Q: Are we going to have questions on finding arc length of the
curve (ex: question 46 from 7.6) on the final?
A: It's certainly possible. We did it both when y = f(x) and
when x = f(t), y = g(t). The formulas are fairly easy to
remember. As a basic rule, anything that might have shown
up on an hour test is a candidate for the final, plus there
will be a hyperbola question.
Q: Will we need the textbook after this semester?
A: It is the text for Math 242 and for Math 243, so you shouldn't
be selling it, unless this is your last calculus class ever. It's
also a good paperweight.
Sa 12/10 -- Reading Day office hour, Saturday December 10 (1-2 pm), 327 Altgeld (nobody showed!)
F 12/9 -- Test 3 returned and discussed. The structure of the Final and of the grading in the course was covered.
W 12/7 -- Test 3, in class.
M 12/5 -- Problem 6 from the review session. A problem in which x(t) and y(t) are given, and you have to find dy/dx and d^2y/dx^2. The ICES forms were distributed. Summary of topics for test 3: one or two questions on the convergence of a power series, including the endpoints, then the material from HW20->25: sections 6.4, 8.1 and 9.1 -> 9.6. There will not be a question on hyperbolas on test 3, although there may be one on the final, in the part where you choose questions to answer. I will answer any questions about the exam up until about 8pm Tuesday night, and will try to post them on the website.
Su 12/4 -- The review session, in 141 Altgeld, except for problem 6, in which I erroneously suggested that an ellipse could have eccentricity equal to 2.
F 12/2 -- More discussion of conic sections and a distribution of review questions for test 3.
W 11/30 -- HW 23 returned, and discussed to some extent. We started on
the very last topic: conic sections, done somewhat differently from
the book.
HW: 9.5 -- 1, 2, 29, 30; 9.6 -- 6, 7, 13, 14, 19, 20, 35, 36, 39, 40.
M 11/28 -- HW's 21 and 22 were returned and a few common errors were
discussed. I talked about 9.4 and part of 9.5. The final hour exam
will be in class on W 12/7, and will cover material through the last
hw, which is due on F 12/2.
HW 9.4 -- 1, 2, 3, 4, 15, 16, 17, 18, 30; 9.5 -- 5, 6, 11, 12.
F 11/18 -- A lot of people came, to get one homework set returned and
somewhat discussed. I also derived the formula for the number of ways
to parenthesize n symbols. For example, if n = 4, the five choices
are:
((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), a(b(cd))
Using the method of generating functions, which uses power series in
a fundamental way, this number is shown to equal (1/n)*C(2n-2,n-1),
where this last expression is a binomial coefficient. I also briefly
talked about counting points in the plane with integer
coordinates. This is actually an important area of mathematical
research. The other thing I did was give a proof of the Cauchy
condensation theorem: if a_n is a positive decreasing sequence, let
b_n = 2^n*a_{2^n}. (Thus, if a_n = 1/n, then b_n = 2^n/2^n = 1.) Then
the sum of a_n is convergent if and only if the sum of b_n is convergent.
W 11/16 -- More on polar coordinates and parametric curves. HW
questions discussed to the extent that people's lagging interests
would allow. This included the parametric form for the unit circle: x
= (1-t^2)/(1+t^2), y = (2t)/(1+t^2), which leads to the
parameterization of Pythagorean triples: a = r^2 - s^2, b = 2rs, c =
r^2+s^2 implies a^2 + b^2 = c^2. HW due M 11/28:
9.2 -- 22, 23, 25, 26, {33, 34, 35, 36 one problem}, 53, 54; 9.3 --
10, 11, 12.
M 11/14 -- Test returned. A beginning of the discussion of polar
coordinates. Keep in mind that the points are unchanged, and all
that is happening is that different names for the same points are
given. If you have graphing software, play with the polar equations.
Nothing I can say at the blackboard will be as useful as your
experiential understanding! HW due W 11/16:
9.1 -- 1, 4, 8, 9, 11, 12, 17, 18, 25, 26; 9.2 -- 1abc, 2abc, 5, 6,
13, 14.
F 11/11 -- Distracted prof means that the test isn't back
today. Discussion of exponential growth and decay and some
simple differential equations, and an impressionistic introduction
to polar coordinates and conic sections. HW due M 11/14:
8.1 -- 1, 2, 4, 5, 7,8, 21, 22, 29, 30, 33, 34
W 11/9 -- Test. Not put on the web.
M 11/7 -- A few questions about the second test, which will be in class on
Wed. 11/9. A quick run through arclength and surface area of
revolution, both of which can be done better in parametric form, and
an introduction to exponential growth and decay. The following
homework is due on F 11/11:
6.4 -- 4, 5, 8, 9, 11, 12, 16, 17, 21, 22, 29, 30
Test two postponed to Wed. Nov 9 in class. Review session will still be Sunday night. The second test, covering chapter 10, will be held Monday Nov. 7, in class. There will be a review session Sunday night from 7 until 8:30 or so, in room 141 Altgeld Hall, which is roughly one floor below the last review session.
F 11/4 -- Unhappy class performance on the last two homeworks led to a delay of test two to next Wednesday. This class period was a slow walk through these homeworks. Monday will have new, unrelated, material, but I'll also answer last-minute questions. The next homework will be due on F 11/11.
W 11/2 -- Some discussion of the last two homeworks, both of which will be returned on Friday. The long-awaited proof of Stirling's formula. Arc-length. Review problems distributed, to be talked about on Sunday.
A student writes: "For problem 66, I got 0.
Here's what I did:
-2/3=(1/3)-1
-2/6=(1/6)-(1/2)
-2/9=(1/9)-(1/3)
...
So I replaced the negative terms and set up two different
series and subtracted them.
They both turned out to be the harmonic series
(1+1/2+1/3+1/4+...), so I got zero.
Am I right, and if I'm not, where should I start?
My reply: Well, try looking at the series up to a specific cutoff, like up to
1/(3n). If you do it your way, you get oo - oo, but you have to be a
little more careful. It's not unlike one of the honors questions.
M 10/31 -- Review of the last three homeworks and a little bit more.
this is the last new material for test two. Unless you bring questions
to class on W 11/2, I will have to start talking about ... section
6.4: Arc length and surface area of revolution. HW due W:
p. 770: 31 -> 40, 66.
Note typos fixed 1:50pm Sunday on problem numbers.
F 10/28 -- More on power series and their endless ability to be
twisted into other power series. A side trip to the Fibonacci numbers.
HW due M 10/31:
10.8 -- 49, 50, 51, 52; 10.9 -- 23, 24, 25, 26; HQ6 10.8 -- 68 (typos fixed.)
W 10/26 -- Power series. Some more review of infinite series. The
earliest the second test could be is M 11/7. There will be
a review, in any case, on Sunday evening 11/6. Time and details to
follow. HW:
10.8 -- 17, 20, 31, 32, 33, 34, 35, 36, 41, 42. HQ6 (due 10/31) --
10.8 -- 68.
M 10/24 -- More on series, and then an introduction to power series.
Read the book, do the homework.
10.8: 1 --> 12, 21 --> 26.
F 10/21 -- Mostly review, recent HW gone over carefully, and a bit of discussion of HQ5. A conditionally convergent series can be rearranged to some to anything you want. The homework for Monday is to do infinite series questions of your choice, which I will then attempt to correct. The second hour exam will cover chapter 10, which means we only have power series to go. The earliest it could be is M 11/7. It will probably be that week in any case, which means there will be another review on Sunday night, November 6.
I mistakenly wrote that the last 4 problems on the HW are from 10.10 -- which is "power series solutions to differential equations", a topic we never covered in class. I hope you realize that I meant to have them in the very next section, "miscellaneous problems".
W 10/19 -- Basically, a review of the content of the last two
homeworks. A little bit informally about power series. The HW for
F 10/21 is mostly review:
10.7 -- 13, 14, 17, 18; Misc. Problems (p. 769) -- 17, 18, 19, 20; HQ5.
M 10/17 -- More on section 10.7, with the absolute convergence rule,
the ratio test and the root test. We'll spend lots of time on
examples in the next few classes. Homework, due W 10/19:
10.7 -- 9, 10, 11, 12, 15, 16, 21, 22, 25, 26, 31, 32, 33, 34 and ...
due F 10/21: HQ5.
Thanks to alert e-mailers, I have corrected two errors in the
homework for Monday. The main set of the problems is from 10.6, on the
comparison test, and HQ4 is from 10.5.
A student writes: I don't understand
what question 50(b) is asking. Are we supposed to prove that
cn is decreasing using the integral given in the problem and
problem 49?
My reply: Yes, that's exactly what you're being asked to prove, and
then explain why
this means that it converges. The explanation is pretty short.
F 10/14 -- Zipping along, past the comparison tests to the alternating
series test and absolute and conditional convergence. Somewhat lighter
homework to account for HQ4.
10.4 -- 19, 36; 10.6 -- 15, 16, 17, 18, 23, 24, 29, 30, 31, 32.
HQ4 is due on Monday: 10.5 -- 49, 50.
W 10/12 -- Integral test and comparison and limit comparison tests
discussed and questions answered. HW due Friday, except for HQ4, which
is due M 10/17:
10.5 -- 27, 28, 31, 32, 35, 36, 39, 40; 10.6 -- 1, 2, 3, 4, 5, 6
HQ4 10.5 --49, 50 (both, including the odd one)
TeX handout on tap for Friday.
M 10/10 -- On to the integral test! Think visual. Homework due W:
10.4 -- 19, 20; 10.5 -- 1, 2, 5, 6, 13, 14, 15, 16, 21, 22, 23, 24.
F 10/7 -- HW 9 returned, with a mulligan. Handout on Taylor
polynomials and exact formulas, both in integral form and in
"mean value" form as in the book. Examples given of the basic
series. Next homework, due Monday, it will be graded for sure:
10.3 -- 25, 26, 45, 46; 10.4 -- 11, 12, 13, 14, 21, 22, 23, 24, 31, 32.
Just for absolute clarity: there will not be a meeting Sunday night,
that's only for the pre-test reviews.
W 10/5 -- First test returned. Full throttle into Taylor polynomials and Taylor series. No written homework for Friday, but spend serious time reading section 10.4!
M 10/3 -- First test.
Test 1, in class, Monday 10/3.
F 9/30 -- A small bit on integration, but the main review will be
Sunday night in 245 Altgeld, starting at 6:30pm, and running until
9:00, or you get tired. We did start chapter 10, with more on
infinite sequences and infinite series. HW9 is due W 10/5:
10.2 -- 5, 8, 9, 10, 27, 28, 39, 40, 57, 58; 10.3 -- 11, 12, 13, 14,
19, 20.
W 9/28 -- A few examples of integration techniques. A bit more on improper integrals. The review questions for the Sunday night session were distributed. We began to discuss infinite sequence and infinite series, with special applications to recurring decimals. No HW yet on them. The review will be Sunday night 6:30-9:00, but staying as long as you need.
M 9/26 -- Discussion of integrals consumed most of the period, with a small introduction to infinite sequences. The homework assignment for Wednesday is for you, if you choose, to do some integrals and ask me to look at whether you've done them correctly. An ungraded assignment.
F 9/23 -- Sorry I forgot to put a note up for the 9/21 class. We
talked about the Gamma function and a little about complex-valued
functions as well. This is the last homework to be covered on the
first exam. Test 1 will be M 9/33 = 10/3, in class. There will be
a sample test distributed on F 9/30 of about twice the length of
a potential exam. There will be a review on Sunday night 10/2,
time and place to be announced. Homework due 9/26;
7.8 -- 1, 2, 3, 4, 5, 6, 17, 18, 21, 22, 39, 40.
W 9/21 -- My enrollment in the UIUC "Early Senility Program" was
discussed. We finished section 7.7 and started 7.8. The homework, due
9/23 is
7.7 -- 3, 4, 7, 8, 15, 16, 21, 28; HQ3 is due on Friday.
M 9/19 -- Anecdotes were discussed and chocolates distributed.
Examples from the homework were given. There is nearly always
more than one way to do an integral and, unfortunately, they are
not always interesting. Quadratic substitutions were discussed.
The HW for 9/21 is
7.6 -- 17, 18, 31, 32, 46. 47, 50; 7.7 -- 1,2; HQ3 (due Friday) 7.7 -- 48ab.
F 9/9 I'm back and alive. The HW for M 9/19 is:
7.5 -- 19, 20; 7.6 -- 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 22, 23.
We started to talk about trig substitutions.
In response to a question, "twice-differentiable" just means that you should assume that f, f' and f'' all exist.
W 9/7 -- HW 3 was collected, HW 4 and HQ 2 are due on Friday. We
did more with partial fractions and started on trigonometric
substitutions. Examples were done in class and questions answered.
I gave out a handout on the substitution u = tan(x/2). Here is
HW 4:
7.5 == 8, 12, 13, 14, 25, 36, 38, 39, HQ2
Note: when I said in class that |r - s| < t means that -t < r - s < t,
I meant that this applies in HQ2e,f, but you have to figure out what
r, s and t are. And, see Workshop Website
for where I'll be next week.
A student writes:
I just had a question about choosing "u" for
integration. I know that there is the order of importance
which is logarithm, inverse trig, power of x, trigometric,
and exponential. but does this rule apply all the time or
are there some exceptions? and if the integral has x & let
say, lnx, then does u = lnx because x is power of x (x^1)?
My reply: This "rule" you mention is just a general guide. Ordinarily,
you want to make a substitution for the most complicated or inner
expression.
A student writes: is there a rule for rational & partial
functions? let say if the exponent of the x from the
denominator is smaller than the exponent of the x from the
numerator, is that when you apply the long division system?
and if it is vice-versa, then do you use the method of
partial function by using A & B? Or is there no such rule?
My reply: I will talk about this in class Wednesday. In general, you
divide the denominator into the numerator by long division, and deal
with the remainder, where the numerator has lower degree than the
denominator, by partial fractions.
F 9/2 -- We got our last new classmember, and decided that I will
make up for being away on 9/12,14,16 by holding extra review sessions
on the Sunday before the exams. The earliest possible date for the
first exam is 10/3. (Note: I know Rosh Hashanah begins at sundown on
10/3; let me know if this is an issue.) I also distributed the
phone/email list and returned HW2 with HQ1. We finished the discussion
of 9.4 and started 9.5. HW3:
7.3 -- 21, 28; 7.4 -- 4,7,19,20,43,44; 7.5 -- 1,2,3,4,5,6 (due 9/7)
and
HQ2, due
Fri. 9/9. If you start working on this before Wednesday, you'll have a
chance to ask questions that I might even answer.
W 8/31 -- A catch-up day. The difference between high school and
college math classes. Retrospective of HW 1. The questions of HW 2
taken slowly. That Honors Question isn't as bad as you all thought
it was. More trigonometric integrals. No homework due 9/2. I'll
make up for it over Labor Day weekend.
M 8/29 -- We went over HW 1 and its solutions (unlinked), discussed
the most commonly listed topics that you feel uncomfortable with;
namely, integral approximations, Newton's method, L'Hopital's Rule
(which we'll be talking about), and centroids and hyperbolic functions
(which we won't be talking about). We zipped into section 7.4 on
trigonometric integrals. The homework for W 8/31 contains an Honors
Question.
7.3 -- 25,30,51,52; 7.4 -- 1,2,11,15,16,18.
On the honors question, what I intended to write in (b) was:
"Take the sum of the inequalities in (*)k for k =
1,...,n-1 to bound the integral from 1 to n of ln x dx."
F 8/26 -- One more handout: a problem solving template
, which is taken from George Polya's great book "How to solve it",
on problem-solving. We briefly covered integration by simple
substitution and then a bunch of examples of integration by parts.
The first homework assignment is due Monday. All problems are
assigned, but only the even ones will be graded:
7.2 -- 9,10,13,18,33,34; 7.3 -- 1,2,5,6,11,14,31,32.
By the way, I think that 32 is much more easily done by substitution
than by parts, and that's ok with me.
W 8/24 -- First day of class. Two handouts:
Course
Organization, and link to Math 230
Syllabus. A quick review of first semester calculus. The first HW
is to read through chapters 1 - 6 of the text and write down two or
three things you do not understand or are not comfortable with. This
is an assignment to turn in, but it obviously can't be graded! There
will be a real assignment for the weekend.
A student writes:
"do we need to bring our
books regularly to the MATH 230 honors class?"
My reply: "It's up to you. I won't expect you to bring them, but I might
occasionally make a reference to a diagram or formula. I will
always bring the book with me."