My students probably struggled with the last few topics covered in the semester as my coverage on those topics was not as thorough as my coverage on the rest of the topics covered in the semester.
On a related note, I occasionally ponder the possibility of covering less, but in greater detail.
On a different note, I think it's strange how many students ask for review sessions, but when I bring up the idea of extending class time or a separate hour (temporarily ignoring the issue of increasing credit) to my peers, there is the idea that students wouldn't agree to it. The problem with office hours is that students either have other classes schedule during that time or are afraid to come. Office hours by appointment is fine, as long as it isn't abused.
Maybe there should be a game. Played by a professor. Every time the professor says something that might be on the exam, students should say some key phrase like "That's magic!" Perhaps every once in a while, and especially the first week, this should be aided with a pause or a hand signal or a sign (much like an "Applause" sign).
Wednesday, May 16, 2012
Next TA Meeting
This general idea of removing bias crossed my mind. How should we, if at all, deal with section to section discrepancies? I like the idea of mixing up the exams and having the person's name on the very back page of the last problem. That way, you can keep the scoring page on the front, and when you hand back the exams, you just turn the pile over and they can look for their names without seeing each other's scores. It also reduces section bias. But undoubtedly there is differences for the homework portion and emphasis of material from section to section. Maybe one of the TA emphasizes a certain type of problem and grades with respect to that emphasis. So even unbiased grading on midterm and final exams will be lopsided. I definitely would either scale each section so that all sections are comparable, or assign grades based on individual sections. TAs might expect a certain student to do well or do poorly.
See also next post.
See also next post.
Other Ideas to Focus On
$\lim a_n=0$ does not imply convergence.
Students use the ratio/root test on the endpoints of a power series. It's hard for them to grasp that those are exactly the points where the ratio/root test gives 1, and hence inconclusive. Apply the ratio/root test is a waste of time. Just as bad is when they apply the ratio/root test to the endpoints and get a value other than 1.
$\lim\left(1+\frac{x}{n}\right)^n=e^x$
Students use the ratio/root test on the endpoints of a power series. It's hard for them to grasp that those are exactly the points where the ratio/root test gives 1, and hence inconclusive. Apply the ratio/root test is a waste of time. Just as bad is when they apply the ratio/root test to the endpoints and get a value other than 1.
$\lim\left(1+\frac{x}{n}\right)^n=e^x$
Saturday, May 12, 2012
Mistakes
THE FOLLOWING IS FALSE:
$$\frac{c}{a+b}=\frac{c}{a}+\frac{c}{b}$$
FAKE EXAMPLE OF THIS ERROR BEING MADE:
Student writes:
$$\frac{x^2}{1+x^2}=\frac{x^2}{1\vphantom{x^2}}+\frac{x^2}{x^2}$$
TRY IT WITH NUMBERS:
$$\frac{2}{1+1}=1$$
$$\frac{2}{1}+\frac{2}{1}=4$$
$$\frac{c}{a+b}=\frac{c}{a}+\frac{c}{b}$$
FAKE EXAMPLE OF THIS ERROR BEING MADE:
Student writes:
$$\frac{x^2}{1+x^2}=\frac{x^2}{1\vphantom{x^2}}+\frac{x^2}{x^2}$$
TRY IT WITH NUMBERS:
$$\frac{2}{1+1}=1$$
$$\frac{2}{1}+\frac{2}{1}=4$$
Tuesday, May 8, 2012
Practice Exam 5
I don't feel it is necessary for me to make another practice test. At this point you should have realized they're just homework problems in disguise. Now it's time for you to make your own practice test. Below is a rough guideline.
Instructions: Do the following. Exceptions to the rule are anything mentioned by Dr. Brown to focus on or not focus on. Please show your work and explain whenever possible. It's better to say what's on your mind (provided it's correct), because the graders are not mind readers. Please simplify answers when reasonable, such as $\sin{2\pi/3}$ or $\log(1)$.
1) Midterm 1
2) Midterm 2
3) One problem similar to every different type of "easy" homework problem. By "easy" homework problem, I mean the less conceptual and abstract ones. Do odd-numbered problem so you can check your solution with the back of the book. Alternate about every twenty minutes between easier and harder sections (different for every student).
4) My four practice tests
5) Learn to find exams in other locations. Skip problems when they don't apply. Let me get you started: http://www.math.jhu.edu/~wsw/F11/.
Bonus (for students looking for the extra challenge): One problem similar to every different type of "hard" homework problem.
Instructions: Do the following. Exceptions to the rule are anything mentioned by Dr. Brown to focus on or not focus on. Please show your work and explain whenever possible. It's better to say what's on your mind (provided it's correct), because the graders are not mind readers. Please simplify answers when reasonable, such as $\sin{2\pi/3}$ or $\log(1)$.
1) Midterm 1
2) Midterm 2
3) One problem similar to every different type of "easy" homework problem. By "easy" homework problem, I mean the less conceptual and abstract ones. Do odd-numbered problem so you can check your solution with the back of the book. Alternate about every twenty minutes between easier and harder sections (different for every student).
4) My four practice tests
5) Learn to find exams in other locations. Skip problems when they don't apply. Let me get you started: http://www.math.jhu.edu/~wsw/F11/.
Bonus (for students looking for the extra challenge): One problem similar to every different type of "hard" homework problem.
Monday, May 7, 2012
Compute the Taylor series for x^4-3x^2+1 around a=1
Compute the Taylor series for $f(x) = x^4-3x^2+1$ around $a=1$.
Method 1 Take derivatives.
Step 1: Determine the series coefficients.
Step 1a: Evaluate the function at 1.
$f(1) = -1$
Step 1b: Take the first derivative and evaluate it at 1.
$f^{\prime}(x) = 4x^3-6x$
$f^{\prime}(1) = -2
Step 1c: Take the second derivative and evaluate it at 1.
$f^{\prime\prime}(x) = 12x^2-6$
$f^{\prime\prime}(1) = 6
Step 1d: And so on...
$f^{\prime\prime\prime}(x) = 24x$
$f^{\prime\prime\prime}(1) = 24$
$f^{\prime\prime\prime\prime}(x) = 24$
$f^{\prime\prime\prime\prime}(1) = 24$
Step 2: Lay out the coefficients in front of the appropriate power of $(x-a)$ and divide by the corresponding factorial.
For example, the coefficient of $(x-1)^2$ is the evaluation of the second derivative (2) divided by the factorial of two.
In tabular form we have the following:
At the end of the day we have $$-1-2(x-1)+3(x-1)^2+4(x-1)^3+(x-1)^4$$
[20120507]
Method 2 Potentially the easiest method provided it's allowed by the instructor.
First we recognize that $f$ can be written as $((x-1)+1)^4-3((x-1)+1)^2+1$.
Then we expand the terms to obtain $((x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$ from the first term, $-3((x-1)^2+2(x-1)+1)$ from the second term, and $1$.
Simplifying we get $(x-4)^4+4(x-1)^3+3(x-1)^2-2(x-1)-1$.
[20191223]
Method 3 Theoretical method; arguably harder.
We know the coefficient of $(x-1)^4$ must be 1, because it is the only term that contributes to the highest degree term of $f$.
We expand $(x-1)^4$ to get x^4-4x^3+6x^2-4x+1 (I computed the expansion with the help of Pascal's triangle).
There are no powers of $x^3$ in $f$ so we need to offset it with $4(x-1)^3$. The expansion here is $4(x^3-3x^2+3x-1)$ or $4x^3-12x^2+12x-4$.
Keeping track we have $-6x^2+8x-3$ to worry about. So we add $3(x-1)^2$. The expansion here is $3(x^2-2x+1)$ or $3x^2-6x+3$.
The remaining amount to worry about is $2x$. So we add $-2(x-1)$ or $-2x+2$.
The remaining about to worry about is $2$. So we add $-1$.
Remark: Throughout this method, I use the term "add." It's useful to think in terms of adding negative "x" instead of subtracting "x." My high school teacher taught my fellow students and me that subtraction (Satan) and division (Devil) are evil and so we should instead "add a negative" and "multiply by the inverse," respectively.
[20191223]
Method 1 Take derivatives.
Step 1: Determine the series coefficients.
Step 1a: Evaluate the function at 1.
$f(1) = -1$
Step 1b: Take the first derivative and evaluate it at 1.
$f^{\prime}(x) = 4x^3-6x$
$f^{\prime}(1) = -2
Step 1c: Take the second derivative and evaluate it at 1.
$f^{\prime\prime}(x) = 12x^2-6$
$f^{\prime\prime}(1) = 6
Step 1d: And so on...
$f^{\prime\prime\prime}(x) = 24x$
$f^{\prime\prime\prime}(1) = 24$
$f^{\prime\prime\prime\prime}(x) = 24$
$f^{\prime\prime\prime\prime}(1) = 24$
Step 2: Lay out the coefficients in front of the appropriate power of $(x-a)$ and divide by the corresponding factorial.
For example, the coefficient of $(x-1)^2$ is the evaluation of the second derivative (2) divided by the factorial of two.
In tabular form we have the following:
| Coefficient | Factorial | x term | All Together |
|---|---|---|---|
| $f^{(0)}(1)=f(1)$ | 0! | $(x-1)^0$ | $\frac{-1}{0!}$ |
| $f^{(1)}(1)=f^{\prime}(1)$ | 1! | $(x-1)^1$ | $\frac{-2}{1!}(x-1)$ |
| $f^{(2)}(1)=f^{\prime\prime}(1)$ | 2! | $(x-1)^2$ | $\frac{6}{2!}(x-1)^2$ |
| $f^{(3)}(1)=f^{\prime\prime\prime}(1)$ | 3! | $(x-1)^3$ | $\frac{24}{3!}(x-1)^3$ |
| $f^{(4)}(1)=f^{\prime\prime\prime\prime}(1)$ | 4! | $(x-1)^4$ | $\frac{24}{4!}(x-1)^4$ |
At the end of the day we have $$-1-2(x-1)+3(x-1)^2+4(x-1)^3+(x-1)^4$$
[20120507]
Method 2 Potentially the easiest method provided it's allowed by the instructor.
First we recognize that $f$ can be written as $((x-1)+1)^4-3((x-1)+1)^2+1$.
Then we expand the terms to obtain $((x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$ from the first term, $-3((x-1)^2+2(x-1)+1)$ from the second term, and $1$.
Simplifying we get $(x-4)^4+4(x-1)^3+3(x-1)^2-2(x-1)-1$.
[20191223]
Method 3 Theoretical method; arguably harder.
We know the coefficient of $(x-1)^4$ must be 1, because it is the only term that contributes to the highest degree term of $f$.
We expand $(x-1)^4$ to get x^4-4x^3+6x^2-4x+1 (I computed the expansion with the help of Pascal's triangle).
There are no powers of $x^3$ in $f$ so we need to offset it with $4(x-1)^3$. The expansion here is $4(x^3-3x^2+3x-1)$ or $4x^3-12x^2+12x-4$.
Keeping track we have $-6x^2+8x-3$ to worry about. So we add $3(x-1)^2$. The expansion here is $3(x^2-2x+1)$ or $3x^2-6x+3$.
The remaining amount to worry about is $2x$. So we add $-2(x-1)$ or $-2x+2$.
The remaining about to worry about is $2$. So we add $-1$.
Remark: Throughout this method, I use the term "add." It's useful to think in terms of adding negative "x" instead of subtracting "x." My high school teacher taught my fellow students and me that subtraction (Satan) and division (Devil) are evil and so we should instead "add a negative" and "multiply by the inverse," respectively.
[20191223]
Saturday, April 28, 2012
Practice Exam 4
Practice Test 4
Made available 20120428 11:59 PM.
1) Determine the tangent line of the polar function $r=6\cos\theta$
at the point $(x,y)=(3,3)\in\mathbb{R}^{2}$.
Made available 20120428 11:59 PM.
1) Determine the tangent line of the polar function $r=6\cos\theta$
at the point $(x,y)=(3,3)\in\mathbb{R}^{2}$.
Friday, April 27, 2012
Practice Exam 3
Practice Test 3
Administered on 20120427. Updated and made available online 20120430 6:15 PM.
1) Solve the initial value problem
\[
y^{\prime}+\frac{1}{x}y-\sin x=0
\]
\[
y(\pi)=0
\]
Administered on 20120427. Updated and made available online 20120430 6:15 PM.
1) Solve the initial value problem
\[
y^{\prime}+\frac{1}{x}y-\sin x=0
\]
\[
y(\pi)=0
\]
Monday, April 23, 2012
Practice Exam 2
Practice Exam 2.
Made on 20120420. Partially distributed on 20120420. Made available online 20120423.
No calculators!
#1) Evaluate the integral.
\[
\int\frac{s^{3}}{\sqrt{s^{2}+4}}ds
\]
Made on 20120420. Partially distributed on 20120420. Made available online 20120423.
No calculators!
#1) Evaluate the integral.
\[
\int\frac{s^{3}}{\sqrt{s^{2}+4}}ds
\]
Practice Exam 1
Practice Exam 1
Made on 20120417. Distributed in class on 20120417. Made available online on 20120423.
#1) Evaluate the integral.
\[
\int\frac{6x+1}{(x+1)(2x-1)}dx
\]
Made on 20120417. Distributed in class on 20120417. Made available online on 20120423.
#1) Evaluate the integral.
\[
\int\frac{6x+1}{(x+1)(2x-1)}dx
\]
Sunday, April 15, 2012
Class Discussion due 20120422
What!?! Spring Fair is this weekend. As such, I'll consider reserving a room and leaving a pile of tests on a table for those who want to come in and sit to take the test. The goal is to let you guys come in freely on Friday 3-7 PM. (4/20/2012). Though I recommend when you come, you commit to staying for an hour block. Perhaps I will come in and out to check. But I hope everybody enjoys Spring Fair, minus the one hour they come.
Then Friday 3-7 PM, we will do it again (4/27/2012). I will make the exam available online for those who can't make it.
In addition, I will make a test that you can work on before each Friday. That's 4 practice tests!
Hint 1: It will be based on your homework. Why? 1) That's most likely how Dr. Brown will write his test. 2) If you get a problem wrong, or don't know how to do a problem, then you'll know which problems you'll need to do more of. 3) In order to write more, I have to somehow make it easy to write.
Hint 2 (for first two practice tests): There will be 8 problems. One of the problems will try to ask something from the first half of the course (in minor preparation for the final). One of the problems will involve polar coordinates (Section 10.4). One of the problems will be an improper integral. The remaining five problems will come from Section 11.1 to 11.7.
Advice: Do all the problems from Section 11.7.
Then Friday 3-7 PM, we will do it again (4/27/2012). I will make the exam available online for those who can't make it.
In addition, I will make a test that you can work on before each Friday. That's 4 practice tests!
Hint 1: It will be based on your homework. Why? 1) That's most likely how Dr. Brown will write his test. 2) If you get a problem wrong, or don't know how to do a problem, then you'll know which problems you'll need to do more of. 3) In order to write more, I have to somehow make it easy to write.
Hint 2 (for first two practice tests): There will be 8 problems. One of the problems will try to ask something from the first half of the course (in minor preparation for the final). One of the problems will involve polar coordinates (Section 10.4). One of the problems will be an improper integral. The remaining five problems will come from Section 11.1 to 11.7.
Advice: Do all the problems from Section 11.7.
Wednesday, April 4, 2012
Class Discussion due 20120410
Check your email regarding the Practice Test Session.
For the Homework Buddy system, you're suppose to put A2D (agree to disagree) in the situation where you've discussed the discrepancy and somehow feel your answer is right and the other person won't listen to you. Otherwise, you should typically convince the other person why your answer is right and get them to change it. Future checks of the use of A2D will be more strict. And in general, listing another person as your Homework Buddy should mean ALL your answers match up. Perhaps not the work, but at least the answers. For now, I will also reduce the Homework Buddy bonus from 5 to 4. Until better use of the system occurs
For the Homework Buddy system, you're suppose to put A2D (agree to disagree) in the situation where you've discussed the discrepancy and somehow feel your answer is right and the other person won't listen to you. Otherwise, you should typically convince the other person why your answer is right and get them to change it. Future checks of the use of A2D will be more strict. And in general, listing another person as your Homework Buddy should mean ALL your answers match up. Perhaps not the work, but at least the answers. For now, I will also reduce the Homework Buddy bonus from 5 to 4. Until better use of the system occurs
Tuesday, March 27, 2012
Problem 10.4.40
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?
We could solve problems like number 40 as follows.
Problem: When do two curves $r_1(\theta)$ and $r_2(\theta)$ intersect?
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 $\frac{dy}{dx}$.
Tuesday, February 28, 2012
Organizing the Work for Euler's method
Trying to figure out a way to organize Euler's method, I thought the following might do a decent job. I saw several nice tables similar to this in submitted homework. Definitely play around and do whatever works best for you.
Thursday, February 23, 2012
Test Your Knowledge 20120223
More random practice problems. If you hesitate on the answer, strongly consider reviewing the relevant material.
Wednesday, February 22, 2012
Tuesday, February 21, 2012
Class Discussion due 20120228
Do some method recognition on your own. Perhaps we'll do odd in class and you can do evens on your own. If you do evens now and if we exhaust the odd ones in class, then by the time we do evens as a class, it'll be half-new. Note that there's also the review section of chapter seven for which we can go through method recognition.
Any feedback on things you find distracting or that can be improved on?
What is something you learned from reading the book?
I can't be sure I can cover all the bases, but at the minimum, try testing your knowledge here (20120209) and here (20120221).
Any feedback on things you find distracting or that can be improved on?
What is something you learned from reading the book?
I can't be sure I can cover all the bases, but at the minimum, try testing your knowledge here (20120209) and here (20120221).
Class Summary 20120221
[Section 1]
9.2.23 About three iterations.
9.2.22 About three iterations.
Emphasized being able to do calculations by hand.
Drew a picture, but didn't have time to explain the equation in full. Thought perhaps it was too conceptual to be repeated in class.
Was trying to just help with problem 9.3.42 but basically did it only without determining the constants.
pg 499 15-25 odd method recognition.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
25 Allowed to discuss the degree and needing to simplify by using partial fractions. While there, I was able to sneak in my blurb about simplifying common factors and Heaviside cover-up method.
[Section 2]
I started with
pg 499 15-25 odd method recognition.
I gave each one it's own column making the work a little more neat.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
A student for 19 made an important simplification before the u-substitution that I overlooked.
$e^{x+e^x}=e^xe^{e^x}$
I couldn't figure out on the spot if the u-substitution $x+e^x$ really works or not, but I didn't linger, as the purpose of the exercise is to test recognition.
I was able to be more thorough about Heaviside cover-up method, throwing in an example of a repeated factor and using an easy to compute $x$ to compute any lingering variables.
After that I went to discus 9.2.23 up to $y_3$ but without computing $y_3$. I spend a moment to explain the meaning of the formula and drew a little picture.
A student asked about multiple solutions and what that means. I first refer to Figure 8. Then I explain there are infinitely many solutions. If you give an initial condition, then you get a particular solution.
Then I talked about level sets but with #5 instead of #3. (I did #3 in Section 1)
After class a student asked about finishing problem 7.5.25, that is after we have divided to make sure the numerator is less than the denominator. I wanted to hand back homework to those who were waiting for it, while he had to leave.
Another student asked about $y(5)$. I suppose the confusion was why we stop at $y_5$. After some discussion and drawing a picture, it was made apparent that it depends on the step size $h$.
Comments:
There's a lot to cover and there's only so much I can make sure students pay attention to. Ultimately it's up to them to ask the questions and do whatever they can to secure their understanding. For example, by reading the book, going to math help room, going to the professor's or TA's office hours, and comparing solutions with a friend.
9.2.23 About three iterations.
9.2.22 About three iterations.
Emphasized being able to do calculations by hand.
Drew a picture, but didn't have time to explain the equation in full. Thought perhaps it was too conceptual to be repeated in class.
Was trying to just help with problem 9.3.42 but basically did it only without determining the constants.
pg 499 15-25 odd method recognition.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
25 Allowed to discuss the degree and needing to simplify by using partial fractions. While there, I was able to sneak in my blurb about simplifying common factors and Heaviside cover-up method.
[Section 2]
I started with
pg 499 15-25 odd method recognition.
I gave each one it's own column making the work a little more neat.
I emphasized that the u-substitution and it's differential comes in a pair. And that you shouldn't forget the differential on the right hand side.
I made sure to bring up DETAIL.
19-23 have the same idea of making the u-substitution that would simplify the problem.
A student for 19 made an important simplification before the u-substitution that I overlooked.
$e^{x+e^x}=e^xe^{e^x}$
I couldn't figure out on the spot if the u-substitution $x+e^x$ really works or not, but I didn't linger, as the purpose of the exercise is to test recognition.
I was able to be more thorough about Heaviside cover-up method, throwing in an example of a repeated factor and using an easy to compute $x$ to compute any lingering variables.
After that I went to discus 9.2.23 up to $y_3$ but without computing $y_3$. I spend a moment to explain the meaning of the formula and drew a little picture.
A student asked about multiple solutions and what that means. I first refer to Figure 8. Then I explain there are infinitely many solutions. If you give an initial condition, then you get a particular solution.
Then I talked about level sets but with #5 instead of #3. (I did #3 in Section 1)
After class a student asked about finishing problem 7.5.25, that is after we have divided to make sure the numerator is less than the denominator. I wanted to hand back homework to those who were waiting for it, while he had to leave.
Another student asked about $y(5)$. I suppose the confusion was why we stop at $y_5$. After some discussion and drawing a picture, it was made apparent that it depends on the step size $h$.
Comments:
There's a lot to cover and there's only so much I can make sure students pay attention to. Ultimately it's up to them to ask the questions and do whatever they can to secure their understanding. For example, by reading the book, going to math help room, going to the professor's or TA's office hours, and comparing solutions with a friend.
Test Your Knowledge 20120221
What is $\tan^{-1}(1)$?
What is $\ln(1)$?
Set up the partial fractions $$\frac{1}{(x-1)^2(x^2+2x+2)^2}$$
Set up and solve the partial fractions, try using the Heaviside cover-up method. $$\frac{x}{(x-2)(x-3)}$$
What is $\int \sec(\theta) d\theta$? Note: The easiest way to do this is just to recite the formula from the book found in section 7.2.
What is $\int \frac{1}{\sqrt{1+x^2}} dx$ Note: Some students get this confused with another important integral. Don't get confused.
What is $\sin(\pi/2)$?
What is $\ln(1)$?
Set up the partial fractions $$\frac{1}{(x-1)^2(x^2+2x+2)^2}$$
Set up and solve the partial fractions, try using the Heaviside cover-up method. $$\frac{x}{(x-2)(x-3)}$$
What is $\int \sec(\theta) d\theta$? Note: The easiest way to do this is just to recite the formula from the book found in section 7.2.
What is $\int \frac{1}{\sqrt{1+x^2}} dx$ Note: Some students get this confused with another important integral. Don't get confused.
What is $\sin(\pi/2)$?
Thursday, February 16, 2012
7.3.18, 7.3.25, and help on 7.3.30
7.3.18
For simplicity, I will assume $a$ and $b$ or positive real numbers.
$$\int \frac{dx}{[(ax)^2-b^2]^{3/2}}$$
We identify that we should use secant.
For simplicity, I will assume $a$ and $b$ or positive real numbers.
$$\int \frac{dx}{[(ax)^2-b^2]^{3/2}}$$
We identify that we should use secant.
Tuesday, February 14, 2012
Class Summary 20120214
[Section 1]
Partial fractions. Covered examples from blog.
7.4.50 and 7.4.44.
method recognition pg 499
1) u = sin
3) try simplifying [leave how to integrate $\frac{1}{\sin(x)}$]
5) $u=t^2$ [maybe partial fraction]
7) $u = \tan^{-1}(x)$
9) by parts
11) u = denom (LEAVE IT OPEN)
13) trig 7.2 (We stopped with this one)
trig recognition pg 483
5-13 odd
we ended up finishing 5 through 11 odd.
[Section 2]
trig recognition pg 483
5-11 odd
Partial fractions. Covered examples from blog.
method recognition pg 499
see above.
In this class we stopped at #11.
[Comments]
I wanted to, but didn't have time to do 18, 25, or a modification of 30.
In section 2, I realized that many people didn't have their books during the method recognition exercise and so I wrote the problems on the board.
Partial fractions. Covered examples from blog.
7.4.50 and 7.4.44.
method recognition pg 499
1) u = sin
3) try simplifying [leave how to integrate $\frac{1}{\sin(x)}$]
5) $u=t^2$ [maybe partial fraction]
7) $u = \tan^{-1}(x)$
9) by parts
11) u = denom (LEAVE IT OPEN)
13) trig 7.2 (We stopped with this one)
trig recognition pg 483
5-13 odd
we ended up finishing 5 through 11 odd.
[Section 2]
trig recognition pg 483
5-11 odd
Partial fractions. Covered examples from blog.
method recognition pg 499
see above.
In this class we stopped at #11.
[Comments]
I wanted to, but didn't have time to do 18, 25, or a modification of 30.
In section 2, I realized that many people didn't have their books during the method recognition exercise and so I wrote the problems on the board.
Class Discussion due 20120221
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.
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.
5 Solutions to Problem 7.2.10
Problem 7.2.10
This took at least two hours to put together and write up. Just letting
you know the effort I'm willing to put in to help you do well.
This took at least two hours to put together and write up. Just letting
you know the effort I'm willing to put in to help you do well.
Thursday, February 9, 2012
Test Your Knowledge 20120209
Look through and see if you can do these. Try to work them out. You may submit questions here and qualify for participation of Class Discussion due 20120214.
If you finished these and would like more, let me know with a comment or e-mail! I need positive feedback that students are making use of such a post. Such a comment will not satisfy participation requirements, unless it provides constructive criticism, such as comments regarding the type of problems which appear.
If you finished these and would like more, let me know with a comment or e-mail! I need positive feedback that students are making use of such a post. Such a comment will not satisfy participation requirements, unless it provides constructive criticism, such as comments regarding the type of problems which appear.
Tuesday, February 7, 2012
Class Discussion due 20120214
Check out the discussion that's already happened here: Class Discussion Due 20120207
If you didn't already participate in that discussion, please do. Participating is easier than you think.
A new thing you can comment on is what you like or don't like about section.
Last week I had you look at Appendix A and Reference page 1 Algebra.
This week, look at the left-hand side of Reference page 2.
Try to finish reading Appendix A and do some problems (every 8 is a good suggestion).
If you tried to post, but couldn't feel free to send your comment by e-mail. Hopefully you'll eventually be able to post on your own. Sometimes it's a matter of web browser compatibility. Personally I use Google Chrome when I browse on www.blogger.com.
If you didn't already participate in that discussion, please do. Participating is easier than you think.
A new thing you can comment on is what you like or don't like about section.
Last week I had you look at Appendix A and Reference page 1 Algebra.
This week, look at the left-hand side of Reference page 2.
Try to finish reading Appendix A and do some problems (every 8 is a good suggestion).
If you tried to post, but couldn't feel free to send your comment by e-mail. Hopefully you'll eventually be able to post on your own. Sometimes it's a matter of web browser compatibility. Personally I use Google Chrome when I browse on www.blogger.com.
Thursday, February 2, 2012
Integral of x*exp(x)*cos(x) with respect to x
This post is a follow-up to Useful Mnemonic: DETAIL where the problem was posed.
Determine $f(x)=\int x e^x \cos (x)dx$.
CHOICE A (Suggested by DETAIL):
Let $u=x\cos(x)$ and $dv= e^x dx$.
Then $du=\cos(x)-x\sin(x)dx$ and $v=e^x$.
Then $f(x)=x\cos(x)e^x - \int e^x \left[ \cos(x)-x\sin(x) \right] dx$
CHOICE B:
Let $u=x e^x$ and $dv=\cos(x)dx$.
Then $du = e^x+x e^x dx$ and $v=\sin(x)$.
Then $f(x)=x e^x \sin(x) - \int \sin(x) \left[ e^x +x e^x\right] dx$
CHOICE C:
Let $u=e^x$ and $dv=x \cos(x)dx$.
Then $du = e^x dx$ and $v=\int x\cos(x)dx$
Let $\hat{u}=x$ and $d\hat{v}=\cos(x)dx$.
Then $\hat{v}=\sin(x)$
Then $\int x\cos(x)dx = x \sin(x)- \int\sin(x)dx = x\sin (x)+\cos(x) +\hat{C}$
We only concern ourself with a particular $v$, so we assume $\hat{C}=0$.
Then $f(x) = e^x \left[ x\sin(x) + \cos (x) \right] -\int \left[x\sin(x)+\cos(x) \right] e^x dx$
What remarks can we make? For this specific problem, regardless of the above three choices for $u$ and $dv$, we will have to do integration by parts again. DETAIL recommended choice A, but choice B was just as short. Though as a personal opinion, I find it easier to make sign errors when determining antiderivatives to trigonometric functions than when determining antiderivatives to exponential functions.
Recall that $\frac{d}{dt} \sin (t) = \cos (t)$, while $\frac {d}{dt} \cos (t)=-\sin(t)$.
Determine $f(x)=\int x e^x \cos (x)dx$.
CHOICE A (Suggested by DETAIL):
Let $u=x\cos(x)$ and $dv= e^x dx$.
Then $du=\cos(x)-x\sin(x)dx$ and $v=e^x$.
Then $f(x)=x\cos(x)e^x - \int e^x \left[ \cos(x)-x\sin(x) \right] dx$
CHOICE B:
Let $u=x e^x$ and $dv=\cos(x)dx$.
Then $du = e^x+x e^x dx$ and $v=\sin(x)$.
Then $f(x)=x e^x \sin(x) - \int \sin(x) \left[ e^x +x e^x\right] dx$
CHOICE C:
Let $u=e^x$ and $dv=x \cos(x)dx$.
Then $du = e^x dx$ and $v=\int x\cos(x)dx$
Let $\hat{u}=x$ and $d\hat{v}=\cos(x)dx$.
Then $\hat{v}=\sin(x)$
Then $\int x\cos(x)dx = x \sin(x)- \int\sin(x)dx = x\sin (x)+\cos(x) +\hat{C}$
We only concern ourself with a particular $v$, so we assume $\hat{C}=0$.
Then $f(x) = e^x \left[ x\sin(x) + \cos (x) \right] -\int \left[x\sin(x)+\cos(x) \right] e^x dx$
What remarks can we make? For this specific problem, regardless of the above three choices for $u$ and $dv$, we will have to do integration by parts again. DETAIL recommended choice A, but choice B was just as short. Though as a personal opinion, I find it easier to make sign errors when determining antiderivatives to trigonometric functions than when determining antiderivatives to exponential functions.
Recall that $\frac{d}{dt} \sin (t) = \cos (t)$, while $\frac {d}{dt} \cos (t)=-\sin(t)$.
Useful Mnemonic: DETAIL
[20120202]
A student had an interesting random thought.
What to do when the integrand is a product of three functions?
In particular he asked about $\int x e^x \cos (x)dx$.
What do you think?
[20120207]
I had you work on this in class. Please complete the problem and ask questions if you have any.
A student had an interesting random thought.
What to do when the integrand is a product of three functions?
In particular he asked about $\int x e^x \cos (x)dx$.
What do you think?
[20120207]
I had you work on this in class. Please complete the problem and ask questions if you have any.
Tuesday, January 31, 2012
Class Discussion due 20120207
#2. In the comment section of the relevant week, submit a question or something you learned every week. These should be mildly relevant to the class. If you posted something along these lines on the Facebook page, you may simply copy and paste what you posted. Other comments such as interesting links, even those not related to math are welcome, but do not count toward the required participation. No duplicates, so the earlier you submit a question or comment, the less you'll have to read of other posts.
Read Appendix A.
Memorize/Learn Reference Page 1: Algebra.
[20120202:] If you don't want to sign with your name, that's fine. Next week in class, I will figure out which user names belong to which people.
Read Appendix A.
Memorize/Learn Reference Page 1: Algebra.
[20120202:] If you don't want to sign with your name, that's fine. Next week in class, I will figure out which user names belong to which people.
Math 109 Spring 2012
#0. My name, email, class website, other website. I drew a map of where my office room and math help room are located.
#1. Staple your homework or else lose a point.
#2. In the comment section of the relevant week, submit a question or something you learned every week. These should be mildly relevant to the class. If you posted something along these lines on the Facebook page, you may simply copy and paste what you posted. Other comments such as interesting links, even those not related to math are welcome, but do not count toward the required participation. No duplicates, so the earlier you submit a question or comment, the less you'll have to read of other posts.
#3. I'd like to try to get you to read sections of the appendix and do extra problems. Sometimes it's some of these basic ideas that students trip up on, creating an obstacle to the ideas the instructor wishes to teach about calculus. Related is reference page 1, 2, 5, 6, 3, and 4.
#4. We will try working in groups and form strong class participation. I'd like to help you learn, so it's up to you to ask questions that help me figure out how to run the section. I prefer a more dynamic feel to class.
#5. I put a strong emphasis on you knowing your trigonometric identities.
#6. Some useful sites: Wikipedia, Mathworld, Wolfram Alpha. But I do explain that we shouldn't let calculators and Wolfram Alpha be crutches. I cry on the inside when I see a student punch in something like 20 divided by 5 into a calculator.
#7. I explain my goal of applying math to more than just it's uses as math, but as about thinking. Like back in the day with geometry proofs.
#8. A pattern... or is it?
1,2,1,2, (1 is most popular, some 3's)
1,2,1,2,3,1,2,3, (4 is most popular, but some 5's)
1,2,1,2,3,1,2,3,5,1,2,3,5,? (7 and 8 are most popular. Good reasons for other answers.)
The point is about expectation of a pattern where there isn't.
[1]#9. pg A10 #70.
$-\pi+\pi=0$
$\sqrt{2}\cdot \sqrt{2}=2$
$\pi\cdot\frac{1}{\pi}=1$
[2]#10. Help Room 10 minutes.
#1. Staple your homework or else lose a point.
#2. In the comment section of the relevant week, submit a question or something you learned every week. These should be mildly relevant to the class. If you posted something along these lines on the Facebook page, you may simply copy and paste what you posted. Other comments such as interesting links, even those not related to math are welcome, but do not count toward the required participation. No duplicates, so the earlier you submit a question or comment, the less you'll have to read of other posts.
#3. I'd like to try to get you to read sections of the appendix and do extra problems. Sometimes it's some of these basic ideas that students trip up on, creating an obstacle to the ideas the instructor wishes to teach about calculus. Related is reference page 1, 2, 5, 6, 3, and 4.
#4. We will try working in groups and form strong class participation. I'd like to help you learn, so it's up to you to ask questions that help me figure out how to run the section. I prefer a more dynamic feel to class.
#5. I put a strong emphasis on you knowing your trigonometric identities.
#6. Some useful sites: Wikipedia, Mathworld, Wolfram Alpha. But I do explain that we shouldn't let calculators and Wolfram Alpha be crutches. I cry on the inside when I see a student punch in something like 20 divided by 5 into a calculator.
#7. I explain my goal of applying math to more than just it's uses as math, but as about thinking. Like back in the day with geometry proofs.
#8. A pattern... or is it?
1,2,1,2, (1 is most popular, some 3's)
1,2,1,2,3,1,2,3, (4 is most popular, but some 5's)
1,2,1,2,3,1,2,3,5,1,2,3,5,? (7 and 8 are most popular. Good reasons for other answers.)
The point is about expectation of a pattern where there isn't.
[1]#9. pg A10 #70.
$-\pi+\pi=0$
$\sqrt{2}\cdot \sqrt{2}=2$
$\pi\cdot\frac{1}{\pi}=1$
[2]#10. Help Room 10 minutes.
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