Due to a mindset that participation happens the day the discussion is due. I've pushed the due date up a day. Though it doesn't really matter. The point is to discuss and ask questions. I will keep track of the week you post, not under which post you make your comment.
Homework Bonus:
Because I can be a tough grader, I will implement a bonus system.
1) Check your work with a friend (they must be in my section, but they can be from either section).
2) Get the same answers. You may agree to disagree, if so, mark the problem with A2D.
3) Underneath your name, write HWBuddy: (Their name). You may have multiple buddies, write HWBuddies: (Name1), (Name2), ... .
Since this is a new idea, I will start by awarding 3 points for successfully doing this. If there is a high enough participation rate, I will increase the amount of bonus points in subsequent weeks. As such, encourage your classmates to participate in this bonus program!
Staple Penalty:
Lose 1 point for not stapling your paper.
Comment Bonus:
Participating at least once every week, ten times, and at least twenty times will gain approximately thirty points. Think of it like dropping another one or two homework scores.
Participating at least once every week, eight times, and at least twelve times will gain approximately fifteen points. Think of it like dropping another half or one homework scores.
[20120307]
Regarding Comment Bonus:
The goal should be to post consistently. This is best displayed by posting once every week. Beyond that, the distribution of how you want to post 20 times is up to you. So if at the end of the semester, I see you posted consistently, and reached at least 20, then you get the higher tier bonus. If you posted slightly less consistently and/or posted at least 12 times, but less than 20, then you get the lower tier bonus.
Regarding Homework Bonus:
So it's Thursday night and your friend hasn't even done their homework. Unfortunately, it doesn't really make sense to check your work with their incomplete work. The purpose of the bonus is to go through your completed homework, preferably on/before Wednesday, so that you have time to figure out and correct your mistakes. In fact, a great time is to check before or after the Wednesday lectures. Grab a trustworthy classmate or two and agree to have your homework completed by then. If he/she doesn't complete it as agreed, don't keep a grudge, but just move on and choose another classmate next time.
Because of the effort I put into seeing if you are checking your answers correctly, I will deduct 2 points for not adhering to the spirit of the Homework Bonus, and I will award 5 points for doing it correctly. There might be points in between.
Homework Bonus is recorded separately. So it doesn't directly affect your homework score. I will write "HB#" to the right of your homework score. It'll be added at the end of the year, after a homework score is dropped. This is good, because theoretically you can get a low score like 5 on the homework, and a homework bonus of 5 (though if you're checking your homework, you should probably be averaging 25). If I kept the scores together, and your lowest combined homework was a 10, then that homework would be dropped and you essentially lose the bonus. By keeping the score recorded separately, your bonus is preserved. Of course by the previous paragraph, you might have a negative total number Homework Bonus points, and in that case I'd treat it like a zero. Though honestly I only expect you to gain points.
Part of the spirit of the Homework Bonus should be that your buddy should theoretically get the same score you get. Ideally you should either both get a problem wrong or both get a problem right.
Another part of the Homework Bonus is to complete all the problems, covering for cases such as accidentally skipping a problem and doing the wrong problem.
And last but not least, the spirit of the Homework Bonus definitely says don't just copy, that's cheating.
I just want to confirm some notes:
ReplyDeleteFor the Integrated factor it should be in the format:
A(x)y'+B(x)y+C(x)=0
which turns into
y'+P(x)y=Q(x)
where P(x)=B(x)/A(x) and Q(x)=-C(x)/A(x)
right? because my notes are a little sloppy
That's right! You need to put the first-order linear term in the form $y^\prime+P(x)y=Q(x)$ for the integrating factor of the form $I(x)=e^{\int P(x) dx}$ to be useful.
DeleteI'm working on the 10.3 homework and my works seems to be a series of equivalent statements and it's taking me a while to organize them in a way to get the final answer. Do you have any advice for making my work more concise? (for example, number 16 that only begins with one known value). Thanks!
ReplyDeleteFor #15-20. You always have $x=r\cos\theta$ and $y=r\sin\theta$. Simply plug the given information into these expressions! Thus for number 16, we get $x=(4\sec\theta)\cos\theta$ and $y=(4\sec\theta)\sin\theta$. Then simplify and amaze yourself!
Delete[Originally posted Mar 3, 2012 03:35 PM]
Wow I don't know how I wasn't seeing that! Thank you!
ReplyDeleteNo problem!
DeleteAs a remark, having $x(\theta)$ and $y(\theta)$ (or in some of those problems $x(r)$ and $y(r)$) would bring you back to sections 10.1 and 10.2. Where the parameter is $\theta$ (or respectively, $r$).
You can choose to deal with understanding the parametric curve as is or by getting rid of the parameter.
I'm having a really tough time graphing the polar equations when I go through section 10.3. Can you go over how to do this in section? I feel like this is a pretty basic thing to do, and I'm just missing a small piece of information.
ReplyDelete-Ashleigh
There are different approaches, especially depending on the problem. One approach is to look at the equation in Cartesian coordinates. See the two replies I made above. Another is to simply plot points. Computing $\frac{dy}{dx}$ as a function of $\theta$ can also be useful.
DeleteIf you have a specific problem you're interested in, I can probably work out a small variation of it.
DeleteI understand how to do the assignments, and I have been doing practice problems to prepare for the test. The problem isn't I don't know how to do the problems, but they take me a long time to complete. I am worried about not finishing the test. Is there any advice you can give me?
ReplyDeleteWhen you are given the test, look at each problem for about ten to fifteen seconds. Then start off doing the ones you think you can do the easiest. The goal is to have your mind process the other problems as you do the easier problems. This might save you some time.
DeleteOtherwise, keep on practicing. If you're lucky, a problem on the test will be one that you've already done. This should hopefully cut down the time required to do the problem.
Also, "slow and steady wins the race." The extra time it takes you to complete a problem might be worth it if you're doing the problem correctly. Sometimes students are able to finish a problem really quickly, but accidentally copied the problem down wrong and ultimately get zero points for their efforts. Speed comes with practice.
What is the structure of the midterm on Wednesday?
ReplyDeleteAfter looking back to certain sections while studying I'm finding that I'm having trouble with 7.4 with Partial fractions. Mostly with finding A,B,C...ect. I'm able to find equations linking them but I'm having trouble solving them. I know it sounds like the simplest part to the problem, but it's the part I'm having most trouble with. Any suggestions on how to approach these partial fraction problems?
ReplyDeleteTrying reading this post: Related to Partial Fractions: Heaviside Cover-up Method
DeleteOtherwise, check that you're cross multiplying correctly. Sometimes students multiply by too much or too little. Also make sure you don't make sign errors. These can be disastrous.
What would you say the most effective way to set up and solve word problems involving solving differential equations is, step by step?
ReplyDeleteFirst, determine the independent variable. Write it down. (Often this is time, but it doesn't have to be)
DeleteSecond, determine the quantity that is changing. Write it down. (Examples include, volume, or the amount of some chemical)
Third, determine the equation for the change of rate. This is done by determining the "rate in" and subtracting the "rate out." By "rate in" I mean an increase in the quantity determined in step two and by "rate out" I mean a decrease.
For example: If it is the change of volume with respect to time then we would determine that $\frac{dV}{dt}=\text{"rate in"-"rate out"}$ where $V$ is the volume in meters cubed and $t$ is the time in seconds.
Hey, because we're studying now and doing practice problems, is there a certain section you would recommend spending more time on? Also I'm finding it takes a long time to draw slope fields. It's not that it's difficult, just time consuming. Will there be questions like that on the exam?
ReplyDeleteI think you have to tailor your studying based on your strength and weaknesses. Perhaps warm-up and cool down with what you're best at; this boosts my confidence. But for bulk of the studying, try and eliminate your weaknesses. At the same time, you need to guess the likelihood of what will be covered on the exam.
DeleteFor example, definitely make sure you know how to deal with setting up and solving a partial fractions problem.
I love guessing what's on the exam and seeing how many I get right. What do you think will be on the test?
If I knew what would be on the exam I would tell you, but unfortunately I don't. The best I can say is save the time consuming parts of the exam for last and if required to draw a slope field, try plotting at least a 3x3 region and if possible a 5x5.
Reviewing the book's four-step strategy on page 495 to 496 should prove useful. Can you do all the problems on page 499 and 500? Not enough time? How about most?
DeleteNote that we did #31 on page 499 in class and determined that one method to solve it is given by Example 5 on page 498.
Do you think hyperbolic functions will be covered on the exam? If so, can we go over them in section because I don't exactly understand what they are/represent.
ReplyDeleteI recommend you read the paragraph on page 482. "NOTE As Example 5 illustrates..."
DeleteDo you think we should study some of the problems at the end of certain sections that ask you to prove theories. Generally these problems don't have any numbers.
ReplyDeleteTypically the more proof oriented problems are there to expose you to thinking more abstractly and ideas slightly beyond the basics. Sometimes professors like to add these types of problems as extra credit. I've never TA'd for Dr. Brown before, and so I don't know his style. You might study a couple, but with a lower priority than other problems.
DeleteI looked over orthogonal trajectory in section 9.3 and I dont understand what it is. I don't know if I should skip over this section because it probably isn't as important.
ReplyDeleteFigures 7, 8, and 9 on page 597 and 598 might be useful in picturing what orthogonal trajectories are about. Example 5 goes through how to find them. As you weren't assigned a homework problem on the topic, there is a less likelihood that they'll be on the exam. However, if you have time, it's worth reading the example.
DeleteI remember Professor Brown saying something about being able to use trigonometric substitutions as long as the function is one to one. Can you elaborate on that?
ReplyDeleteSure, let's say our bound was $x=0$ to $x=1$ and we want to make the substitution $x=a \sin\theta$, like on page 478.
DeleteWithout restricting the domain of sine, then $\sin(\theta)$ would not be one-to-one and we'd be unable to unambiguously choose bounds for $\theta$. One such choice is $\theta=0$ to $\theta=\pi/2$ while another is $\theta=\pi$ to $\theta=5\pi/2$.
Thus, we restrict the domain of sine in order to make $a\sin \theta$ one-to-one. See page 478.
Why is it that the graph of a parametrized function may be differentiable but the original x,y function may not be? or vice versa
ReplyDeleteFirst, given a function $f(x)$, we can turn its graph into a parametric curve as $(t,f(t))$.
DeleteSecond, it helps to think of a parametric curve as describing the motion of a particle. $(x(t),y(t))$. We can break down motion in the xy-plane into motion in the x-direction and motion in the y-direction. Thus differentiability of the motion with respect to time depends on the differentiability of the individual functions $x(t)$ and $y(t)$.
Thus the parametric curve $(t,f(t))$ is differentiable if and only if $f$ is differentiable.
Thus while $(t,f(t))$ might be differentiable, there's no reason why an arbitrary parametrization $(x(t),y(t))$ needs to be differentiable.
Most parametric curves you've encountered are nice. But theoretically, a parametric curve could suddenly double back on itself, or stop at a point for a length of time $t$.
When finding the length of a curve do we need to convert the limits?
ReplyDeleteIn general, the bounds for a definite integral should belong to the variable in which you are integrating.
DeleteAlways keep track of which variable the bounds are in.
Example:
$\int _{2}^{5} \frac{1}{x\ln(x)}dx$
$=\int _{x=2}^{x=5} \frac{1}{x\ln( x )}dx$
$=\int _{x=2}^{x=5} \frac{1}{u}du$
$= \left. \ln (u) \right|_{x=2} ^{x=5} $
Because I kept track of the variable of the bounds, I won't accidentally plug in 2 and 5 for $u$. At this point we can change the bounds for $u$, or plug $u$ back in.
When finding the length of the curve, if you're finding the length of the curve by using $y$ as a function of $x$, then the bounds will be in the $x$ variable. If it's a parametric curve with parameter $t$, then the bounds will be in $t$.
Hopefully that makes sense.
Professor Brown mentioned that we should study problems that resemble our homework, so does that mean we do not need to know orthogonal trajectories, or a word problem for differential equations such as the mixing problem in the book?
ReplyDeleteIf I was a student and if that's what he mentioned, then because neither orthogonal trajectories, nor word problems for differential equations were assigned for homework, then I would not worry about them.
DeleteActually, different professors have different styles, and this is the first time I've TA'd for Dr. Brown, so I don't know his style. As an undergrad, I've encountered professors who give practice exams that are basically the same as the real exam. Others give strong hints as to what is on the exam, but don't give everything away. And there are some that simply mislead their students entirely, such as lecture on one topic and test entirely from the book.
DeleteAs such, a safe route is to play it both ways.
In this situation, this would mean to focus some of your time on problems that resemble your homework and another portion of time on topics which Dr. Brown seemed to spend time on.
It was somewhat unclear as to what identities and formulas we will be given on the test?
ReplyDeleteDo you know exactly which ones we will be given? are you allowed to say? I think this is very important because if we are not given them then there is a huge emphasis on memorizing them as some of the identities will undoubtedly be the key to solving some of the problems we are given.
Ryan Schneider
I stand by what I said in section today. The likelihood is that if you need an identity, then it'll be on the list. It would otherwise be cruel to tell students you're going to provide them with trigonometric identities and omit the ones they need.
DeleteBut apart from this, I would say the more you know, the less time you spend looking at a formula sheet and the more time you spend trying to solve the problem.
Speaking of which, always make sure that you've arrived at the answer to the question being asked.
While doing the homework for section 10.3 I'm not really understanding problem like 16 and 20 where you have to change r=(constant)(trig function) into a Cartesian equation. The book doesn't explain it very well and doesn't give very helpful examples. Any hints?
ReplyDeleteWhen I saw this post, I was sure I had replied to it and was wondering if I would need to go through all my explanation again. Thankfully I did answer it already. See the same post on Class Discussion due 20120311. CTRL+F "MaggieFruehan".
DeleteCan we go over how to determine the bounds when asked to find the area or length of part of a polar or parametric curve? Thanks!
ReplyDeleteI would suggest you come to section prepared with a specific problem for the TA to answer. If she does not get to your question, feel free to inquire on this blog.
DeleteFor problems like #8 section 10.3 where we need to sketch a polar region (it gives us limits on r and theta), there is no equation so do we just give any value of r (within the limit) any value of theta (within the limit) and then graph it? And how many points is enough because I don't know if we're trying to obtain a specific shape.
ReplyDeleteI don't have the book on me, but while there might not be an equation, there are certainly inequalities, and when there are inequalities, one can come up with equalities.
DeleteExample:
Sketch the polar region $1\leq r \leq 2$ and $\pi/4\leq\theta\leq3\pi/4$.
One attempt at going about the problem is to figure out the equalities and draw those first.
$r=1$ is the circle of radius 1. $r=2$ is the circle of radius 2. $\theta= \pi/4$ is the line $y=x$. $\theta = 3\pi/4$ is the line $y=-x$. These for sides, together with the inequalities, give us the picture as I've generated here on Wolfram Alpha.
Note: I could have had strict inequalities when defining the polar region. In which case, the line that creates that boundary in the region should be dotted, to indicate such points are included in the region.
DeleteHow do we choose which function should be g(x) (comparison method) for problem 52? I've only figured out how to do the problem graphically.
ReplyDeleteI'm not sure how to begin problem 38. I was thinking the comparison method but again I don't know which function to choose as g(x).
ReplyDeleteIs it possible for me to use the square root of x as g(x) in order to compare it to the integral in problem 40?
ReplyDelete