The answers to the final exam are up. The curve for the final is as follows:
55+ A
40-55 B
25-40 C
15-25 D
0-15 F
Friday, December 09, 2005
Friday, December 02, 2005
Solutions to those practice problems
1. Let A be a square matrix with A*A = Identity. Let M be a square matrix, such that AMA = M. Let v be an eigenvector of M with eigenvalue e.
a) Show that Av is one too.
ANSWER.
We want to show MAv = eAv.
We don't know much about MA, but we do about AMA:
AMAv = Mv = ev.
That's ev on the right, but we want eAv or Aev. So left multiply by A:
AAMAv = Aev = eAv.
Now we use A*A = identity to cancel the AA at the beginning.
b) Assuming that, show that v+Av, v-Av are too.
ANSWER. This is easier.
M(v+Av) = Mv + MAv = ev + eAv = e(v+Av)
Same thing with minuses.
2. Find the exponential of t times the matrix M
[2 0 3 9 0]
[0 2 1 6 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 3]
Hint: it would be easier if you subtracted 2 * Identity... how do you make use of that?
ANSWER.
exp(A+B) = exp(A)exp(B) if A and B commute. For example if A = 2t * Identity.
Breaking this up that way, and looking at exp(B) = 1 + B + B^2/2! + B^3/3! + B^4/4!
exp(Bt) = [please ignore the periods, only there to keep Blogger from collapsing the line]
[1 0 0 0 0] . [0 0 3 9 0] .. [0 0 0 0 0] ....... [0 0 0 0 0]
[0 1 0 0 0] . [0 0 1 6 0] .. [0 0 0 0 0] ....... [0 0 0 0 0]
[0 0 1 0 0] + [0 0 0 0 0]t + [0 0 0 0 0]t^2/2! + [0 0 0 0 0]t^3/3! + ...
[0 0 0 1 0] . [0 0 0 0 0] .. [0 0 0 0 0] ....... [0 0 0 0 0]
[0 0 0 0 1] . [0 0 0 0 1] .. [0 0 0 0 1] ....... [0 0 0 0 1]
Higher powers only contribute in the lower right corner, giving exactly
the exponential series for exp(t). So the total is
Q. Solve the ODE system y' = My for arbitrary initial conditions.
ANSWER. y = exp(tM)c. So if c =
[f]
[g]
[h]
[j]
[k],
then exp(tM)y = exp(2t) times
[f + 3th + 9tj]
[g + th + 6tj]
[h]
[j]
[exp(t)k].
3. Let y' = f(t,y).
a) What is the general form of f, if this equation is separable and linear homogeneous?
ANSWER. If it's linear homogeneous, then f(t,y) = g(t)y. In which case it's
automatically separable.
b) What is the general form of f, if this equation is separable and linear but not necessarily homogeneous?
ANSWER. If it's linear inhomogeneous, then f(t,y) = g(t)y + h(t) = g * (y + h/g).
To be separable, that h(t)/g(t) should only depend on y, not on t. So it doesn't depend on y, nor on t; it must be some constant C. Hence f(t,y) = g(t) * (y+C).
4. Let c_0 = 0, c_{n+1} = c_n + 2n + 1. Find c_0,c_1,c_2,c_3,c_4,c_5 and guess a general formula for c_n. Then determine the radius of convergence of the power series \sum_n c_n x^n.
ANSWER.
c_0..5 = 0,1,4,9,16,25. By the way, one way to prove that c_n = n^2 (though you weren't asked to) is to think of the numbers being added on like this:
or if you weren't taking a test, you could look it up here.
Anyway, the ratio test or the root test tell you that the radius of convergence is 1.
5. Assume that
f(x) y dx + (x+y) dy = 0
is exact. Determine the possibilities for f, and solve this ODE. (You should have two constants in your answer: one because you don't know f precisely, and the other from solving the ODE.)
ANSWER. Oops, actually you do know f precisely; I was thinking of a slightly harder problem. Probably a good one for the final.
Anyway, exactness says partial/partial_y (f(x) y) = partial/partial_x (x+y),
or f(x) = 1.
Now we do the integrals to determine F(x,y) such that y dx + (x+y) dy = dF:
= xy + 1/2 y^2.
So the solutions are given by xy + 1/2 y^2 = C.
a) Show that Av is one too.
ANSWER.
We want to show MAv = eAv.
We don't know much about MA, but we do about AMA:
AMAv = Mv = ev.
That's ev on the right, but we want eAv or Aev. So left multiply by A:
AAMAv = Aev = eAv.
Now we use A*A = identity to cancel the AA at the beginning.
b) Assuming that, show that v+Av, v-Av are too.
ANSWER. This is easier.
M(v+Av) = Mv + MAv = ev + eAv = e(v+Av)
Same thing with minuses.
2. Find the exponential of t times the matrix M
[2 0 3 9 0]
[0 2 1 6 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 3]
Hint: it would be easier if you subtracted 2 * Identity... how do you make use of that?
ANSWER.
exp(A+B) = exp(A)exp(B) if A and B commute. For example if A = 2t * Identity.
Breaking this up that way, and looking at exp(B) = 1 + B + B^2/2! + B^3/3! + B^4/4!
exp(Bt) = [please ignore the periods, only there to keep Blogger from collapsing the line]
[1 0 0 0 0] . [0 0 3 9 0] .. [0 0 0 0 0] ....... [0 0 0 0 0]
[0 1 0 0 0] . [0 0 1 6 0] .. [0 0 0 0 0] ....... [0 0 0 0 0]
[0 0 1 0 0] + [0 0 0 0 0]t + [0 0 0 0 0]t^2/2! + [0 0 0 0 0]t^3/3! + ...
[0 0 0 1 0] . [0 0 0 0 0] .. [0 0 0 0 0] ....... [0 0 0 0 0]
[0 0 0 0 1] . [0 0 0 0 1] .. [0 0 0 0 1] ....... [0 0 0 0 1]
Higher powers only contribute in the lower right corner, giving exactly
the exponential series for exp(t). So the total is
| 1 | 0 | 3t | 9t | 0 |
| 0 | 1 | t | 6t | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | exp(t) |
Q. Solve the ODE system y' = My for arbitrary initial conditions.
ANSWER. y = exp(tM)c. So if c =
[f]
[g]
[h]
[j]
[k],
then exp(tM)y = exp(2t) times
[f + 3th + 9tj]
[g + th + 6tj]
[h]
[j]
[exp(t)k].
3. Let y' = f(t,y).
a) What is the general form of f, if this equation is separable and linear homogeneous?
ANSWER. If it's linear homogeneous, then f(t,y) = g(t)y. In which case it's
automatically separable.
b) What is the general form of f, if this equation is separable and linear but not necessarily homogeneous?
ANSWER. If it's linear inhomogeneous, then f(t,y) = g(t)y + h(t) = g * (y + h/g).
To be separable, that h(t)/g(t) should only depend on y, not on t. So it doesn't depend on y, nor on t; it must be some constant C. Hence f(t,y) = g(t) * (y+C).
4. Let c_0 = 0, c_{n+1} = c_n + 2n + 1. Find c_0,c_1,c_2,c_3,c_4,c_5 and guess a general formula for c_n. Then determine the radius of convergence of the power series \sum_n c_n x^n.
ANSWER.
c_0..5 = 0,1,4,9,16,25. By the way, one way to prove that c_n = n^2 (though you weren't asked to) is to think of the numbers being added on like this:
1 | 12 22 | 123 223 333 | 1234 2234 3334 4444 | ... |
or if you weren't taking a test, you could look it up here.
Anyway, the ratio test or the root test tell you that the radius of convergence is 1.
5. Assume that
f(x) y dx + (x+y) dy = 0
is exact. Determine the possibilities for f, and solve this ODE. (You should have two constants in your answer: one because you don't know f precisely, and the other from solving the ODE.)
ANSWER. Oops, actually you do know f precisely; I was thinking of a slightly harder problem. Probably a good one for the final.
Anyway, exactness says partial/partial_y (f(x) y) = partial/partial_x (x+y),
or f(x) = 1.
Now we do the integrals to determine F(x,y) such that y dx + (x+y) dy = dF:
| integral_{t=0}^x P(t,0) dt | + | integral_{s=0}^y Q(x,s) ds | |
| = | integral_{t=0}^x 0 dt | + | integral_{s=0}^y (x+s) ds |
| = | 0 | + | (xs + 1/2 s^2) from s=0 to y |
= xy + 1/2 y^2.
So the solutions are given by xy + 1/2 y^2 = C.
Wednesday, November 30, 2005
Sections on the final, and some practice problems
The book sections covered on the final will be a subset of those covered in the homework, the last one of which was due Monday.
In particular, there is no homework nor will there be final exam questions about Laplace transforms nor series solutions to differential equations.
Here are some questions on most of the topics on the final to get you started.
1. Let A be a square matrix with A*A = Identity. Let M be a square matrix, such that AMA = M. Let v be an eigenvector of M with eigenvalue e.
a) Show that Av is one too.
b) Assuming that, show that v+Av, v-Av are too.
2. Find the exponential of t times the matrix M
[2 0 3 9 0]
[0 2 1 6 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 3]
Hint: it would be easier if you subtracted 2 * Identity... how do you make use of that?
Solve the ODE system y' = My for arbitrary initial conditions.
3. Let y' = f(t,y).
a) What is the general form of f, if this equation is separable and linear homogeneous?
b) What is the general form of f, if this equation is separable and linear but not necessarily homogeneous?
4. Let c_0 = 0, c_{n+1} = c_n + 2n + 1. Find c_0,c_1,c_2,c_3,c_4,c_5 and guess a general formula for c_n. Then determine the radius of convergence of the power series \sum_n c_n x^n.
5. Assume that
f(x) y dx + (x+y) dy = 0
is exact. Determine the possibilities for f, and solve this ODE. (You should have two constants in your answer: one because you don't know f precisely, and the other from solving the ODE.)
In particular, there is no homework nor will there be final exam questions about Laplace transforms nor series solutions to differential equations.
Here are some questions on most of the topics on the final to get you started.
1. Let A be a square matrix with A*A = Identity. Let M be a square matrix, such that AMA = M. Let v be an eigenvector of M with eigenvalue e.
a) Show that Av is one too.
b) Assuming that, show that v+Av, v-Av are too.
2. Find the exponential of t times the matrix M
[2 0 3 9 0]
[0 2 1 6 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 3]
Hint: it would be easier if you subtracted 2 * Identity... how do you make use of that?
Solve the ODE system y' = My for arbitrary initial conditions.
3. Let y' = f(t,y).
a) What is the general form of f, if this equation is separable and linear homogeneous?
b) What is the general form of f, if this equation is separable and linear but not necessarily homogeneous?
4. Let c_0 = 0, c_{n+1} = c_n + 2n + 1. Find c_0,c_1,c_2,c_3,c_4,c_5 and guess a general formula for c_n. Then determine the radius of convergence of the power series \sum_n c_n x^n.
5. Assume that
f(x) y dx + (x+y) dy = 0
is exact. Determine the possibilities for f, and solve this ODE. (You should have two constants in your answer: one because you don't know f precisely, and the other from solving the ODE.)
Tuesday, November 22, 2005
HW #7 due Monday Nov 28th (revised)
4.2 #12 11,13-14,20 ("characteristic" = "eigen")
4.3 #2-3
4.5 #1,2,7
4.3 #2-3
4.5 #1,2,7
Monday, November 14, 2005
And now for something completely different
If you're curious what I do in the other half of my professor gig, on Thursday I'm giving an undergrad talk on these:
Puzzles, straightening honeycombs, and adding matrices
Given the three kinds of puzzle pieces pictured above, define a puzzle to be a decomposition of a triangle into puzzle pieces (such that the edges match up, like in a jigsaw puzzle). Call a puzzle ``rigid'' if there is no other puzzle with the same outer boundary. A lot can be proven about puzzles (we'll do (1) and (2) in the talk):
1. The number of 0s on one side equals the number of 0s on each of the other two sides - see if you can prove this one before the talk!
2. The lines in the puzzle pieces can all be simultaneously straightened (as in the right-hand picture) if and only if the puzzle is rigid
3. There is an easy 1:1 correspondence between rigid puzzles and inequalities on the eigenvalues of the sum of two Hermitian matrices
4. The statement ``Given four generic lines in space, there are exactly two others that touch all four,'' and others like it, can be turned into puzzle-counting statements.
Thursday, November 17, 12:00 pm, AP&M 2402
Refreshments provided!
Puzzles, straightening honeycombs, and adding matrices
Given the three kinds of puzzle pieces pictured above, define a puzzle to be a decomposition of a triangle into puzzle pieces (such that the edges match up, like in a jigsaw puzzle). Call a puzzle ``rigid'' if there is no other puzzle with the same outer boundary. A lot can be proven about puzzles (we'll do (1) and (2) in the talk):
1. The number of 0s on one side equals the number of 0s on each of the other two sides - see if you can prove this one before the talk!
2. The lines in the puzzle pieces can all be simultaneously straightened (as in the right-hand picture) if and only if the puzzle is rigid
3. There is an easy 1:1 correspondence between rigid puzzles and inequalities on the eigenvalues of the sum of two Hermitian matrices
4. The statement ``Given four generic lines in space, there are exactly two others that touch all four,'' and others like it, can be turned into puzzle-counting statements.
Thursday, November 17, 12:00 pm, AP&M 2402
Refreshments provided!
Sunday, November 13, 2005
Dropping one homework
This is something I meant to declare a few weeks ago; silly of me.
We're going to drop one homework for each person when calculating grades.
We're going to drop one homework for each person when calculating grades.
Wednesday, November 09, 2005
Tests are graded
People definitely did better on this test than the last one. Still unpleasantly many uses of the "universal distributive law", e.g. "e to the a+b equals e to the a
plus e to the b", or "(x^2 + y^2)^n = x^2n + y^2n". Algebra misteakakes are a real killer in this (and basically any other) math class.
The grade boundaries (remember these are approximate, and that the eventual grade is calculated from the number, not the letter):
00-10 F
10-20 D
20-35 C
35-50 B
50-up A
Way more As and fewer Fs this time.
I'll have the answers posted later in the evening, and sometime soon the grading scheme for each problem. The tests will be back tomorrow.
plus e to the b", or "(x^2 + y^2)^n = x^2n + y^2n". Algebra mist
The grade boundaries (remember these are approximate, and that the eventual grade is calculated from the number, not the letter):
00-10 F
10-20 D
20-35 C
35-50 B
50-up A
Way more As and fewer Fs this time.
I'll have the answers posted later in the evening, and sometime soon the grading scheme for each problem. The tests will be back tomorrow.
Sunday, November 06, 2005
Book questions to prepare for the midterm
Conrad
1.1 #14-21
1.2 #5-7, 8-15 (not the graphing part so much)
1.3 #13-16, 26-30 (again, skip the graphing)
2.1 (skip the friction stuff, which we didn't discuss) #8-15
2.2 #1-14, 22
2.4 #1-7, 12-21
Truly, I am sorry about the delay in getting this stuff on the web. I do have a doctor's note.
1.1 #14-21
1.2 #5-7, 8-15 (not the graphing part so much)
1.3 #13-16, 26-30 (again, skip the graphing)
2.1 (skip the friction stuff, which we didn't discuss) #8-15
2.2 #1-14, 22
2.4 #1-7, 12-21
Truly, I am sorry about the delay in getting this stuff on the web. I do have a doctor's note.
Saturday, November 05, 2005
Midterm #2 Wednesday
Sorry for the delay in posting this stuff about midterm #2; some family stuff has occupied me continuously since 2 PM yesterday. I will not be able to post a list of sample problems until morning afternoon.
There will be no Matlab this Tuesday; rather, the TAs will be in their offices for additional office hours during that time. Nor shall there be HW due Friday (Veteran's Day); instead, the homework due the following Friday will be slightly longer.
The sections covered will be all in Conrad (no sequences/series):
1.1, 1.2, 1.3, 2.1, 2.2, 2.4. We've also done a little graphical analysis of ODEs in class that might show up (or might not).
Same setup: no books (including blue), no calculator, one double-sided handwritten sheet that you shouldn't bother filling with integral tables. And as on the last test, learn the rules of exponents or you may hurt your chances very badly.
There will be no Matlab this Tuesday; rather, the TAs will be in their offices for additional office hours during that time. Nor shall there be HW due Friday (Veteran's Day); instead, the homework due the following Friday will be slightly longer.
The sections covered will be all in Conrad (no sequences/series):
1.1, 1.2, 1.3, 2.1, 2.2, 2.4. We've also done a little graphical analysis of ODEs in class that might show up (or might not).
Same setup: no books (including blue), no calculator, one double-sided handwritten sheet that you shouldn't bother filling with integral tables. And as on the last test, learn the rules of exponents or you may hurt your chances very badly.
Monday, October 31, 2005
Saturday, October 29, 2005
HW #5
Sorry for the delay. I really do mean to get these out on Friday, not Saturday (or later!). Happy Hallowe'en.
2.1 #1,8,9,10,11
2.2 #1,2,5,10,13
2.4 #2,3,6,17,18
2.1 #1,8,9,10,11
2.2 #1,2,5,10,13
2.4 #2,3,6,17,18
I'm turning comments off
Not that more than two or three people care (and spambots), but it's probably better that I turn off comments, otherwise I give the false impression that it'll occur to me to read them. If you have questions, email your TA and me.
Wednesday, October 19, 2005
How embarassing!
While sleepily writing up the answers I dropped a factor of n in the question involving sin n. (Don't worry, we didn't grade using that as an answer key.) The corrected version is up now.
Part of HW #4
This is due Oct 28, nine days hence. I will probably add a couple of problems based on what we do Friday.
1.1 #14,15,16,17
1.2 #1,2,7
1.3 #1,2,3,4,14,20,23
1.1 #14,15,16,17
1.2 #1,2,7
1.3 #1,2,3,4,14,20,23
Midterm #1 answers and curve
The answers to midterm #1 are here.
Here's the curve (do read below for what this means):
0-20 F
20-30 D
30-40 C
40-50 B
50+ A
Yes, it was a hard exam; no, it wasn't that hard to get an A on.
The final grade will not be computed by averaging letters together. It will be computed by adding up numbers (book HW + Matlab HW + midterm 1 + midterm 1 + final).
If we say "40 was a B on midterm 1, 60 was a B on midterm 2" etc. and you get B,B,B,B,B then you'll get a B.
People ask "Is there hope for me to pull up my grade?" Absolutely! This is a very important benefit of having the numerical grades closer to 50 than to 90 -- there's much more room at the top. If the averages are 95 and you get 60 on one test and 99 on the next, the 99 doesn't help much. If the averages are 20 and you get 0 on one test, the second test gives you a lot more chance to get back on top.
Here's the curve (do read below for what this means):
0-20 F
20-30 D
30-40 C
40-50 B
50+ A
Yes, it was a hard exam; no, it wasn't that hard to get an A on.
The final grade will not be computed by averaging letters together. It will be computed by adding up numbers (book HW + Matlab HW + midterm 1 + midterm 1 + final).
If we say "40 was a B on midterm 1, 60 was a B on midterm 2" etc. and you get B,B,B,B,B then you'll get a B.
People ask "Is there hope for me to pull up my grade?" Absolutely! This is a very important benefit of having the numerical grades closer to 50 than to 90 -- there's much more room at the top. If the averages are 95 and you get 60 on one test and 99 on the next, the 99 doesn't help much. If the averages are 20 and you get 0 on one test, the second test gives you a lot more chance to get back on top.
Monday, October 17, 2005
Definitely back by Friday
We haven't finished the grading yet (Monday night), but we will definitely be returning the midterms Thursday, before Drop Day on Friday. (That was part of the reason for having a midterm so early.)
Saturday, October 15, 2005
A note from the homework grader
On the midterm, I'm going to try my hardest to be very specific about what constitutes a complete answer, even moreso since people have been asking me about homework grading.
This is what the homework grader has to say about it:
"Here is what I like to see.
1. If a theorem is used, it should be cited (especially with
convergent/divergent series problems since there are so many tests).
2. Non-trivial integrations (especially ones by parts) can't be omitted.
For example, f(x) = ln(x)/x^2 from the last hw. They can't just write
Integral(f(x)) = -ln(x)/x - 1/x with no intermediate steps.
3. In general, when in doubt, show the work.
Those are the main things for me. Stating theorems and showing work."
This is what the homework grader has to say about it:
"Here is what I like to see.
1. If a theorem is used, it should be cited (especially with
convergent/divergent series problems since there are so many tests).
2. Non-trivial integrations (especially ones by parts) can't be omitted.
For example, f(x) = ln(x)/x^2 from the last hw. They can't just write
Integral(f(x)) = -ln(x)/x - 1/x with no intermediate steps.
3. In general, when in doubt, show the work.
Those are the main things for me. Stating theorems and showing work."
Some book questions to help prepare for midterm #1
We had homework on sections 11.1,2,3,4,6,8,9,10.
We discussed some of the material from 11.5 (alternating series) and 11.7 (strategies for testing for convergence) in class; you should know these too.
11.11, 11.12 you can ignore (for now, at least).
Sequences aren't as important for us as series, especially power series, so pay more attention to the later sections.
Basically, you should be able to do any question in 11.7, and at least one of these will be on the test.
The questions 11.8 #3-28 are another good group to be comfortable with.
#29 is tricky and you should be able to do it. The only way to answer a "If you know A, does it follow that B?" question with "No" is to say "No; here's a series for which A is true, but B is not true." If all you say is "I don't see how to get B from A" you leave open the possibility that someone else might. This is one reason it's worth memorizing examples of weird behavior.
11.9 #9,10,15-18,38. (This last one's a little more work than you'll actually see on the test.)
11.10 #3-18. For Taylor series centered at a different from 0, you evaluate derivatives at a instead of 0. Know your sin from your cos!
#23-32 are good too; you should be able to plug in, multiply, differentiate.
We discussed some of the material from 11.5 (alternating series) and 11.7 (strategies for testing for convergence) in class; you should know these too.
11.11, 11.12 you can ignore (for now, at least).
Sequences aren't as important for us as series, especially power series, so pay more attention to the later sections.
Basically, you should be able to do any question in 11.7, and at least one of these will be on the test.
The questions 11.8 #3-28 are another good group to be comfortable with.
#29 is tricky and you should be able to do it. The only way to answer a "If you know A, does it follow that B?" question with "No" is to say "No; here's a series for which A is true, but B is not true." If all you say is "I don't see how to get B from A" you leave open the possibility that someone else might. This is one reason it's worth memorizing examples of weird behavior.
11.9 #9,10,15-18,38. (This last one's a little more work than you'll actually see on the test.)
11.10 #3-18. For Taylor series centered at a different from 0, you evaluate derivatives at a instead of 0. Know your sin from your cos!
#23-32 are good too; you should be able to plug in, multiply, differentiate.
Friday, October 14, 2005
Some stuff about midterm 1, coming Monday
Turn off your phone.
No books or calculators.
Don't bring a blue book.
One double-sided sheet of handwritten notes.
Turn off your phone.
Material only what's been on the homeworks so far or discussed in class (e.g. alternating series).
Please raise your hand and ask questions during the test if you need to!
I'm hoping for an average of about 60/100, so that grades aren't assigned based on random fluctuation. So once you think you got half, breathe easy!
No really gross integrals. Some basic limits. I don't want it to be a speed test.
Turn off your phone!!!
I will post a big list of study problems tomorrow before noon.
Basically a bit of sequences, more about series, some Taylor series stuff.
No books or calculators.
Don't bring a blue book.
One double-sided sheet of handwritten notes.
Turn off your phone.
Material only what's been on the homeworks so far or discussed in class (e.g. alternating series).
Please raise your hand and ask questions during the test if you need to!
I'm hoping for an average of about 60/100, so that grades aren't assigned based on random fluctuation. So once you think you got half, breathe easy!
No really gross integrals. Some basic limits. I don't want it to be a speed test.
Turn off your phone!!!
I will post a big list of study problems tomorrow before noon.
Basically a bit of sequences, more about series, some Taylor series stuff.
Sunday, October 09, 2005
HW #3
Look at 11.7 for more practice with convergence vs. divergence.
11.8 #3,8,9,10,35.
For this you need the "interval of convergence". We didn't discuss this in class -- it just means, find the radius of convergence, then test what happens at x=R and x=-R. (I.e. whether they should be included as the endpoints of the interval of real numbers within the radius of convergence.)
11.9 #7,8,14,35
11.10 #7,8,9,10,27,28
11.8 #3,8,9,10,35.
For this you need the "interval of convergence". We didn't discuss this in class -- it just means, find the radius of convergence, then test what happens at x=R and x=-R. (I.e. whether they should be included as the endpoints of the interval of real numbers within the radius of convergence.)
11.9 #7,8,14,35
11.10 #7,8,9,10,27,28
Friday, September 30, 2005
Class 9/30 + HW #2
Alternating series theorem, ratio test, root test.
HW #2:
11.2 #9,10,29,62
.3 #15,16,19,20
.4 #11,13,14,30
.6 #2,3,10
HW #2:
11.2 #9,10,29,62
.3 #15,16,19,20
.4 #11,13,14,30
.6 #2,3,10
Wednesday, September 28, 2005
Where to turn in homework
I said today in class to turn in homework in section. You can still do that if you want to.
However, henceforth, the preferred method is to stick them in specially marked boxes on the 2nd floor of AP&M.
1. The homework drop boxes are wooden boxes attached to the walls on the 2nd floor of AP&M, clearly labeled "HOMEWORK TURN-IN BOX". Look for the four slots labeled 20D Knutson MATLAB and 20D Knutson HW (two of each type).
2. Written HW is to be turned in to the drop boxes before Friday at 12:00pm, and make sure to put your written HW in a slot labeled 20D Knutson HW. It doesn't matter which of the two HW slots you use; if one is obviously full please use the other.
3. MATLAB is to be turned in to the drop boxes before Wednesday at 12:00pm. Be sure to put your MATLAB in one of the two slots labeled 20D Knutson MATLAB.
However, henceforth, the preferred method is to stick them in specially marked boxes on the 2nd floor of AP&M.
1. The homework drop boxes are wooden boxes attached to the walls on the 2nd floor of AP&M, clearly labeled "HOMEWORK TURN-IN BOX". Look for the four slots labeled 20D Knutson MATLAB and 20D Knutson HW (two of each type).
2. Written HW is to be turned in to the drop boxes before Friday at 12:00pm, and make sure to put your written HW in a slot labeled 20D Knutson HW. It doesn't matter which of the two HW slots you use; if one is obviously full please use the other.
3. MATLAB is to be turned in to the drop boxes before Wednesday at 12:00pm. Be sure to put your MATLAB in one of the two slots labeled 20D Knutson MATLAB.
Friday, September 23, 2005
HW #1
Some of these are odd-even pairs, so you can assure yourself that you've got the right idea with the odd ones (with the answers in the back).
Stewart 11.1:
4,5,7,8, 16,17,19,29,30, 39,40, 49,50,51, 61,64,68.
Stewart 11.1:
4,5,7,8, 16,17,19,29,30, 39,40, 49,50,51, 61,64,68.
No MATLAB session on Sep 27
No MATLAB session next Tuesday (Sep 27). Everything else about the sessions will be explained by TAs on Thursday 29th. The MATLAB assignments are all here by the way.
Class 9/23
In today's class, we defined sequence, bounded, convergent to A, and just convergent.
These two are related, of course: we proved if a sequence converges to a limit A, then it converges.
These two are related, of course: we proved if a sequence converges to a limit A, then it converges.
Wednesday, September 21, 2005
I'm Prof. Allen Knutson, of UCSD Mathematics.
This fall, 2005, I'll be teaching Math 20d on series and differential equations.
I've set up this blog the better to reach people in my class.
Largely it will mirror the course website (which itself needs setting up). One benefit of having it here also is that it can be dropped into RSS aggregators (I like bloglines).
I'm hoping that the comment section will also prove useful. Stranger things have happened!
This fall, 2005, I'll be teaching Math 20d on series and differential equations.
I've set up this blog the better to reach people in my class.
Largely it will mirror the course website (which itself needs setting up). One benefit of having it here also is that it can be dropped into RSS aggregators (I like bloglines).
I'm hoping that the comment section will also prove useful. Stranger things have happened!
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