reasons for the short amount of time you were given to process and answer the question.
DeleteFirst, I believe that you all have the ability, if you choose to study more, to answer the questions I pose in the time I give you.
Second, the old style would just be for me to put the problem on the board and continue doing it. I believe giving you a couple extra seconds to consider the problem helps engage you while I proceed do the problem, and if you're lucky you have already solved it.
Third, the time in class was only 50 minutes, and I had more prepared than we were able to cover. I probably could have skimmed down on the method recognition a little bit. But I thought that was an important exercise to get you to do at home, so I emphasized doing it in class.
Fourth, some of the examples I put on the board (in particular the partial fraction examples) were already displayed in the comments of this blog. You could have already attempted them prior to class.
Thanks again, Thomas! Hope to hear more feedback from you and others again! :)
Tim,
ReplyDeleteI had a hard time with 7.3.40 in this past homework. It would be good if you clarify during session next week.
Thanks,
Marcelo
Were you able to draw the picture? Drawing the picture is often an overlooked step. Were you able to set up the integral? Part of setting up the integral is determining where the two curves intersect. Did you decide to integrate along $x$ or $y$. Sometimes you have a choice and sometimes one way is easier than the other. Note that you only have to find the area of one region, and the area of the other region is just the difference between the area of the circle and the value you found.
DeleteHello Tim,
ReplyDeleteI was wondering if for our next session you could explain Euler's method.
Thank you,
nahyr
I enjoyed working in groups to set up the problems this week. It helped me to see the patterns. However, I also like going through example problems as a class. Maybe there could be a good mix of both during section? Starting with working in groups for a few, then moving on to class problems, which I think prepare us well for homework.
ReplyDeleteI also would appreciate an explanation of Euler's method.
-Ashleigh Thomas
Nothing to say here that hasn't been already said by the above commenters, with the exception of my continued dislike for group problems as opposed to class problems. I'm thinking along the lines that people can get together in a group to do math problems at any time, but that the TA session is organized to ensure the explanation of concepts on a more in-depth level than we usually have time for in class.
ReplyDeleteOnce again, just my two cents.
I was also having problems understanding exactly what euler's method is and why it is used. I was therefore especially having some trouble with homework questions 22 and 24 of section 9.2.
ReplyDeleteMichelle Bohrson
I know there is a relatively organized way of working through an Euler's method problem. I just remember being able to structure all the X_n, Y_n, etc on a chart, and it made it a lot easier to work through. Does anyone know what I'm trying to describe hah?
ReplyDeleteYes, I've also been trying to figure out a way to structure it too. This is what I came up with: Organizing the Work for Euler's Method
DeleteTim,
ReplyDeleteMy concern is similar to Michelle's. I also don't fully understand Euler's method. If you could go over 22 it would be very helpful.
Thanks,
Marcelo
I still do not fully understand how you find the integrating variable when you are doing linear ODEs. Could you clarify.
ReplyDeleteI'm assuming you mean integrating factor. The book explains why you want to find an integrating factor on page 616 to 617. As to how, then just use the formula on page 617.
DeleteStep 1: Put your linear ODE into the following form: $$\frac{dy}{dx}+P(x)y=Q(x)$$
Step 2: Determine the integrating factor. When possible, simplify it. $$I(x)=e^{\int P(x) dx}$$
Step 3: Multiply both sides by $I(x)$. You will get (by design of the integrating factor's job) $$\frac{d}{dx}\left( I(x)y \right)=I(x)Q(x)$$
Step 4: Integrate both sides with respect to $x$. $$ I(x)y =\int I(x)Q(x) dx$$
Step 5: Finish the problem.
I think the discussion section is helpful and keeps people engaged. Also, I was having some trouble understanding how to do questions 24 and 42 in section 9.3 of the homework and it would be really helpful if we could go over those.
ReplyDeleteThanks!
i htink it would be a goo idea to discuss the concepts an logistic behind euler's method. examples of it make sense, but recognizing situations to use it, and getting started is extremely difficult.
ReplyDelete-max aserlind
I really enjoyed last weeks session because we cover lots of problems and you helped us with important identities that we should know for integrating. Thank you!
ReplyDeleteHi Tim,
ReplyDeleteIn last Friday's lecture, we cross-multiplied terms in the form dy/dx to move all of the y terms on one side and the x terms on another side. I thought dy/dx was Leibniz notation for an integral, and I was somewhat confused when we did that in lecture. It would be very helpful if you could clarify that for me. Thanks!
B. Arda Ozilgen
This is a wonderful inquiry and I have posted a reply here: Regarding the use of dy/dx
DeleteIt all makes sense now! Thank you...
DeleteI like working practice problems in lecture. You hit us with a pretty good number and we don't spend a lot of time on one if we get stumped because you show us the process. I'm not a big fan of working with partners. I also had a question regarding linear diff. eq. A simple example in lecture was y'= x+y. To me it looks like a product. We first had to find the integrating factor to solve it. It seemed to me that x + y is a pretty simple product though. (dy/dx)xy= x+y right? or would it be (dy/dx)(xy)= xdy+y, which is why we had to use the integrating factor?
ReplyDeleteI suppose ultimately those who enjoy working with partners work with partners and those who like to talk it out will do that. But the important part is to have a moment to process a problem yourself.
DeleteAs for your question, one thing to keep in mind that $y$ is a function of $x$.
Thus $\frac{d}{dx}(xy)=y \frac{d}{dx}(x)+ x \frac{d}{dx}(y)=y+x \frac{dy}{dx}$.
The reason we need an integrating factor is to make the equation look like $\frac{d}{dx}[F(x)\cdot y]=G(x)$. Under that situation, we could integrate both sides with respect to $x$ and solve for $y$. We'd obtain $y=\frac{1}{F(x)}\int G(x) dx$.
What is the integrating factor for the simple example you gave?
By the way, it might be convenient to put the equation in the following form: $y^\prime -y = x$
That is $y^\prime + P(x) y = Q(x)$. See page 617.
Does anyone know what significance β has? As in "sinβx"? Is it just a variable representing an arbitrary constant?
ReplyDeleteAny symbol can have any significance depending on the context!
DeleteTry to compute the following three integrals:
$$\int \sin (\beta x) dx$$
$$\int \sin (\beta x) d\beta $$
$$\int \sin (\beta x) dy$$
My comment for this week is about Differential Equations. It occured to me that ODEs are like mathematical riddles. They bascially ask "find me a function whose derivative plus the function equals the function" (or something like that). The difference with this is that we have methods to solve a mathematical riddle, whereas a metaphorical riddle can't necessarily be solved by some kind of method. Just food for thought.
ReplyDeleteWhat would a real world application of slope fields be?
ReplyDeleteSomething that occurred to me when you put the variety of integrals on the board for us to solve: If we have
ReplyDelete(x^2)/(x+2)^4
would we need to do long division before we attempted partial fractions? If we expanded the denominator, its degree would be higher than the numerators, so I'm not sure if that is necessary.
We wouldn't, because the degree of the numerator is less than the degree of the denominator. We only need to use long division if the degree of the numerator is greater than the degree of the denominator.
DeleteSorry, I realized that's a terrible example. One that better illustrates my problem is:
Delete(x^5)/(2+x^3)^2
I don't see anything different. In your first example, the degree of the denominator is 4 and the degree of the numerator is 2. 2 is less than 4. In your second example, the degree of the denominator is 6 and the degree of the numerator is 5. 5 is less than 6. Thus in both examples, you don't need to do long division.
DeleteIn any case, you should try to set-up and solve both. Try to solve it traditionally and with the method of plugging in values.
Just thought it was pretty cool how parametric equations can be used to sketch a curve that doesn't have to be a funtion.
ReplyDelete