#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.
Tuesday, January 31, 2012
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.
Tuesday, December 6, 2011
Regarding Remainder of an Alternating Series
Alternating series test (Wikipedia.org)
Definition of an alternating series:
A series of the form $$\sum_{k=0}^\infty (-1)^k a_k$$
where all the $a_k$ are non-negative. The emphasis here is on the non-negative.
We know the absolute value of the remainder $R_n=\sum_{k=n+1}^\infty (-1)^k a_k$ for an alternating series will be less than the next term in the series, $a_{n+1}$.
So what's the point of all this? Well, some power series might look like alternating series, but they'll only be alternating series for certain values of $x$. Thus, you can only estimate the remainder using the above estimate for values of $x$ for which the power series turns into an alternating series.
[Remark: It sort of follows from the above, that a power series that doesn't look like an alternating series, might be an alternating series for certain values of $x$]
Definition of an alternating series:
A series of the form $$\sum_{k=0}^\infty (-1)^k a_k$$
where all the $a_k$ are non-negative. The emphasis here is on the non-negative.
We know the absolute value of the remainder $R_n=\sum_{k=n+1}^\infty (-1)^k a_k$ for an alternating series will be less than the next term in the series, $a_{n+1}$.
So what's the point of all this? Well, some power series might look like alternating series, but they'll only be alternating series for certain values of $x$. Thus, you can only estimate the remainder using the above estimate for values of $x$ for which the power series turns into an alternating series.
[Remark: It sort of follows from the above, that a power series that doesn't look like an alternating series, might be an alternating series for certain values of $x$]
Answering a student's email questions
For question 6, you should set $x=0$ and solve for $t$. You'll get two values of $t$, but only the one that gives you $y\geq 3$ will work. Then set $y=3$ and solve for $t$. Again you'll get two values of $t$, but only one gives you $x\geq 0$.
For question 7, the furthest distance from the origin is when $r$ is greatest. Thus we are trying to maximize $r$. Then from one variable calculus, just take the derivative with respect to $\theta$ to find critical points. $\frac{dr}{d\theta}=0$. There's a geometric interpretation to this, but it's much easier to explain through pictures than through words. If requested, I'll explain this.
Question 8 is asking for when $r=0$. No derivatives required. Let me know if I'm wrong on this.
For question 10, your mistake might have been that the integral should go from $0$ to $\pi$ and not from $0$ to $2\pi$.
For question 7, the furthest distance from the origin is when $r$ is greatest. Thus we are trying to maximize $r$. Then from one variable calculus, just take the derivative with respect to $\theta$ to find critical points. $\frac{dr}{d\theta}=0$. There's a geometric interpretation to this, but it's much easier to explain through pictures than through words. If requested, I'll explain this.
Question 8 is asking for when $r=0$. No derivatives required. Let me know if I'm wrong on this.
For question 10, your mistake might have been that the integral should go from $0$ to $\pi$ and not from $0$ to $2\pi$.
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