Course: Math 231/Math 249, Honors
Calculus II, Section E1H/Q1H
Time: MWF 1:00--1:50am
Location: 143 Atlgeld
Hall
Instructor: Prof. Leininger
Phone: 265-6763
Email: clein (at) math.uiuc.edu
Office: 366 Altgeld Hall
Office hours (my office): Monday 4:00--5:00, Tuesday 8:10--9:00, Thursday 11:00--11:50.
Text: Stewart, Calculus: Early Transcendentals, 6th edition
About the honors course: Although the official syllabus for this class is the same as the usual Math 231, we will be delving deeper into certain aspects of the material. In particular, we will be doing some proofs. This semester the recurring theme will be the complex numbers.
I will be pushing many of you to your limitations in the hopes that you will gain a better understanding of calculus. If you are not willing to put in the extra effort required to succeed, I strongly urge you to switch to one of the regular sections of Math 231. On the other hand, you will not be punished for enrolling in the honors section, and I will be reasonable when it comes to grading more difficult problems.
Grades: Grades for Math 231 will be based on quizzes [20%], three
midterm exams [20%,20%,20% each], and a final exam [20%].
The grade for the one credit hour Math 249 will be assigned based on your
completion of the ``honors problems'' (I expect there to be between 5 and
7 of these). These are assignments made throughout the
semester that will investigate some topics outside the regular course
material. These will be collected and graded.
I will curve an exam score if absolutely necessary (e.g. the problems were
legitimately too difficult or the exam
was too long). The final grades will be assigned on the usual scale:
90--100 A, 80--90 B, 70--80 C, 60--70 D,
below 60 Failing.
Homework and quizzes: I will be assigning homework daily in
addition to the honors problems, but it will not be collected. Instead, I
will have approximately two quizzes per
week which will be similar to (or the same as) what is in the homework
(one will be about 10 minutes, the other longer, between 15 and 25
minutes).
These will not always be on the same day each week, and some weeks we may
only have one quiz. In general, these will take place at the beginning of
class.
I encourage you to work together on homework. The best approach is to attempt all the problems on your own before discussing it with your classmates.
Missed exams/quizzes: I will drop the lowest 4 quiz grades for the
semester, so don't come to class when you're sick (although an email
letting me know you won't make it is always appreciated). A missed exam
will count as ZERO, unless there is a valid excuse. Make-up exams for the
midterms will not be given, and a make-up for the final will occur only
under extreme circumstances. In the situation that there is a valid reason to miss an exam (my discretion), I expect prior notification of the absence if it is humanly possible. In the case of a midterm, the average of the other exam score and the final exam will be used to replace the missing score. Honors problems will not be accepted late under any circumstances.
Attendance: The quizzes serve the secondary purpose of providing a
basic attendance record. This could be relevant in making grade decisions
for borderline cases.
Exams: The three midterm exams will be 50 minutes each, during
class.
Midterm 1: Monday, September 19, usual time and classroom.
Midterm 2: Friday, October 21, usual time and classroom.
Midterm 3: Friday, December 2, usual time and classroom.
Final Exam: Friday, December 16, 1:30--4:30 usual classroom.
Assignments and other useful information:
- Monday 8/22: Lecture notes. Make sure you read Section 10.3 on polar coordinates! Homework: §5.5 problems 7, 11, 13, 17, 19, 23, 29, 39, 43; §10.3 problems 1--6.
- Honors problem 1. This will be due Friday 9/9/11 (but you should get started on it now!) Please send me any comments or corrections!
- Wednesday 8/24: Lecture notes. Homework: §7.1 problems 3, 9, 11, 21, 23, 29, 43, 57, 65. Quiz.
- Friday 8/26: Lecture notes. Homework: §7.2 problems 5, 9, 17, 19, 23, 25, 35, 45, 51. Quiz.
- Monday 8/29: Lecture notes. Homework: §7.3 problems 7, 15, 17, 19, 25, 31 (for this last, see solution 2 of Example 5 from this section). Quiz.
- Wednesday 8/31/Friday 9/2: Lecture notes. Homework: §7.4 problems 7, 9, 11, 15, 17, 21, 25, 29, 31,
35.
- Wednesday 9/7: Lecture notes. (We got behind a day, so the date of these notes is off). These notes contain my solutions to the integrals on the worksheet. Homework: §7.5, all odd problems. I don't expect you to have done all of these by next class, but you should do 4 or 5 every day for practice. When you finish these, practice with the odd integrals from the previous chapters, too.
- Friday 9/9: Lecture notes. Homework: § 7.7 problems 7, 11, 13, 15, 19, 21. Quiz.
- Monday 9/12: Lecture notes. Homework: §7.8 problems 5, 11, 19, 25, 29 (compare example 7 from the book), 49, 51, 53.
- Honors problem 2. This is due Monday, 9/19 before you take your midterm.
- Here are some notes on Honors problem 1.
- Wednesday 9/14: Lecture notes. Homework: §8.1 problems 3, 5, 7, 9, 13, 17, 35, 37. Quiz.
- Friday 9/16: Lecture notes. Homework: §8.2 problems 7, 9, 13, 15, 25, 29, 31.
- Extra office hours Friday 9/16: I will be available between 2:00 and 3:30 on 9/16.
- Midterm review problems (see also homework from 9/7 above): Chapter 7 review problems 1--49 odd, 63, 65, 69, 71, 73. Make sure you know how to compute arc length and surface area for surfaces of revolution. Study complex numbers and complex valued functions as in the first honors problem.
- Another practice problem: Suppose L is the line through the origin making an angle θ with the positive x-axis (with 0 < θ < π/2) and suppose f(x) is a function on the interval [a,b] whose graph is above this line. This means that f(x) > tan(θ) x. Find the surface area of the surface obtained by revolving the graph y=f(x) around the line L.
- Monday 9/19: Midterm 1: §7.1--7.5, 7.7, 7.8, 8.1, 8.2.
- Midterm 1 solutions.
- Honors problem 2 solutions.
- Honors problem 3: § 11.1, problem 80. Due Wednesday, September 28.
Hints: For part (a) you should use the ε N definition of convergence. For part (b) there are several steps: (1) after computing the first 8 terms you should see that it appears that the even terms decrease and the odd terms increase. You should prove this by induction. More precisely, assuming a_{2n} > a_{2n+2}, prove that a_{2n+1} < a_{2n+3} and a_{2n+2} > a_{2n+4}. (2) You should prove that the even and odd sequences each converge by proving that they are bounded and appealing to the bounded monotone sequence theorem. (3) You should prove that the limits L_{e} and L_{o} of the even and odd sequences, respectively, are the same. For this, show that the identity a_{n+1} = 1 + 1/(1+a_{n}) for the terms gives you TWO identity for L_{e} and L_{o}.
There are very likely other ways to do this problem!!
- Wednesday 9/21: Lecture notes. Homework: §11.1 problems 3, 5, 7, 9, 11, 13, 15. Also, for each of the sequences below, decide whether it converges, and prove it (so, for a given ε > 0 find an N so that...):
A. {n/(3n+3)} B. {n^{2}/(300n+2n^{-1})}
C. {(-1)^{n}n/(n^{2}+n + 5)} D.
{2^{n}/n!} Solutions.
- Friday 9/23: Lecture
notes. Homework: §11.1 problems 17--45 odd, 57, 61, 63, 65, 77.
Quiz.
- Honors problem 4: §11.2 problems 64, 73, 76. Due Wednesday, October 5.
- Monday 9/26: Lecture notes. Homework: §11.2 problems 11, 17, 21, 25, 35, 39, 41, 43, 55. (for 41, 43 see example 4 from book).
- Wednesday 9/28: Lecture notes. Homework: §11.3 problems 3, 7, 15, 19, 23, 25, 27, 35. Quiz: FALSE!!!
- Solutions to Honors problem 3.
- Friday 9/30: Lecture
notes. Homework: §11.4 problems 3-31 odd, 37,
43. Quiz.
- Monday 10/3: Lecture notes. Homework: §11.5 problems 1--19 odd. Quiz.
- Honors problem 5: §11.3 problem 40 and §11.5 problem 36. Due Friday, October 14.
- Wednesday 10/5: Lecture notes. Homework: §11.5 problems 23--29 odd §11.6 problems 1--19 odd, 25, 27, 29, 31, 33.
- Thursday 10/6: office hours cancelled!! you can see me between 3 and 4 Wednesday 10/5.
- Friday 10/7: No class. I will hold additional office hours on Monday 10/17 from 4:00 until 5:30 to make up for the missed class.
- Solutions to honors problem 4.
- Monday 10/10: Lecture notes. Homework: §11.6 problems 21, 23; §11.7 problems 1-38 (lots of practice for the test next week!) Series Worksheet solution.
- Wednesday 10/12: Lecture notes. Homework: §11.8 problems 3---13 odd, 27, 29, 31. Also, keep working on the series from previous sections! Quiz.
- Friday 10/14: Lecture
notes. Homework: §11.8 problems 1-31 odd. Continue to work on
series
from previous sections...Quiz
- For midterm 2, we will cover section 11.1--11.8. As preparation:
- Make sure you understand the definition of the limit of a sequence. Given ε make sure you can find N (for practice, try this for §11.1 problems 18, 19, 25, 29, and 35 for example). Be able to compute using limit laws, too.
- Know geometric series, when they converge, what their sums are, and how to find sums via algebraic manipulation (e.g. sums and differences; telescoping series).
- Make sure you know how all the tests work, what the hypotheses are, and be able to use them effectively.
- Go through §11.2--11.7 and decide convergence/divergence for every series in the problem sets, as well as absolute versus conditional convergence if the series converges.
- Be able to estimate the sum of a series which converges by applying the integral test and the alternating series test, and how to get an estimate with a certain accuracy (see pages 700--702 and §11.3 problems 32--38; and pages 712--713 and §11.5 problems 23--30).
- Be able to detect a geometric series in a problem (as in honors problem 4) and compute the sum.
- Be able to find the radius and interval of convergence of a power series.
- Monday 10/17: We continued the discussion from §11.9
proving that a power series was differentiable and that its deriviative is the ``obvious'' power series.
- Wednesday 10/19: I answered questions and we reviewed.
- Office hours reminder: 10/20 My office hours Thursday, 10/20, have been rescheduled for 10:00 -- 11:00.
- Friday 10/21: Midterm 2! Solutions. I just spotted a typo on the solutions, too.. the limit for number 2 should be ``1/5'' instead of ``1''.
- Here are solutions to honors problem 5.
- Monday 10/24: Lecture notes. Homework: §11.9 problems 3--9 odd, 15, 17, 27.
- Office hours 10/24--10/28 This week's office hours will have
to be shuffled a little bit. They are: Tuesday 8:45--9:45, Thursday
10:00--11:00 (in particular, Monday afternoon's office hours are
canceled).
- Wednesday 10/26: Lecture notes. Homework §11.10 problems 1--15 odd. Quiz.
- Honors problem 6. §11.10 problem 70(a). See the lecture notes from 10/26 for what to do... Due Wednesday, 11/2. Solutions.
- Friday 10/28: We went over the second midterm.
- Monday 10/31: Lecture notes. Homework §11.10 problems 17--27 odd.
- Wednesday 11/2: Lecture notes, Homework
§11.10 problems 29--37 odd, 43, 47, 49, 53, 55, 57.
- Honors problem 7 is here. Last
one! Due Friday 11/18 or Monday 11/28... whatever you prefer.
- Friday 11/4: Lecture
notes. No new homework. Use the weekend to catch up and to work on
Honors problem 7. Quiz.
- Monday 11/7: I described how, given a function f(x), the Taylor polynomial T_{n}(x) at x = a is a good approximation to f(x) at a: specifically, it is the unique degree n polynomial so that
lim_{x → a} ^{(f(x)-Tn(x))}/_{(x-a)n} = 0. After that, I followed the book's examples from § 11.11. Homework §11.11 problems 13, 15, 17 (parts a,b for these three), 25, 31, 33.
- Wednesday 11/9: Lecture notes. Homework §10.1 problems 5, 7, 13, 15, 17, 21, 31, 41.
- Friday 11/11: Lecture notes. Homework §10.1 problems 24, 27, 28. §10.2 problems 3, 5, 9, 17, 19, 31. Quiz.
- Monday 11/14: Lecture notes. Homework §10.2 problems 33, 37, 41, 45, 53, 57, 61, 65. Compute the area bounded by the curve given by x = (1+cos(t))sin(t), y = (1+sin(t))cos(t), t ∈ [0,2π], Quiz.
- Wednesday 11/16: Lecture notes. Homework §10.3 problems 21, 31, 33, 35, 56, 57, 59, 67
- Friday 11/18: Homework §10.3 problems 37, 39, 41. §10.4 problems 1, 9, 13, 17, 19, 23, 27, 30, 45.
- Midterm 3 will cover sections 10.1--10.4, 11.8--11.11. In preparation:
- Know how to sketch a curve given by parametric equations and find parametric equations given a description of the curve.
- Be able to find the tangent line to, length of, and area bounded by a parametric curve
- Know how to switch between polar and rectangular coordinates.
- Be able to sketch polar curves, find areas, lengths, tangent lines.
- Know what a power series is, how to find its radius and interval of convergence.
- Be able to compute derivative and integrals for power series.
- Be able to find power series representations for functions with Taylor/Maclaurin series. Know the basic Maclaurin series (page 743).
- Know Taylor's inequality and how to use it to prove that a function is equal to its power series.
- Be able to estimate the sum of a power series with its Taylor polynomials using Taylor's inequality and/or the alternating series test.
- Monday 11/28: Quiz.
- Here are solutions to honors problem 7.
- Friday 12/2: Midterm 3!! Solutions.
- Review problems for Final exam
- §7.1 1--37 odd
- §7.2 1--49 odd
- §7.3 1--29 odd
- §7.4 1--51 odd
- §7.5 1--79 odd
- §7.7 7--17 odd
- §7.8 1--41 odd, 49--59 odd
- §8.1 1--17 odd
- §8.2 1--15 odd, 25--33 odd
- §8.3 1--13 odd, 23--33 odd
- §10.1 1--27 odd, 24, 28, 40--44 all
- §10.2 1--9 odd, 17--21 odd, 25--35 odd, 36, 37--43 odd, 57--65 odd
- §10.3 15--25 odd, 29--47 odd, 55--69 odd
- §10.4 1--13 odd, 17--41 odd
- §11.1 3--45 odd, 55-65 odd (for 17--45, if the sequence converges, try to prove it using the definition).
- §11.2 1, 11--51 odd, 55, 59, 63, 71.
- §11.3 3--29 odd, 33--37 odd
- §11.4 3--31 odd, 37, 39, 43, 45
- §11.5 3--29 odd, 33
- §11.6 1--31 odd
- §11.7 1--37 odd
- §11.8 1--31 odd, 37
- §11.9 1--17 odd, 23--29 odd, 31
- §11.10 3--37 odd, 43, 47--57 odd
- §11.11 13--21 odd (parts (a),(b) only), 25, 27, 29
- Review honors problems 1 and 7 --- there will be one problem about complex numbers...
- Office hours during finals: Monday 10--11, 2--3, Tuesday 10--11,
3--4, Wednesday 10--11, 3--4, Thursday 10--11, 2--3.
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