I made a big mistake in my first section and a small mistake in my second section.
We could solve problems like number 40 as follows.
Problem: When do two curves $r_1(\theta)$ and $r_2(\theta)$ intersect?
Tuesday, March 27, 2012
Class Discussion due 20120401
Again never a due date. Keep asking questions.
Things are getting busy for me, but that doesn't mean I won't be there to help! With that being said, consider the many other useful options. Graduate students in the math help room should usually be able to help with conceptual questions as well as homework questions. Dr. Brown also has office hours. I rarely see anybody during mine, so do drop by!
But I like these tips a high school teacher once told the class:
"College is NOT a big high school."
: He literally emphasized the word 'not'.
"Work smart, not hard."
: Don't take this literally. You should work hard, but you should work hard on the right things. In other words, you don't go into the forest and cut down a bunch of trees if you want to build a brick house. For example, pay attention to concepts which Dr. Brown emphasizes.
Things are getting busy for me, but that doesn't mean I won't be there to help! With that being said, consider the many other useful options. Graduate students in the math help room should usually be able to help with conceptual questions as well as homework questions. Dr. Brown also has office hours. I rarely see anybody during mine, so do drop by!
But I like these tips a high school teacher once told the class:
"College is NOT a big high school."
: He literally emphasized the word 'not'.
"Work smart, not hard."
: Don't take this literally. You should work hard, but you should work hard on the right things. In other words, you don't go into the forest and cut down a bunch of trees if you want to build a brick house. For example, pay attention to concepts which Dr. Brown emphasizes.
Sunday, March 18, 2012
Class Discussion due 20120325
There's actually no due date. Just post.
I'll do my best to address finding bounds, but ultimately you just need to practice a lot. Also, I know there's a lot of posts, but it might help to read through questions that other students have come up with, and my replies to them.
I know it's not easy to do some of the above, since in general I find it hard to follow my own advice. But I feel there's no harm in giving you ideals which I strive to follow.
Reiterate Some Goals with the Commenting System:
(1) I want students to comment consistently and abundantly.
(2) Students wait until its too late to ask questions.
(3) It's good to ask questions, but sometimes we need to ask the right questions. Something I'm still learning to do myself.
(4) Realize that it's okay to ask the TA and professor questions.
(5) Realize that it helps to reflect on what you've learned.
(6) Helping each other.
I'll do my best to address finding bounds, but ultimately you just need to practice a lot. Also, I know there's a lot of posts, but it might help to read through questions that other students have come up with, and my replies to them.
I know it's not easy to do some of the above, since in general I find it hard to follow my own advice. But I feel there's no harm in giving you ideals which I strive to follow.
Reiterate Some Goals with the Commenting System:
(1) I want students to comment consistently and abundantly.
(2) Students wait until its too late to ask questions.
(3) It's good to ask questions, but sometimes we need to ask the right questions. Something I'm still learning to do myself.
(4) Realize that it's okay to ask the TA and professor questions.
(5) Realize that it helps to reflect on what you've learned.
(6) Helping each other.
Wednesday, March 14, 2012
Practice Test Session Proposal
A student wrote:
I was wondering if we could hold a practice test session for the next test that is timed. I know it would help me along with others to get into the habit of solving problems under a timed environmemt.
Thank you!
My initial thoughts:
Great idea in theory, but typically there is low interest or two many people are busy. I also try similar things as it is with little response.
My reply:
I tried to address this prior to the first midterm in several ways. I posted questions on the blog, titled "Test Your Knowledge."
I did method recognition in class, which was meant to getting you to realize you can do many problems quickly, and/or you have the ability to practice and improve your abilities. It was also meant to get you to doing such on your own.
If you can get at least 30 people from my section (that's half my students) to agree to come to such a practice test session, I will write problems and monitor such a session.
Action:
Sign-up here to semi-commit to coming to such a practice test session. If more than 30 people sign up then a date will be determined. Once the date of the practice test session is determined, you must cancel a week before the planned date, or a really good reason for not coming, otherwise lose 10 points. Bonus 1 point for coming. It'll be less than 50 minutes long. Spending about 15 minutes after to go over and grade it.
I was wondering if we could hold a practice test session for the next test that is timed. I know it would help me along with others to get into the habit of solving problems under a timed environmemt.
Thank you!
My initial thoughts:
Great idea in theory, but typically there is low interest or two many people are busy. I also try similar things as it is with little response.
My reply:
I tried to address this prior to the first midterm in several ways. I posted questions on the blog, titled "Test Your Knowledge."
I did method recognition in class, which was meant to getting you to realize you can do many problems quickly, and/or you have the ability to practice and improve your abilities. It was also meant to get you to doing such on your own.
If you can get at least 30 people from my section (that's half my students) to agree to come to such a practice test session, I will write problems and monitor such a session.
Action:
Sign-up here to semi-commit to coming to such a practice test session. If more than 30 people sign up then a date will be determined. Once the date of the practice test session is determined, you must cancel a week before the planned date, or a really good reason for not coming, otherwise lose 10 points. Bonus 1 point for coming. It'll be less than 50 minutes long. Spending about 15 minutes after to go over and grade it.
Class Discussion due 20120318
Write stuff. Try to reply to each other's questions.
There's always time for learning material that has already been covered. Professors occasionally like to put a question missed by students on the first midterm back on the second. Or from midterm to final.
I make a due date more of an incentive and reminder. In general, I don't want you to try to only post when you need to make a due date. The goal is that you post without worrying about the artificial due date. Similarly, the goal is that you learn without worrying about your grade. Theoretically, when you meet your goal, the other follows.
Similar philosophy would be to consider, "You shouldn't study to improve your grade; you should improve your grade, because you study."
These are closely related to, "It's not the destination that's important, it's the journey."
[20120319]
Reiterate Some Goals with the Commenting System:
(1) I want students to comment consistently and abundantly.
(2) Students wait until its too late to ask questions.
(3) It's good to ask questions, but sometimes we need to ask the right questions. Something I'm still learning to do myself.
(4) Realize that it's okay to ask the TA and professor questions.
(5) Realize that it helps to reflect on what you've learned.
(6) Helping each other.
There's always time for learning material that has already been covered. Professors occasionally like to put a question missed by students on the first midterm back on the second. Or from midterm to final.
I make a due date more of an incentive and reminder. In general, I don't want you to try to only post when you need to make a due date. The goal is that you post without worrying about the artificial due date. Similarly, the goal is that you learn without worrying about your grade. Theoretically, when you meet your goal, the other follows.
Similar philosophy would be to consider, "You shouldn't study to improve your grade; you should improve your grade, because you study."
These are closely related to, "It's not the destination that's important, it's the journey."
[20120319]
Reiterate Some Goals with the Commenting System:
(1) I want students to comment consistently and abundantly.
(2) Students wait until its too late to ask questions.
(3) It's good to ask questions, but sometimes we need to ask the right questions. Something I'm still learning to do myself.
(4) Realize that it's okay to ask the TA and professor questions.
(5) Realize that it helps to reflect on what you've learned.
(6) Helping each other.
Monday, March 5, 2012
Problem 7.5.29
I wasn't able to solve this problem in the first section, but we were on the right track! A student in my second section came up with the solution.
Problem:
$$\int \ln(x+\sqrt{x^2-1})dx$$
Problem:
$$\int \ln(x+\sqrt{x^2-1})dx$$
Class Discussion due 20120311
Remember that you may comment on pretty much anything related to the course. What you like about section/lecture, what you don't like about section/lecture, a problem you would like done, a question about homework, and something you learned in section/lecture are some of the choices you have to comment on.
Perhaps not enough people have been reflecting on what they've learned, and overlook how useful a learning tool this is.
[20120301, Updated 20120307]
Homework Bonus:
Because I can be a tough grader, I will implement a bonus system.
0) Have completed the homework.
1) Check your work with a friend (they must be in section 1 or 2, i.e., either of my two sections).
2) Get the same answers. After checking your work, and still disagree on which of you is correct, you may agree to disagree. However, in most cases, one or both of you is wrong, so the mistake should be found and corrected.
3) Underneath your name, write HWBuddy: (Their name). You may have multiple buddies, write HWBuddies: (Name1), (Name2), ... .
I will award between -2 and 5 points. See my comments below. If I get a higher response, I will increase the possible bonuses even more, so encourage your classmates to participate in this bonus program! (For HW4 the max was 4)
Staple Penalty:
Lose 1 point for not stapling your paper.
Comment Bonus:
Tier 1: At least ten posts. Six of which belong to different weeks of the year. A week is Sunday to Saturday. Result: Approximately fifteen points.
Tier 2: At least twenty posts. Eight of which belong to different weeks of the year. A week is Sunday to Saturday. Result: Double Tier 2.
Tier 3: At least thirty posts. Ten of which belong to different weeks of the year. A week is Sunday to Saturday. Result: Triple Tier 3.
[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 many a week you want to post is up to you.
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.
Perhaps not enough people have been reflecting on what they've learned, and overlook how useful a learning tool this is.
[20120301, Updated 20120307]
Homework Bonus:
Because I can be a tough grader, I will implement a bonus system.
0) Have completed the homework.
1) Check your work with a friend (they must be in section 1 or 2, i.e., either of my two sections).
2) Get the same answers. After checking your work, and still disagree on which of you is correct, you may agree to disagree. However, in most cases, one or both of you is wrong, so the mistake should be found and corrected.
3) Underneath your name, write HWBuddy: (Their name). You may have multiple buddies, write HWBuddies: (Name1), (Name2), ... .
I will award between -2 and 5 points. See my comments below. If I get a higher response, I will increase the possible bonuses even more, so encourage your classmates to participate in this bonus program! (For HW4 the max was 4)
Staple Penalty:
Lose 1 point for not stapling your paper.
Comment Bonus:
Tier 1: At least ten posts. Six of which belong to different weeks of the year. A week is Sunday to Saturday. Result: Approximately fifteen points.
Tier 2: At least twenty posts. Eight of which belong to different weeks of the year. A week is Sunday to Saturday. Result: Double Tier 2.
Tier 3: At least thirty posts. Ten of which belong to different weeks of the year. A week is Sunday to Saturday. Result: Triple Tier 3.
[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 many a week you want to post is up to you.
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.
Sunday, March 4, 2012
Word of Caution on Parametric Equations and Polar Coordinates
This is so important it deserves its own post!
Don't make the mistake of misjudging the period (when the graph repeats itself) of a polar graph.
For example: You'd kick yourself if you found out you got a problem wrong, because you thought the bounds were $0$ to $2\pi$ when it was actually $0$ to $\pi$. Or vice versa.
I'll try to get more information on parametric equations up, but I feel this was worth mentioning ASAP.
Don't make the mistake of misjudging the period (when the graph repeats itself) of a polar graph.
For example: You'd kick yourself if you found out you got a problem wrong, because you thought the bounds were $0$ to $2\pi$ when it was actually $0$ to $\pi$. Or vice versa.
I'll try to get more information on parametric equations up, but I feel this was worth mentioning ASAP.
Thursday, March 1, 2012
Case Study: Interpreting Wolfram Alpha's Graphical Output
A student inquired about the output to the input
See the input and output here.
The first perplexing part of the output was the statement "(t from -1.78 to 1.78)." Most students will know that $t$ needs to be greater than or equal to 1.
Next, we can easily compute that at time $t=1$, we get the point $(\sqrt{2},0)$, but the picture doesn't look like it goes through this point.
After the student left, I realized from the labels that the graph were simply shifted.
As a remark, one can specify the range of the parameter. Enter the input
See the input and output here.
parametric plot (sqrt(t+1), sqrt(t-1))on Wolfram Alpha.
See the input and output here.
The first perplexing part of the output was the statement "(t from -1.78 to 1.78)." Most students will know that $t$ needs to be greater than or equal to 1.
Next, we can easily compute that at time $t=1$, we get the point $(\sqrt{2},0)$, but the picture doesn't look like it goes through this point.
After the student left, I realized from the labels that the graph were simply shifted.
As a remark, one can specify the range of the parameter. Enter the input
parametric plot (sqrt(t+1), sqrt(t-1)) t=1..2.
See the input and output here.
Class Discussion due 20120305
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.
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.
Integrating Factor
You can save some trouble with computation if you understand the purpose of an integrating factor.
IMPORTANT: Before computing the integrating factor, put the first-order linear equation into the form $y^\prime + P(x) y = Q(X)$.
When you multiply by the integrating factor $I(x)$, you get $$I(x)y^\prime+I(x)P(x)y=I(x)Q(X)$$.
By design of $I(x)$, this becomes $$\left[I(x)y\right]^\prime =I(x)Q(x)$$.
Side Remark: Keep in mind that $y$ is a function of $x$ when differentiating.
Another side remark: Remember that $y^\prime$ is notation for $\frac{dy}{dx}$.
IMPORTANT: Before computing the integrating factor, put the first-order linear equation into the form $y^\prime + P(x) y = Q(X)$.
When you multiply by the integrating factor $I(x)$, you get $$I(x)y^\prime+I(x)P(x)y=I(x)Q(X)$$.
By design of $I(x)$, this becomes $$\left[I(x)y\right]^\prime =I(x)Q(x)$$.
Side Remark: Keep in mind that $y$ is a function of $x$ when differentiating.
Another side remark: Remember that $y^\prime$ is notation for $\frac{dy}{dx}$.
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