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## Visiting Professor## Office:713 Evans Hall University of California at Berkeley Berkeley, CA 94720-3840 Tel: 510-642-3768 Fax: 510-642-8204 Email: stankova@math.berkeley.edu Office Hours TTh 9:30-10:50am in Evans 713. Webpage: http://math.berkeley.edu/~stankova Click below for the syllabus: ## Course Syllabus for MATH 55, Fall 2014.GSIs finalized office hours were revised in the syllabus on 9/6/2014. Check them out and keep them with at all times. You are welcome to visit any GSI's office hours, along with the professor's office hours. ## For enrollment questions:Do NOT
contact the instructor or the GSIs. We have no control over
enrollment. Contact instead ## Email is ONLY for emergencies (e. g., medical and family emergencies).Email is NOT for resolving enrollment questions, or asking for letters, or for discussion of midterm results, or for any discussion about how the student is doing in the class or how to improve. I have received lately a number of such emails, to all of which the response is to come see me in person in office hours (bringing all necessary documentation with you). You are also welcome to visit any GSI's office hours to discuss math questions or how to improve in the course; the GSIs are very qualified to discuss any math question. This is clearly written in the syllabus and was discussed in detail during the first lecture. ## Homework Assignments and Notes:HW Solutions are posted about a day before the quiz and will be taken off the web in a week. Do NOT ask for solutions to be posted earlier: you must attempt to do your homework without help from posted solutions. If you are late copying them, or you lose them, or some other thing happens: do NOT ask us for the files of the previous solution since we do NOT distribute electronic files of the HW solutions. Instead, ask your classmates for the HW solution files. Make sure you download and save the solutions as soon as they are posted, to avoid having to ask your classmates later on for them. If not specified odd or even exercises, it is assumed only even exercises, e.g., #2-8 means 2,4,6,8. ## Homework 4A, due Wed, Sept 24:Read 2.3. Write #2,6(a)(b)(d),12,14(a)(b)(e),20,22(b)(c),26,30(d),40,50,54,64,74(b). Extra challenge: 76. Read 2.5. Write #2,4,6,8,10,16,18,20(use bijections),24. ## Homework 3B, due Wed, Sept 17:Read 2.1. Write #10,12,16,18,20,22,24,26,32(b)(d),38. Read 2.2. Write #2,4,12,14,16(d),18(c),24,26(b),30,44. ## Homework 3A, due Wed, Sept 17:Read 1.8. Write #4,6,10,14,18,22,30,32,34,36,42,44. This HW will be tough for those who are not experienced with problem solving and proofs, which is natural at this level. Hence some hints are listed here. Do your best.
## Homework 2B, due Wed, Sept 10:Read 1.6. Write #2,4,8,12,16,18,24,28. Challenge: #35*. Read 1.7. Write #6,8*,10,12,18,20,24,26,30,34. ## Homework 2A, due Wed, Sept 10:Read 1.3. Write #6,8,10(b)(c),14,24,30,32,60,62(a). Read 1.4. Write #6,10,16,20,32,50,60. Read 1.5. Write #6,14,20,24,30,40,46.
## Homework 1, due Wed, Sept 3:Read 1.1. Write #2,4,8,12,14,18,28,30,34,38. Extra Challenge: #40, 49. Read 1.2. Write #2,12,16,24,28,40,42. Extra Challenge: #36, 38.
## Berkeley Math Circle |

TTh 3:30-5:00pm, 105 Stanley Hall

**Course Syllabus: **PDF
format , Postscript
format

**Homework Assignments and Notes: **

HW1, due January 27: PDF format, Postscript format

HW2, due February 3, revised 1/29/09: PDF format, Postscript format

HW3, due February 10: PDF format

HW4, due February 17, revised 02/12/09: PDF format.

HW5, due February 24: PDF format

HW6A and HW6B, both due March 3: HW6A PDF format HW6B PDF format

HW7, due March 10: PDF format.

HW8, due March 17: PDF format.

HW9, due March 31: PDF format.

HW10, due April 7: PDF format.

HW11; due April 14: PDF format.

HW12; due April 21: PDF format.

HW13; due April 28: PDF format.

HW14; due May 5: PDF format.

HW15; due May 12: PDF format.

HW15 Solutions: to be taken off the web in a week: PDF format

by Siara Hunt (with GSI: Sarah Iveson) PDF format.

by Maura Torres (with GSI: Theo) PDF format.

Note that some of the pages in the reviews are scanned upside-down: I can't correct this since my scanner rebelled this morning and... decided to randomly rotate the pages! So, print out the reviews and flip the physical pages to correct the situation.

The award for the best final exam review will be presented at the final exam on May 21, Thursday, 12:30-3:30PM in room 155 Dwinelle Hall.

PDF
format. Note that we solved and discussed a number of these
problems during the last office hour/review session in class. Please,
be aware that this mock-final is exactly what its name indicates,
i.e., it does NOT necessarily cover all topics that could appear on
the actual final exam; the mock-final is also not necessarily of the
same format, length or difficulty as the actual final exam. The
mock-final simply contains 10 problems for you to try out to solve as
part of your preparation for the final exam; however, the mock-final
does NOT replace thorough study of all topics for the final exam. The
final will concentrate on topics covered after Midterm 2; but what we
have studied beforehand cannot be forgotten since parts of anything
may pop up within the solutions of the final exam problems. **No
solutions to the mock-final will be posted. Please, do NOT e-mail me
asking for such solutions. If you wish to discuss these problems, you
should go to the office hours listed below (you can go to any GSI's
office hours below). **

**Zvezdelina** **Stankova: **Tuesday May
12, 4:00-5:00pm, 713 Evans.

**Sarah Iveson: **Monday, May 18,
10am-12noon, 743 Evans.

**Shenghao** **Sun: **Tuesday May 19,
4:00-6:00pm, 1004 Evans.

**Gregory Igusa: **regular office hours this
week: Mon, May 11, 2-3pm, and Wed, May 13, 1:30-2:30pm; extra hours
Wed, May 20, 3-4pm, 1020 Evans. 1:30-2:30

**Theo Johnson-Freyd: **by appointment only
with his PDP students, 1058 Evans.

I'll post here more office hours if and when they become available.

**Matthew Satriano: **Monday 3-4pm, Wednesday
3-4pm, 1004 Evans.

Please, remember that e-mail to the instructor is only for emergency situations. Questions about "where and when the final is", "what topics will be included on the final", "what is the format of the final" and so on do NOT constitute an emergency. What is written in the syllabus about exams and what I have said in lecture about the final exam is all I am going to say about the final. It particular, the same rules apply to the final exam as to the midterms, including the cheat sheet. Please, do NOT e-mail me with the above and alike questions. Concentrate instead on learning and preparing for the final exam.

Students in the Calculus 1A class are invited to submit their own reviews for the final exam. The top 3 reviews will be posted on the class website, and the very best review will be rewarded with a book "A Decade of the Berkeley Math Circle - the American Experience, Volume I", edited by Zvezdelina Stankova and Tom Rike, published by the American Mathematical Society, 2009. Here are the rules for the "contest".

**Each review for the final must:**

1. be written by a single student in the class or by a group of students in the class. No help outside of the class is permitted.

2. be either typed, or very legibly and neatly written on white regular sheets of paper 8"x11";

3. be no longer than 15 pages.

4. address the following items: concepts, theorems and problem-solving techniques (with examples). The material to be covered on the review must encompass everything we covered after Midterm 2 and till the end of the semester. Thus, start with the topics we covered AFTER slant asymptotes, and finish with 6.2 Volumes and 6.5 Average Value of a Function. Sections 6.3 and 6.4 will NOT be covered in this course and should not be included in the review.

5. be submitted in **two hard copies directly
to your GSI on Monday, May 11 in discussion sections **(this will
be the last day of discussion sections). No reviews submitted after
that will be accepted.

6. on top of the first page, write legibly
**"Draft of Final Review for Calculus 1A, spring 2009, written
by (put here the name of the student), with GSI (put here the name of
the GSI)".** If there are more than one student involved in
the writing of the review, please, select only one name to be written
on top of the review: this will be the contact student to receive the
prize.

The GSIs and I will then select the top three reviews and post them on the web soon thereafter. The very top review will receive the book prize at the final exam.

get more insight into what you're learning

work alone or connect with peers in a group

share tips for success

talk to grad students ready to help

crazy math questions welcome!

**Every Thursday from 5-7pm in 1015 Evans **

The events are sponsored by Unbounded Representation, the graduate student group for diversity in Math. They already have lots of grads signed up to come, so please invite undergrads!

Please contact Lynn Scow (lynn@math.berkeley.edu) or Shawn McDougal (shawn@math.berkeley.edu) if you have any questions about these events.

**Midterm 2 Review Handout: **PDF
format. Note that, even though I have tried to include everything
we have studied after Midterm1 on this very long review handout, the
material is a lot and some things may not be included. You are
responsible for reviewing everything we covered since Midterm 1; at
the same time, you cannot "forget" what happened before
Midterm 1 since some problems and reasoning on Midterm 2 may depend
on that previous knowledge. Same rules apply for Midterm 2 as they
did for Midterm 1.

**Exam Instructions: Please, read before the exam.** PDF
format

**Instructions for receiving your midterms 2: **We expect that
the midterms will be graded by Monday. Please, keep in mind that you
can look at your midterm in the presence of your GSI (no writing or
erasing is allowed!) and if you have a complaint about the grading of
a problem, you must immediately give BACK your exam to your GSI and
explain what you are complaining about. If the GSI decides that your
complaint is reasonable for review, the GSI will bring your midterm
to me and you must come to my office hours next week to discuss the
problem. I will not review anyone's exam unless the student comes to
my office hours within one week of the exam. However, if you do not
turn it back to your GSI and leave with your exam, you CANNOT
complain about the grading of your problems and I will not review
such exams: for an exam to be reviewed, I must receive it back from
your GSI with comments on which problem you are complaining about.

**Midterm 1 Review Handout: **PDF
format

The Student Learning Center (SLC) offers free Math and Statistics tutoring to all registered undergraduate students. We are located in the Chavez Student Center in the Southwest corner of the Atrium. For more information, visit slc.berkeley.edu

HW6, due March 3: PDF format, Postscript format

HW7, due March 10: PDF format, Postscript format

HW8, due March 17: PDF format, Postscript format

HW9, due March 31: PDF format, Postscript format

HW10, due April 7: PDF format, Postscript format

HW11, due April 14: PDF format, Postscript format

HW12, due April 21: PDF format, Postscript format

HW13, due April 28: PDF format, Postscript format

HW14, due May 5: PDF format, Postscript format

HW15, due May 12: PDF format, Postscript format

**Midterm 1 Review Handout : **Postscript
, PDF

**Midterms and Exams Instructions: Please, read before the exam.**
Postscript
, PDF

**Midterm Extra Reviews: 3 GSIs, Arun Sharma, John Voight and
Michael West, are planning on having extra review sessions for the
midterm: **

**Monday evening (John Voight): 5:00-8:00 p.m.
at 105 North Gate Hall **

**Tuesday afternoon (Michael West): 4-5:30+pm,
105 Latimer. **

**Tuesday evening (Arun Sharma): 6:00-9:00pm,
101 Barker Hall. **

**When the rooms are assigned, the GSIs will post place of the
review sessions on their office doors and on my office door for
anyone who would like to attend. Everyone is welcome to attend the
review sessions: not just students belonging to the corresponding
GSIs' sections. **

**Midterm 2 Review Handout : Postscript
, PDF
(pictures may not appear in the PDF format) **

**Mock Midterm 2: Postscript
, PDF
**

**Midterm 2 Extra Reviews: **

**John Voight: Friday (April 2) from
5:00-7:00pm in 141 McCone. **

**Jeff Brown: Monday (April 5) 6:00-7:20pm in
60 Evans. **

**Arun** **Sharma: Monday (April 5)
7:00-10:00pm in 3113 Etcheverry. **

**Meghan Anderson: Tuesday (April 6)
4:00-5:30pm in 2040 VLSB. **

**When the rooms are assigned, the GSIs will post place of the
review sessions on their office doors and on my office door for
anyone who would like to attend. Everyone is welcome to attend the
review sessions: not just students belonging to the corresponding
GSIs' sections. **

**Final Exam Review Handout: Postscript
, PDF
**

**Mock Final: Postscript
, PDF
**

**Final Extra Reviews: **

**John Voight: Thursday, May 13th, from 2:00 -
5:00 p.m. in 70 Evans. **

**Michael West: Thursday, May 20th, from
3:00-5:00 p.m. in 3 Evans. **

**Arun** **Sharma: Wednesday and Thursday,
May 19th-20th, 4:00-6:00 p.m. in 20 Barrows. **

**Jeff Brown: Thursday, May 20th, 5:00-6:30 pm,
in 60 Evans. **

**Meghan Anderson: Tuesday (April 6)
4:00-5:30pm in 2040 VLSB. **

**Notes: If not specified odd or even exercises, it is assumed
only even exercises, e.g. #2-8 means 2,4,6,8. Asterick * usually
means that the problem is hard/tricky. **

**HW1, Aug. 28, From 0.1.** **Functions and Their Graphs.****
#10,20,22,25,26,30,56; ****bonus: ****#50,58. From 0.2.
****Some Important Functions.**** #8,12,16,32; ****bonus:
****#18,20. **

**Aug. 30, From 0.3.** **The Algebra of
Functions ****#24,28,32,34,42; ****bonus: ****#36,38.
From 0.4. ****Some Important Functions.****
#8,10,20,28,34; ****bonus: ****#38,40. **

**HW2, Sep. 4, From 0.5.** **Exponents and Power Functions.****
#30,48,50,66,68,70,72,86,90,92,94; ****bonus: ****#74,96.
From 0.6. ****Functions and Graphs in Applications.
****10,16,18,22,24,37-40; ****bonus: ****#46,48,50.
**

**Sep. 6, From 1.1.** **The Slope of a
Straight Line.**** #6,8,10,16,20,24,26,52; ****bonus:
****#32,58. From 1.2. ****The Slope of a Curve at a
Point.**** #12-18,24,28; ****bonus: ****#32,36,38.
**

**Announcements, Sep. 3: **

**1. The lectures will move to 105 Stanley
Hall, starting Sep. 4. **

**2. For changes in rooms for workshops,
please, follow the telebears. **

**3. Felice Le's office hours will be in Evans
868, on Tuesdays 10:15am-12:15pm. **

**4. HW solutions will normally appear by
Tuesday morning. Please, do not bug your GSI's to post the solutions
earlier. The reason that I do NOT want students to look at the
solutions before Tuesday is to give you ample opportunity to try to
solve them on your own. Otherwise, one starts depending on the
solutions, and often gives up in trying to solve the problems when
encountering difficulties -- looking instead at the solutions doesn't
teach you much problem-solving, unfortunately. So, psychologically,
it is a good reason not to have the solutions lying around while
solving the HW - just like on exams. Finally, students who really
need an answer key can do first the odd "twin" problems
(most even-numbered problems are preceeded by a very similar
odd-numbered problem) and compare their answers with the textbook
answers. The difficulty of the odd-numbered exercises will be about
the same as in the even-numbered HW exercises. After that, students
can attack the even HW problems without any answer key. We shall not
discuss this issue anymore, so try to do your best with the HW's on
your own or working together with your friends, and then you can look
at the solutions on Tuesday mornings. **

**5. Please, read the syllabus carefully and
follow the instructions there for asking various questions. I have
received several e-mails asking me to place students in various
sections and help them get off the waitlist for the class, and in
addition math questions have arrived on e-mail. If you do send such
questions on e-mail, this shows that you have not read the syllabus
carefully or that you are ignoring it, and in both cases your answers
will be unnecessarily delayed. Use my and the GSIs office hours
effectively - these are golden opportunities for you to get all your
math questions answered. Be organized and set aside free time to
visit office hours. I shall not address these questions in the
future. **

**6. The HW1 solutions were corrected by the
GSI and a new version was posted this morning. **

**7. In general, if a HW problem asks for a
calculator (as in the "technology exercises") but a
solution can be produced WITHOUT a calculator, students should
attempt to solve the problem without a calculator (as much as
possible). It is hard to do, for instance, #42 from 0.3 without a
calculator, but certain parts of it, e.g. calculating the
compositions f(g(x)) and g(f(x)) CAN and should be performed WITHOUT
a calculator. Such exercises will be rare in the HW assignments, they
will be primarily in the "Technology exercises" section,
and often such problems are solvable WITHOUT a calculator. Keep in
mind that neither quizzes nor exams will contain problems requiring
the use of calculators - instead you will need to perform all
calculations by hand. **

**HW3, Sep. 11, From 1.3.** **The Derivative.****
#8,14,16,18,24,26,30,36,46,48; ****bonus: ****#52,54,56.
From 1.4. ****Limits and the Derivative.****
#10,12,14,18,20,34,36; ****bonus: ****#40,42,44. **

**Sep. 13, From 1.5.** **Differentiability
and Continuity.**** #1-12 (all exercises),16,18,20,22,26
****bonus: ****#28,32,34. **

**HW 4, Sep. 18, From 1.6.** **Some Rules for
Differentiation.**** #4,8,20,24,28,30,34,36,38,44,48,54,56;
****bonus: ****#40,42,44. From 1.7. ****More About
Derivatives. ****#8,10,14,22,24,28,30; ****bonus:
****#32,34. **

**Sep. 20, From 1.8.** **The Derivative
as a Rate of Change.**** #4,6,10,12,16,18,20,26,28 ****bonus:
****#24,32. **

**Announcements, Sep. 17: **

**1. Review for Midterm 1: PDF
format; PS
format. The material for Midterm 1 will encompass everything
through section 1.7 (inclusive). Midterm 1 will NOT include the
material from section 1.8, which will be covered this Thursday.
Midterm 1 will NOT include concepts #27-30 on the Review handout.
These concepts will be deferred to Midterm 2. **

**2. Instructions for Midterm 1: PDF
format; PS
format. Read these instructions carefully, so that you don't
spend time during the exam reading them. **

**3. DSP students: So far only one student has
provided me with a hard copy of a DSP official letter, asking for
special accommodations. I have not received any other DSP letters,
whether through e-mail or other means. According to the class
syllabus and as we discussed so thoroughly in lecture, to receive
special accommodations, students must present me with the DSP
official letters at least 10 days prior to the exam to make
arrangements. At the current moment, I am going to assume that there
is only one DSP student and make arrangements for only one student.
Please, be aware that there is no way for us to make quick
arrangements for special accommodations on a short notice, and there
is no way to allow for such accommodations without having received
the DSP notice prior to and well in advance of the exam. Promises of
"future DSP letters" will NOT be honored - please, don't
ask for exception to this policy. **

**4. The student who gave me the DSP letter
must contact me immediately via e-mail to finalize arrangements for
the midterm 1 . **

**Announcements, Sep. 19: **

**Extra Office Hours: Koushik (one of the GSIs)
is offering extra office hours this Friday and Monday from 1-4 pm, in
addition to his regular hours. **

**Announcements, Sep. 20: **

**Extra Office Hours: Jae-Young (one of the
GSIs) is offering extra office hours this Thursday 5-6pm and Friday
4-6pm, in addition to his regular hours. **

**Announcements, Sep. 23: **

**Come early to Midterm 1: Come 10 minutes
early for the Midterm 1 on Tuesday, Sep. 25, so that everyone is
settled down and ready to start exactly at 12:40pm. **

**Read midterm instructions: Please, read the
class syllabus, midterm review and the midterm instructions, BEFORE
sending me e-mails on questions that are already answered there. As
the students who ignored this found out, I do not answer such
questions in e-mails. Be responsible and follow the directions, which
are so clearly spelled out in your review materials and class
syllabus. **

**HW 5, Sep. 27, From 2.1.** **Describing Graphs of
Functions ****#6,8,10,12,14,16,18,22,24,30,35,38,39; ****bonus:
****#32,34,40. **

**Announcements, Sep. 27: **

**If you have a complaint about your midterm
grading: you return your midterm to your GSI in section after the 10
minutes of midterm viewing, and come to my office hours within a
week, the latest by Thursday, Oct. 4. No exams will be reconsidered
after that date. Check out my office hours from the course syllabus. **

**HW 6, Oct. 2, From 2.2.** **The First and Second
Derivative Rules. ****#2,4,6,12,18,20,24,40,44; ****bonus:
****#36,38,42. From 2.3. ****The First and Second
Derivative Tests and Curve Sketching.****
#6,12,18,24,26,34,38,42 ****bonus: ****#40,44. **

**Oct. 4, From 2.4.** **Curve Sketching.****
#7,8,16,20,26,28,31,32 ****bonus: ****#30,38(skip
38(a)). From 2.5. ****Optimization Problems.****
#2,4,6,8,10,13,16; ****bonus: ****#22,26,30. **

**Announcements, Oct. 2: **

**Last time in lecture we agreed on the
following conventions: **

**A function f(x) is increasing on
[a,b] if f(x)<=f(y) for any x < y in [a,b]. The
function f(x) is strictly increasing on [a,b] if f(x)
< f(y) for any x < y in [a,b]. Similarly for
decreasing and strictly decreasing .
Thus, for instance, a constant function is both increasing and
decreasing, but not strictly increasing and not strictly decreasing.
On the other hand, the function y=x is strictly increasing on
all of R , and it is also true that y=x is (just)
increasing on R . **

**Announcements, Oct. 9: **

**The following problems will be relevant to
the upcoming quiz this Thursday, Oct. 11: from HW6, everything from
Sections 2.2, 2.3, 2.4, and from Section 2.5: only problems #2 and
#4. The remaining HW6 problems, #6-8-10-13-16 from Section 2.5, shall
be fair material for the quiz on Thursday, Oct. 18, along with
anything else from HW7. **

**Let's agree on the following conventions: a
concave up function is one whose tangent slopes strictly increase,
and similarly, a concave down function is one whose tangent slopes
strictly decrease. Thus, a line doesn't qualify as either
concave up or concave down (as its tangent slopes remain constant
throughout), and hence we agree that a line has no inflection points.
**

**Recall that a constant function, on the other
hand, is both increasing and decreasing, but not strictly increasing
and not strictly decreasing. Every point on a constant function is a
local maximum and a local minimum (except for endpoints); and every
point on a constant function (including endpoints) is an absolute
maximum and an absolute minimum. **

**HW 7, Oct. 9-11, From 2.6.** **Further Optimization
Problems.**** #2,4,6,10,11,12,14,16,18,20,22,26,27,28; ****bonus:
****#8,24,29,30. Plus: review HW6 problems #6-8-10-13-16 from
Section 2.5. Any of these problems may be tested on the quiz on Oct.
18. **

**HW 8, Oct. 16, From 2.7.** **Applications of Derivatives
to Business and Economics.**** #2,4,6,10,12,14,18; ****bonus:
****#20,22. From 3.1. ****The Product and Quotient
Rules.**** #4,8,10,14,18,24,28,30,34,36 ****bonus:
****#42,44,62. **

**Oct. 18, From 3.2.** **The Chain Rule
and the General Power Rule.****
#2,4,8,16,20,24,30,36,40,41,46,50 ****bonus: ****#58,64.
**

**Announcements, Oct. 21: The GSI who wrote the solutions to HW7
has made some minor corrections to two problems there. The revised
solutions were posted above. **

**Announcements, Oct. 22: Note that there are two drawings in the
Review materials: they will appear only in the PS file. However, one
of the drawings refers to the class problem of the circular flower
and square vegetable gardens, so you have this drawing in your notes.
The other drawing refers to a problem whose statement is written in
the review materials: you can easily draw it yourself. **

**2. Midterm 2 will include: everything on the
Review handout for Midterm 2, except for concepts #19 (exponential
function) and #20 (differential equation) in the Definitions and
anything related to them later on. In addition, as written above,
concepts #27-30 on the Review handout for Midterm 1 can also be
covered by Midterm 2, so study that previous review handout too. **

**3. Come early to Midterm 2: Come 10 minutes
early for the Midterm 2 on Tuesday, Oct. 30, so that everyone is
settled down and ready to start exactly at 12:40pm. **

**4. If you are a DSP student who has given me
the official document but have not yet e-mailed me to arrange
accommodations for Midterm 2, please, e-mail me ASAP. All DSP
students who have e-mailed me to arrange for accommodations for
Midterm 2, must come the latest by 12 noon on Oct. 30 to my office
(Evans 713): late arrivals will miss the GSIs who will take the DSP
students to specially reserved rooms. **

**5. Koushik, one of the GSIs, has been
hospitalized after a sports accident. He will be operated on his leg
tomorrow. He is feeling OK, under the circumstances, but he will need
a couple of weeks to recover from the operation. **

**The Math Department has put in a request for
larger rooms for the next two weeks, so we can temporarily merge
Koushik's 8-9:30am, 11-12:30am and 3:30-5pm sections with the
corresponding sections of Farmer(8-9:30am), Jae-Young(11-12:30am) and
Jae Young(3:30-5pm). **

**Please, everyone look Wed evening on this
class website: for updates on the room assignments for your sections.
Be aware that even if you are not in Koushik's sections, the room for
your section may be changed. We plan for all quizzes (including for
Koushik's students) to proceed as normal, proctored by the other 3
GSIs, so make sure you continue going to your sections and check out
the website for updates on room changes. **

**HW 9, Oct. 23, From 3.3.** **Implicit Differentiation and
Related Rates.****
#2,4,8,12,14,16,18,22,24,26,28,30,36,38,42,46; ****bonus:
****#40,44. Note: Material from this HW will be relevant to
Midterm 2 and possibly the Final exam. **

**Oct. 25, From 4.1.** **Exponential
Functions.**** #4,8,14,16,24,30,42 ****bonus: ****#28,40.
From 4.2. ****The Exponential Function e^x. ****#2,6,10,21,36,40
****bonus: ****#42,46. From 4.3. ****Differentiation
of Exponential Functions.****
#4,10,14,16,18,20,24,26,32(+graph!),34,36,40 ****bonus: ****#44,48.
Note: Material from this HW will be relevant to the Final exam. **

**Announcements, Oct. 23: The room arrangements for sections for
the next two weeks, Thursdays 10/25/07 and 11/01/07, are as follows. **

**1. All 8am-9:30am sections (Koushik's and
Farmer's) will be held in EVANS 70. To be taught by Farmer. **

**2. All 11am-12:30pm sections (Koushik's and
Jae-Young's) will be held in WURSTER 101. To be taught by Jae-Young. **

**3. All 3:30pm-5:00pm sections (Koushik's and
Jae-Young's) will be held in EVANS 9. To be taught by Jae-Young. Note
that this is the original room for Jae-Young's 3:30-5pm section. **

**4. All remaining sections meet at their usual
places. **

**Note that all sections will have quizzes so
make sure you know ahead of time where your section's room is. **

**HW 10, Nov. 1, From 4.4.** **The Natural Logarithm
Function.**** # 4,6,10,28,31,38,40,48; ****bonus:
****#44,46. From 4.5. ****The Derivative of ln(x).****
#6,8,18,20,24,26,30,34, ****bonus: ****#32,36. From 4.6.
****Properties of the Natural Logarithmic Function.****
#2,4,6,8,10,12,14,22,36,46, ****bonus: ****#50,52,54. **

**HW 11, Nov. 6, From 5.1.** **Exponential Growth and
Decay.**** #2,4,6,10,14,16,22,28,31; ****bonus: ****#27,30.
Nov. 8, From 5.2. ****Compound Interest.****
#2,4,8,10,18,20,24,26,28; ****bonus: ****#16,22. From
5.3. ****Applications of the Natural Logarithm Function to
Economics.**** ****(up to Elasticity of demand)
****#2,4,6,8,10,12; ****bonus: ****#9,11. **

**Announcements, Nov. 6: **

**If you have a complaint about your Midterm 2
grading: you return your midterm to your GSI in section after the 10
minutes of midterm viewing, and come to my office hours within a
week, the latest by Thursday, Nov. 8 No exams will be reconsidered
after that date. Check out my office hours from the course syllabus. **

**The room arrangements for sections for this
week, Thursday 11/08/07, are the same as for the past 2 weeks since
Koushik hasn't recovered completely yet. I have listed again the same
room arrangements, but I am waiting for confirmation from the
scheduling services. **

**1. All 8am-9:30am sections (Koushik's and
Farmer's) will be held in EVANS 70. To be taught by Farmer. **

**2. All 11am-12:30pm sections (Koushik's and
Jae-Young's) will be held in WURSTER 101. To be taught by Jae-Young. **

**3. All 3:30pm-5:00pm sections (Koushik's and
Jae-Young's) will be held in EVANS 9. To be taught by Jae-Young. Note
that this is the original room for Jae-Young's 3:30-5pm section. **

**4. All remaining sections meet at their usual
places. **

**Note that all sections will have quizzes so
make sure you know ahead of time where your section's room is. **

**Announcements, Nov. 7: No files of old HWs shall be sent out to
individual students. Please, don't ask the instructor or the GSIs
about old HWs: there won't be any exceptions regardless of what
reasons you give. You are responsiblie for taking each HW solution
off the web within a week while it is posted, for printing it out,
saving it on your computer, etc. -- whatever you do. If you computer
or software crashes, if you lose the HWs, or whatever else happens,
ask your classmates to xerox their old HW solutions. **

**HW 12, Nov. 13, From 6.1.** **Antiderivatives.****
#2-36(all even exercises),42; ****bonus: ****#48,54,56.
( ****Note**** : don't fall in the "trap" set
for you in #12; #16 will involve some guessing, some functions in
#2-24 must first be rewritten in a different equivalent form for you
to be able to find their antiderivatives; #26-36 are actually easier:
they give you the form of the antiderivative on the RHS, you should
take the derivative of this RHS and set it equal to the function on
the LHS, then solve for ****k ****.) Nov. 15, From 6.2.
****Areas and Riemann Sums. ****#4-24(all even
exercises); ****bonus: ****#19,21. **

**Announcements, Nov. 12: **

**Sections this Thursday, Nov. 15: Koushik is
back, not yet 100% recovered but moving OK, and will be able to teach
his 8am-9:30am and 11am-12:30pm sections. Jae-Young will still take
over the two 3:30-5:00pm sections in Evans 9. Thus, this Thursday: **

**1. All sections are back to normal (as
originally assigned with your own GSI), except : **

**2. Koushik's 3:30-5pm section: this section
will still be in Evans 9, grouped together with Jae-Young's section
and taught by Jae-Young. **

**3. Everyone, make sure you know which room
your section is meeting (original room, or Evans 9 for 3:30-5pm), so
that you arrive on time. Quizzes will be given as usual. **

**Koushik's** **office hours for this week
will be only on Thursday, Nov. 15, 9:45-10:45am. **

**Instructor's office hours on Tuesday, Nov. 13
are split between 11:30-12:30pm and 2-2:30pm. Due to a medical
procedure, the instructor's office hours on Thursday, Nov. 15 are
cancelled. However, lectures this week, Nov. 13-15 proceed as usual.
There will a substitute instructor on Thursday. **

**Quizzes this week, Thursday, Nov. 15 will
cover the material only from 5.1 and 5.2 (excluding Present Value: p.
291) -- this adjustment is done because of the shortened lecture last
week due to the fire alarm. **

**However, the remaining material from 5.2 and
5.3 (covered on HW 11) will be relevant for future quizzes and the
final exam, so make sure you learn that material too (i.e. present
value, relative rates of change and logarithmic derivative; but
elasticity of demand in 5.3 is not mandatory). **

**HW 13, Nov. 20, From 6.3.** **Definite Integrals and the
Fundamental Theorem of Calculus.****
#2,6,10,12,24,26,30,36,40,42; ****bonus: ****#46,48,50. **

**Happy Thanksgiving! A healthy balance of food and math over the
holidays might be a really good idea! :) **

**HW 14, Nov. 27, From 6.4.** **Areas in the xy-Plane.****
#2,4,6,8,10,12,14,20,22; ****bonus: ****#24,26,28. Nov.
29, From 6.5. ****Applications of the Definite Integrals
(average value, volumes of solids of revolution). ****#1-10,30-36(all
exercises). **

**Announcements, Nov. 27: **

**Sections this week, Thursdays Nov.29: are
back to normal. Koushik will be teaching all of his sections at the
regular rooms. **

**Koushik** **has office hours this week:
Thursday, Nov. 29, 2-3:15pm. **

**HW 15, Dec. 4, From 6.5.** **Applications of the Definite
Integrals.**** All odd and even exercises: #15-20 and #37-42;
****bonus: ****#27,28. **

**Announcements, Dec. 5: **

**1. Don't forget about quizzes this week in
the last section meetings, Thursday, Dec. 6. **

**2. Review materials for Final Exam: PDF
format. **

**3. The last lecture on Thursday, Dec. 6 will
be dedicated to reviewing for Final Exam. **

**4. The extra material covered in lecture on
Tuesday, Dec. 4: can appear as bonus problems on the Final Exam .
Read the review for the final carefully. **

**Announcements, Dec. 11: **

**Please, do NOT e-mail me with questions where Valley LSB is:
check the campus map. The exam will start at 12:40pm sharp, so be
seated in the auditorium at least 5 minutes early. **

**Felice, one of the GSIs, will be holding
extra office hours this Wednesday, 12/12, 4:30-5:30pm, and next
Monday, 12/17, 2-4pm in the Free Speech Movement Cafe in the Moffitt
Library. She will also hold her regular office hours this week and
next week: Tuesday 10:15-12:15 in Evans 866. Everyone is welcome to
attend her office hours. **

**As discussed in lecture, the same rules apply
for the final as for the midterms, including the one-sided cheat
sheet, no calculators or other aids, etc... No e-mails shall be
answered regarding these rules. Please, do NOT bother the GSIs with
questions related to the rules for the final in hopes that the rules
will be relaxed: the rules for final exam will be exactly the same as
for the midterms. No questions on the number, topic or difficulty of
the exam problems will be answered. So, instead of wasting your time
asking such questions, do your best in studying for the final. Is the
final comprehensive? Whoever asks this question has not read the
class syllabus and has not paid attention to discussions in class. **

**Please, do not e-mail me asking me for
special treatment regarding your final grade, e.g. to drop more of
your quizzes, to change the weight of the exams and what-not: no such
things will be done. Everyone will be treated exactly in the same way
and we shall be equally fair to all students in the class. Please,
refer to the class syllabus for exact information on grading
policies. Any e-mails to me asking about grading policies shall be
ignored. Please, do NOT bother the GSIs about grading policies
either. **

**All DSP students and students requiring
special accommodations must have communicated with me via e-mail
already. All such students must come the latest by 11:55am on Dec. 19
to Evans 798 (this is Paulo deSouza's office), and Koushik (one of
the GSIs) will take them from there to the exam room. I have received
confirmation about these arrangements from everyone except one
student: please, contact me immediately to confirm these
arrangements. **

**Announcements, Dec. 13: **

**HW15 solutions: PDF
format; PS
format; to be taken down after the final exam. **

**Two typos were corrected in the
review for the final. Review materials for Final Exam: PDF
format. **

**1. As a rule, the instructor and the GSIs do
NOT mail, e-mail or communicate in any way the final grades to the
students. The final grades are distributed to the students through
the regular university channels. There will be NO exceptions to this
rule regardless of how anxious you are to see your grade, regardless
of students' computer problems accessing bearfacts, regardless of
students' delinquency in paying bills and hence postponing the
posting of their grades on bearfacts, regardless of the fact that
students may be registered through UCB Extension and may receive
their grades later. Please, do NOT e-mail the instructor or the GSIs
asking for your grade: the GSIs are instructed not to communicate any
grades, and any such e-mails will be ignored by the instructor. **

**2. If a student wishes to dispute the final
grade, he/she has to e-mail the instructor after January 15, 2008,
when the spring semester starts, and before February 15, 2008.
However, the student must be aware that **

**(a) the final scores are calculated by the
percentages announced in the course syllabus, and they are calculated
for everyone in the same way: just like in the syllabus. Do NOT ask
for exceptions to "twitch your score a bit" so as to give
you a higher grade: this won't happen. The grading system is created
so that it is fair to everyone. And it will stay that way. **

**(b) a final grade case will be reopened only
if a student has a convincing reason that his/her final grade was
miscalculated. Final grade cases will NOT be reopened just to check
if the final grade was calculated correctly. The assumption is that
all final grades are calculated correctly, unless a convincing
evidence to the contrary is brought forward. **

**(c) a final grade case, once opened, will be
fully reviewed, and the final score and final grade will be fully
recalculated. This means that, if there is an error, the final grade
could go down as well as up. If the final grade calculation reduces
the grade, I will have no choice but to change the final grade to the
new lower grade and there will be no turning back. **

**3. If a student wishes to know facts about
the final exam (e.g. median, etc): this is internal information and
we do NOT communicate it to the students. I can only tell you that
the final exam was fairly easy with a high median; thus, doing well
on the final exam alone is NO guarantee of an increased final grade.
No further questions about statistics, level of difficulty and other
info about final exam will be distributed, so don't ask for such. **

**4. If a student wishes to view his/her final
exam: the final exams are out of our hands and into the hands of the
main office on the 9th floor in Evans Hall. The student must go there
and follow the rules for viewing the final exams within the allowed
viewing period. The student must also be aware that if he/she wishes
to contest the final exam score, the student CANNOT leave the main
office with the final exam under any circumstances: the student can
view the final only in the presence of the main office staff. Then,
if the student wants to contest the final exam score, the student
must e-mail the instructor after the spring semester'08 starts,
January 15 2008, and before February 15 2008, and I will pull the
final exam from the main office. A student CANNOT take the exam from
the office and bring it to me: if by any chance this happens, the
student will be disqualified from the right to contest his final exam
score. No exceptions. **

**5. If a student didn't take the final exam,
he/she gets an automatic F, as announced in the syllabus. If the
student has a documented medical or family emergency and wishes to
petition for an incomplete, all such petitions will be considered
after the spring semester'08 starts, January 15 2008, and before
February 15 2008. **

**Calculus 1B PDF
format: to be taken down in a week, by next Monday evening. **

**HW1, Jan. 17. From 5.5. Review of Substitution. #2-20;
From 7.1. Integration by Parts. #1-10 (odd and even). **

**Jan. 19. From 7.1. Integration by Parts.
#12-40; From 7.2. Trigonometric Integrals.
#2-20,42,44*,46*,56,62. (For #56 and #62 you need to review 6.1.
Areas between curves and 6.2 Volumes of Solids of Revolution.) **

**Extended office hours of Head TA Stephen Canon: 1097 Evans,
scanon@math.berkeley.edu; Monday 10-noon, Wednesday 1-3 pm, Thursday
3-5 pm, Friday 10-noon. **

**For students wishing to review Calculus 1A integration
material: Mock Final (harder than the actual Final Exam in 1A)
Postscript
, PDF
**

**Change in GSIs: Qin Li's sections will be taught by Charles
Smart, smart@math.berkeley.edu, office 1041 Evans, office hours MW
2-3:30pm. **

**HW2, Jan. 24. From 7.3. Trigonometric Substitution.
#2,4,6,10,12,14,16,20,22,24,26,28: Perform the trigonometric
substitution (also possibly completion to a perfect square) and
simplify to get the integrals to the form of trigonometric integrals:
a product of powers of sines and cosines. If need be, transform any
tangents, cotangents, secants and cosecants into sines and cosines.
No need to evaluate the trigonometric integrals at this point: you
can do that too for practice, but this is not the main point of
working on these particular exercises. As bonus work: some of the
assigned exercises can be done without trig. substitution, but with
ordinary u-substitutions: which are these exercises and can
you evaluate the integrals there more quickly? From 7.4. Integration
by Partial Fractions I . #1(a),3(a),10,12,14,32. **

**Jan. 26. From 7.4. Partial Fractions II.
#16-50,60,62. Note that #40-50 are harder since they may require
two methods of integration, applied one after the other. **

**HW3, Jan 31. From 7.5. Strategies for Integration.
#4,6,10,12,14,16,18,20,24,26,30,32,36,40,42,44,46,50,54,56,58,64,66,68,70.
Note that many of these problems require several methods one after
the other, and some problems will require you to do "something
unexpected", so don't be surprised if you find this HW
relatively harder than the previous HWs. The level of difficulty of
this HW is naturally higher than the others because you are NOT given
which method to use, but you have to make a choice on your own:
hence, you may need occasionally to try several different methods
until you find out which works for each specific problem. Keep in
mind that the more integration problems you solve, the easier it will
become to solve even more problems since you will start noticing
patterns and relations between problems and will be able more and
more to reduce to previously solved problems and apply previously
seen ideas. **

**Feb 2. From 7.7. Approximate Integration.
#2,6,8,14,16,20,22,26,28,30,34,38,39,42,44. **

**HW4, Feb 7. From 7.8. Improper Integrals.
#2,3,6,8,14,18,20,22,26,32,34,36,38,40,50,52,54,56*,58,78*. (Problems
with * are harder, and usually require two or more methods/techniques
to be combined in the solution, or they use some new idea.) **

**Feb 9. From 8.1. Arc Length.
#2,4,8,10,12,14,16,18,30. From 8.2. Surface of Revolution.
#2,4,6,8,10,14,26,28. As bonus extra hard exercises, you can try #36
from 8.1 and #30 from 8.2. **

**The professor's office hours on Feb. 7 are
cancelled due to participating in a movie about the Berkeley Math
Circle. For any emergencies on Feb. 7: I can be found in Evans 959
between 2-5pm, and Evans 81 between 5-6pm. If you are wondering what
the Berkeley Math Circle is: click on the link at the bottom of this
page. **

**For the exams: There are NO sample exams on the web written by
me specifically for this class. Please, do NOT ask for such sample
exams. There will be one discussion section before the exam dedicated
to review for the exam, and there you will practice problems. If you
want to practice on your own: the HW assignments are the best
practice ever for the exams. In due time, I will put a review handout
for the exam on the web: if and when such a review is posted, it will
be posted right here by the HW assignments. There is no need to send
me e-mails asking for sample exams and review sheets: I won't hide
any materials from you, and whatever is intended for you will be
posted right here on the web. **

**Further, please do NOT ask for any different rules about the
cheat sheet: it is only one-sided, regular 8.5x11 inch paper,
handwritten written by you personally - no typing, no xeroxing; ...
and no magnifying glasses during the exam. :) Read the Midterm I
instructions below. **

**I shall not reply to any more e-mails inquiring about the exam
rules: you will receive all information in due time. I don't recall
having missed notifying my classes of any important rules, so just
calm down and concentrate on what is important: coming to class and
discussion sections, reading your notes and textbook, doing your
homeworks, and participating in discussion sections. No review sheets
or sample exams can replace diligent homework and study throughout
the semester. No kidding: I never in my life got one review sheet or
sample exam questions; instead, I prepared myself my own review
sheets and made my sample exams by assembling various HW problems. It
worked splendidly. Whatever you make yourselves will be 10 times more
useful and memorable than anything someone else gives you. So, take a
good look at your HW assignments and notes: everything you need for
the exams is there and it is up to you to summarize it in a form that
YOU will understand, remember and put to use in preparation for
exams. **

**In order to save time on the midterm, please, read carefully
these istructions: they will be printed on the front page of your
midterm. Postscript
format , PDF
format **

**Postscript
format , PDF
format **

**Many of these problems are harder than the problems on the
midterm. Some of these problems will be discussed in sections before
the midterm. The problems here are only suggested as preparation for
the midterm: they are not representative of all types of problems
that may appear on the midterm. Postscript
format , PDF
format Note that this list of problems does not include
probability problems. For the latter, try any problem from the
corresponding homework assignment from 8.5. **

**HW5, Feb 14. From 8.5 Probability #2-12, bonus #14, From
Chapter 8 Review #21, bonus #20. **

**Feb 16. From 11.1 Sequences #4-26,
28*-36*, 42-44 (skip the graphing if you don't have a graphing
calculator),54-56,62*. Be aware that many of the problems on
sequences are hard since they require not only understanding the
concepts well, but also using different problem solving techniques,
e.g. finding a function f(x) which gives the sequence and taking
first derivative f'(x) to determine if f(x) is monotonic; using
L'Hospital's Rule appropriately for finding limits; noting that for a
continuous function f(x), the limit and the function can interchange
orders: e.g. lim(f(g(x))= f(lim(g(x)) and hence applying LH only to
g(x) inside instead of the whole expression; applying simplifying
algebraic manipulations or rationalizing numerator/denominator; using
Proposition 8 in 11.1 about convergence of exponential sequences r^n;
using Sandwich Theorem, Absolute Value Theorem or MBT; using
inductive arguments for recursive sequences; and behind all of this:
being able to use the given formulas for your sequences and write the
first several terms, write the nth term a_n, the (n-1)st term a_(n-1)
and the (n+1)st term a_(n+1). Whether you are comfortable and
efficient with sequences will determine how you will perform on all
of the series topics (Midterm 2 will be based on series). **

**Due to the holiday on Monday, Feb 20, the review for Midterm I
is moved to Friday, Feb 17, in sections, and in order to give you
time to talk about sequences in sections next week, the quiz is moved
from Wed. Feb 22 to Friday Feb 24. Here is the schedule of events: **

**0. Thursday Feb 16: Matthew Gagliardi is
holding extra office hours 3:30-6pm in 1044 Evans Hall. **

**1. Friday Feb 17: Review for Midterm 1 in
sections. **

**2. Sunday Feb 19: Midterm Review for everyone
with Patrick Barrow in 2 LeConte Hall, 2pm-5pm . **

**3. Monday Feb 20: President's Day, no
sections. Prepare for the midterm. **

**4. Tuesday Feb 21: Midterm 1, in-class; come
to the lecture hall at least 5 minutes early, by 12:35pm. The earlier
you come, the more time you'll have to prepare properly for the exam,
and the better chance for all of us to start the exam on time. **

**5. Wednesday Feb 22: No quiz. We hope that
the midterms will be graded by then and returned in sections. You
will be given 10 minutes to look over your exams in sections. If you
wish to contest the grading on some problem, you must return the exam
straight to your GSI after the 10 minutes are up; the GSI will give
your exam to me personally; and you must come to my office hours in
person to explain your complaint. No exam complaints will be accepted
if the exam doesn't come to me through your GSI in the way described
above. **

**6. Thursday Feb 23: lecture, probably more on
sequences or beginning of series. **

**7. Friday Feb 24: Quiz on sequences in
sections. **

**Homework 5 Solutions: Due to the Monday holiday, you will
receive the HW5 solutions on Wednesday, Feb 22. If you wish to check
out the solutions beforehand, here is most of HW5 (with possible
exception of a couple of problems from Section 11.1): Postscript
format, PDF
format. Note that these solutions will be taken off the web
within a week of the posting. **

**HW6, Feb 23. From 11.1 Sequences
#38,40,46,48,50,52,58,60,64. This is already a formidable
sequence of problems on sequences :), but for the die-hards: try #70.
**

**A note to the multitude of formulas for area
of surface of revolution on page p.556.** **We proved in class
formula 4 (rotation happens about the x-axis). The textbook doesn't
explain how to obtain formula 6 from formula 4, but it is not a hard
algebraic manipulation to do that. Along with these two formulas,
there are "twin formulas" 4' and 6', which refer to the
situations when the rotation happens about the y-axis. To summarize: **

**(a) use formula (4) when rotation is about the x-axis and your
function is given in terms of x: y=f(x); **

**(b) use formula (6) when rotation is about the x-axis and your
function is given in terms of y: x=g(y); **

**(c) in formula (4) replace x by y and f(x) by g(y); call this
new formula (4'): use (4') when rotation is about the y-axis and your
function is given in terms of y: x=g(y); **

**(d) in formula (6) switch the places of x and y; call this new
formula (6'): use (6') when rotation is about the y-axis and your
function is given in terms of x: y=f(x). **

**I hope I got all of these right! The bottom line is that
formulas (4) and (6) are interchangeable and can be used on the same
problem when rotation is about the x-axis, but sometimes it is easier
to integrate the stuff in one of the formulas than in the other
formula, and it is hard to say which is easier ahead of time.
Similarly for formulas (4') and (6'). For the midterm, concentrate on
using just (4) and (4'): the stuff we proved in class. If you want
more advanced bonus practice: learn also how to use (6) and (6').
Note that the last HW solutions used all formulas. **

**will** **go as usual 2-2:40pm in my office Evans 713. At
2:40pm I have to leave to proctor another exam (The Bay Area
Mathematical Olympiad) in Evans 959. If you would like to see me
3-4pm on that day, please, knock on 959 and I'll come out to see you
(do not attempt to enter 959, since high school and middle school
kids will be taking a long 4 hour exam inside and we don't want to
interrupt them.) **

**On Thursday, March 2, office hours will go as
usual 2-3:30pm in Evans 713. Last chance to talk about your Midterm 1
grading is Thursday, March 2. After that day, no exams will be
considered for regrading. **

**HW 7, Feb 28. From 11.2. Series. #4,6,8,12-38; For extra
challenge, try 42,44,48,50. **

**There are various Problem Solving Techniques (PST) that can be
used in the assigned problems. **

**(a) First check to see if the series
resembles one of the two basic series: geometric or harmonic. For
geometric series, write out the first two terms of the series, a_1
and a_2; a_1 is a , and the ratio r is a_2 divided by
a_1, as given in formula 4 on p.715. If |r|<1 , then your
series converges with sum given in formula 4: a/(1-r); compare with
Example 3 on p.716 and note that some algebraic manipulations of the
terms of the series are necessary before one recognizes that indeed
this is a geometric series. If |r|> 1 or |r|=1 ,
then your series diverges. If you are given number as a periodic
decimal, this is again a geometric series in disguise: see how they
do Example 4 on p.716. **

**(b) If the series looks like the harmonic
series, but, say, missing the first several terms, or it is the
harmonic series all multiplied by 4, then your series diverges. **

**(c) If the terms of the series a_n are ratios
of polynomials, then either Test for Divergence applies (TD, Theorem
7 on p.718), or the telescoping method; TD always concludes divergent
series: check that limit of the terms a_n is not 0 or does not exist,
before applying the conclusion of TD, compare with Example 8; the
telescoping method usually concludes convergence, but you have to go
through the motions of partial fractions, calculating your A and B,
writing each term a_1,a_2,a_3,...,a_n in terms of your general
formula (difference of two fractions), then add up all
a_1+a_2+...+a_n to see the cancellation, and finally compute the
limit of this sum when n goes to infinity; compare with Example 6. **

**(d) Finally, let's say you can't apply any of
the above methods because your series looks like combining two or
more series of different types. What do you do? Try separating the
various series from one another, as done in Example 9: that is, use
the laws for series given in Theorem 8 on p.719. Be aware that if
your series is the sum of one convergent and one divergent series,
then your original series diverges (just like with improper
integrals). If your series is the sum of two convergent series, then
it also converges. However, if you series is the sum of two divergent
series do NOT immediately jump to the conclusion that the original
series diverges but look closely to see what is going on. **

**March 2. From 11.3. Integral Test. #2-24.
(After you finish these exercises, look carefully at all examples and
see if there isn't another method faster than the Integral Test; so,
after you do the IT, try something else too); #26-30,34; for extra
challenge, try #38*. From 11.4. Comparison Tests. #2-28.
(Again, after you apply CT or LCT, check to see if another faster
method is also applicable. The more ways you can solve a problem, the
deeper understanding of the material you acquire: you start looking
at examples in a more flexible way, considering more options and more
view points, and thus, seeing the "bigger picture" instead
of thinking "locally"); #34,36 (these require reading
carefully the end of section 11.4 on Estimating Sums pp.733-734.) For
extra challenge, try #30*,31*,32*,40*. (Why doesn't #40 contradict
the LCT and the discussion in class on LCT?) **

**HW8, March 7. From 11.5. Alternating Test. #2-20;
#22,26,28(for the last 3 exercises, one needs to read Estimating Sums
on pp.738-739: for alternating series, it is easier to estimate the
error of the partial sums than, say, when using Integral Test(IT) or
Comparison Test(CT)); #32,34,35*. For extra challenge, try #36*. From
11.6. Absolute Convergence and Ratio Tests. 2-18,22,28*,31.
For extra challenge, try #32*,34*,36*. Be aware that in this HW as
well as any following HWs on series, problems can be done in multiple
ways using different test, and quite often within the same approach,
more than one test must be used: e.g. you start applying the Absolute
Value Test (AVT), and then have to follow it by CT, or IT, or
something else. The more ways you can do a problem, the better
understanding of the problem and the series tests you will achieve. **

**March 9. From 11.6. Root Test and Rearrangements.
#20,24,30*. For extra challenge for the die-hards: try #39*,40*.
From 11.7. Summary for Testing Series. #2-38. I strongly
advise to do ALL problems #1-38, including the odd-numbered problems,
whose answers are in the back of the textbook. **

**Notes on sequences: Some students may have
difficulties finding limits of sequences that alternate or
"oscillate" between several values: e.g.
(-1)^n={-1,1-1,1,...}, or {0,1,0,1,...}, or (-1)^n/(n+1)=
{-1/2,1/3,-1/4,1/5,-1/6,...}, or n(-1)^n={-1,2,-3,4,-5,6,...}, or
[2+(-1)^n/(n+1)]={2-1/2,2+1/3,2-1/4,2+1/5,2-1/6,...}. I have been
told that students "don't like" or even "fear"
splitting sequences into two (or more) subsequences and arguing what
happens for each subsequence. Once a GSI told me: "Students
want to keep the sequence as a whole and make a conclusion about
everything at the same time. What can we do to convince them
that splitting sequences into subsequences is OK, and sometimes
necessary as the only way to reason rigorously?" Well, we
don't live in a world where all of our wishes come true
(unfortunately!), so we have to abide by the rules of our world and
change our problem solving techniques (PST) when previous PSTs just
don't work: don't try to apply methods and tests to problems where
they don't fit! You must try something that fits and will give you
the correct solution. To all of the 5 examples above we can apply the
following simple but powerful PST : we notice that for n-even each
sequence follows one pattern, and for n-odd each sequence follows a
different pattern. So, we split the sequence into two subsequences:
{a_{2n}} (the even-indexed terms) and {a_{2n+1}} (the odd-indexed
terms), and find the limits in each case. **

**For example: for the sequence
(-1)^n={-1,1-1,1,...}, the even-indexed terms are given by the simple
formula {a_{2n}=1}={1,1,1,1,...} This is a constant sequence of 1's,
hence its limit is 1. The odd-indexed terms are given by the simple
formula {a_{2n+1}=-1}={-1,-1,-1,...}, hence the limit of this
constant subsequence is -1. So, part of our original sequence
converges to 1, and another part converges to -1. Since 1 is not
equal to -1, we see that the whole sequence "has a split
personality" and "can't make up its mind what limit to
approach", i.e. to put it precisely as it should be written on
the exams: because there are two subsequences with different limits 1
and -1, the whole sequence doesn't have a limit . **

**Try the other examples too and show that for
{0,1,0,1,...} the two resulting subsequences give different limits 0
and 1, hence the whole sequence diverges (no limit!); for
(-1)^n/(n+1)= {-1/2,1/3,-1/4,1/5,-1/6,...} the two resulting
subsequences give the same limits 0 and 0, hence the whole sequence
does converge to 0; for n(-1)^n={-1,2,-3,4,-5,6,...} the two
resulting subsequences give different limits -infinity and +infinity,
hence the whole sequence diverges (no limit!); for the final
sequence: [2+(-1)^n/(n+1)]={2-1/2,2+1/3,2-1/4,2+1/5,2-1/6,...} the
two resulting subsequences give the same limits 2 and 2, hence the
whole sequence does converge to 2. **

**For those who would like to know why this
method is correct, read on. **

**Theorem (Limit Sequence).** **If a
sequence {a_n} has limit L, then any subsequence of it also has limit
L. **

**So, if it happens that two subsequences have different limits
L1 and L2, how could the whole sequence converge? To what limit? If
the whole sequence converged to some limit L, by our Limit Sequence
theorem above, each subsequence must converge to the same L, but we
have two specific subsequences in mind that don't abide to this rule:
one converges to some L1 and another converges to some another L2.
This is a blatant contradiction, hence the whole sequence has no
chance to have a limit in this case. We formulate **

**Theorem (Subsequences Limits 1).** **If a
sequence {a_n} has two (or more) subsequences that converge to
different limits, then the whole sequence does not have a limit, i.e.
{a_n} diverges. If a sequence {a_n} has some subsequence which
diverges, then the whole sequence {a_n} also diverges. **

**This theorem explains the reasoning in Examples 1,2 and 4
above. **

**Theorem (Subsequences Limits 2).** **If a
sequence {a_n} can be split into two (or three, or four, or finitely
many) subsequences, each of which converges to the same limit L, then
the whole sequence {a_n} converges to that same common limit L. **

**This last theorem explains the reasoning in Examples 3 and 5
above. **

**I hope this thorough explanation settles all
disputes and unhappiness about similar problems, and I expect that
the GSIs will report successful completion of similar problems in all
sections. **

**Postscript
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format **

**HW9, March 14. From 11.8. Power Series. #4-30,35,36; For
extra challenge, try #33(a)(find the radius and interval of
convergence of the series),#39,40. From 11.9. Functions as Power
Series. #1,2-8,12-28,36*,37,38*; For extra challenge, try
32,35,39*,40*. **

**As you will see when doing the HW, the problems from 11.8 are,
by-and-large, more straightforward than those in 11.9. The reason:
in 11.8 you are given for free the power series and you are asked to
find its radius and/or interval of convergence, and occasionally to
identify the sum-function of the series; thus, you can utilize your
previous knowledge of tests for convergence/divergence series, mainly
by applying Ratio and Root Tests and checking the endpoints of the
resulting intervals by other tests. **

**However, going the other direction in 11.9 is considerably
harder: you are given a function f(x) and you are asked to find a
power series which equals f(x). At this point, we don't have a
uniform approach to this problem (we'll study it under the name
Taylor series in 11.10), so we must improvise: we must start
with a power series that we already know, and manipulate it to get in
the end the series that we need. One approach is to start with the
standard geometric series and substitute in it something appropriate
for x (cf. Ex.1,2,3,8 in 11.9); this method gives also the interval
of convergence of the new series almost for free: substitute the same
stuff for x in |x|<1 and solve. **

**A second approach is to start with the given function f(x),
differentiate f(x) (resp. integrate f(x)) to obtain a function g(x)
which is known to equal to some power series S(x); then reverse your
operations, i.e. integrate term-by-term the power series S(x) (resp.
differentiate S(x)) and you will obtain a new series Q(x) that equals
the original f(x); this method is the essence of Theorem 2 in 11.9
Term-by-term differentiation and integration (cf. Ex.5,6,7 in
11.9). Note that your power series Q(x) for f(x) will have the same
radius of convergence as S(x), but not necessarily the same interval
of convergence: the endpoints must be checked separately. **

**The trickiest thing to understand here is that, even if Q(x)
converges at one endpoint, say, at x=a+R to a sum Q(a+R), it is not
necessarily true that the sum Q(a+R) actually equals f(a+R). In the
two classical examples we tackle in class: pi/4=1-1/3+1/5-1/7+1/9-...
(Ex.7 in 11.9) and ln 2=1-1/2+1/3-1/4+1/5-..., it happens to be true
that for the endpoint x=1 (resp. x=-1) of each associated power
series, the sum of the power series actually equals the value of the
function involved (arctan(1), resp. ln(2)). These facts need proofs
in order to justify them. As of now, when using term-by-term
differentiation or integration, we can find the radius and interval
of convergence of our new series Q(x), but we will know that
Q(x)=f(x) for sure only inside (a-R,a+R). For x=a-R and x=a+R, we can
determine if Q(x) converges, but we won't be able to conclude without
further proof that Q(a+R)=f(a+R), or that Q(a-R)=f(a-R). **

**A third approach is to use partial fractions: split the
function f(x) into simpler fractions; use the geometric series on
each fraction (making sure all geometric series have the same
center); add up the resulting two or more series, and find the
correct interval of convergence. Each involved geometric series will
give its own interval of convergence: intersect all such intervals in
order to find the interval of convergence of the whole series (cf.
Exercise #12 in 11.9). For instance, if one geometric series gives 2<
x< 4 and another geometric series gives 2.3 < x < 3.7, this
means that the whole geometric series will converge on the smaller
interval (2.3,3.7). **

**Example 8 in 11.9 is important for enumerative purposes: how to
estimate the value of an integral using power series. Represent your
integrand f(x) (the function to be integrated) via a power series
S(x) using any of the methods above: f(x)=S(x). Find the radius of
convergence of S(x). Integrate S(x) term-by-term to obtain a series
Q(x), which equals the desired integral of f(x). Apply the given
bounds A and B of integration to Q(x) (make sure that the interval of
integration [A,B] is entirely within the interval (a-R,a+R)). You
will obtain an ordinary series Q(B)-Q(A). Use an appropriate method
for estimating the remainder of series in order to evaluate Q(B)-Q(A)
within the desired error. Follow this discussion while reading
Example 8, and while doing Exercise #28 in 11.9. **

**HW9, March 16. From 11.10. Taylor Series. #2-20,24-36.
Read section 11.10 up to end of p.767. We will cover multiplication
of series and applications of Taylor series in calculating integrals
and limits probably next week. However, problems like #8 and 28
require very simple multiplication of power series similar to what
we'll do in class on Tuesday for x^3/(x+2) centered at 0. In #30:
obviously, use something similar as in #29. In #32: replace sin x by
its Taylor series and see if any cancellations occur. In all
problems, you can use any methods for power series we have seen so
far: substitution and manipulation of a previously shown power
series, TT-integration or TT-differentiation on previously shown
power series, partial fractions and then geometric series, and the
Taylor series approach via a table and looking for a pattern. Note
that by "previously shown power series" we refer to the
table at the end of p.767, plus ln(1-x) in Example 6 on p.757 (you
may use that ln(1-x) equals its Taylor series for x in [-1,1)). You
may want to plug into this (-x) instead of x and use instead
ln(1+x)=x-x^2/2+x^3/3-x^4/4+... for x in (-1,1]. **

**HW10, March 21. From 11.10. Taylor series. #38-60. For
extra challenge, try #62*: it provides a classic example of a
function f(x) which has a Taylor series at x=0, but does not equal to
it. In problems #56-60, you have to reason backwards: which famous
Taylor series does your series resemble, what x was substituted and
is that x within the interval of convergence of the Taylor series?
From 11.11. Binomial Series. #2-10,14-18. For extra challenge,
try 19*: this problem offers an alternative elegant (but hard to come
up with!) proof that the binomial series equals to its function
(1+x)^k. For the other exercises: study well the Examples in 11.11
and in your classnotes on Binomial Series. **

**March 23. From 11.12 (Applications to Physics
optional). Applications of Taylor Polynomials.
#2-10,14-20,24-28. From Chapter 11. Review. #40-58. As you can
see from all Review exercises, anything here can be used as practice
for Midterm 2. I have assigned in this HW only those exercises that
refer to power series. Yet, between now and Midterm 2, you should do
all exercises #1-58, plus the Concept Check and True-False Quiz in
the Review section. For extra challenge, try from Problems Plus
Section :
#6*,11*,16*,17*,22* (these are only for the real die-hards!) **

**Postscript
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**Midterm II Reviews : **

**Thursday, March 23, 5pm, 50 Birge Hall, with Matt. **

**Monday, April 3, 5-8pm, 50 Birge Hall, with Pat. **

**HW11, April 6. From 9.1. Modeling with Differential
Equations #2-12. From 9.2. Direction Fields (read up to
Euler's Method on p.596) #2,3,6,8,10,12,18. **

**HW12, April 11. From 9.3. Separable Equations
#2-22,28,30,38,40. **

**April 13. From 9.4. Exponential Growth and
Decay: #2-6,10-20. For extra challenge, try #11,21,22. **

**HW13. April 18. From 9.5. Logistics
Differential Equations: #2-10. For extra challenge, try #11,13.
For #4: read and mimick Example 4 on p.628. From 9.6. Linear
Differential Equations: #2-20, #30. For extra challenge, try
#23-24. **

**April 20. From 9.7. Predator-Prey Systems:
#2-10. From Chapter 9 Review Section: #2,6-20,21,24,25,26.
For extra challenge, try #22,23,27. **

**On the "hot-shot" problems from lecture today,
Thursday, April 18: **

**Problem 1.** **dP/dt=2P(1-P/800). Find the
inflection points of all solutions P(t). **

**Notes: Differentiate both sides of the given
DE in order to find where P"(t)=0. Be careful to differentiate
both sides of with respect to the same variable, t , i.e. use
the Chain Rule on the RHS: **

**P"(t)=[2(1-P/800)-2P/800]P'(t)=(2-4P/800)2P(1-P/800).
**

**The last two factors were obtained from using the given DE and
substituting for P'(t). Thus, P"(t)=P(400-P)(800-P)/80000, so
P"(t)=0 if P=0,P=400 or P=800. One can now check that for 0 <
P < 400 or P > 800: P"(t) > 0 hence P(t) is concave up
there; for P < 0 or 400 < P < 800: P"(t)< 0 hence
P(t) is concave down there. Since no solutions cross the equilibria
c=0 or c=800, the only change in concavity of the curves P(t) happen
when P=400. Therefore, the only inflection points in the solutions to
our DE happen on the horizontal line P(t)=400. **

**Problem 2.** **dP/dt=2P(1-P/1000)(1-5/P).
What are the solutions here, what are the equilibria, and what
happens with the solutions when P drops below 5? **

**Notes: After multiplying out P(1-5/P), the DE
simplifies to dP/dt=(1000-P)(P-5)/500. The only equilibria (i.e.
constant solutions P(t)) are P=1000 and P=5. Note that P=0 is NOT an
equilibrium since P cannot be 0 from the initial DE (there is 5/P in
the DE). So, in this situation, the previous basic logistic equation
equilibrium P=0 is replaced with another equilibrium P=5, and
analogous picture for the DE solutions P(t) applies in this situation
too. Check it out! **

**Note on equilibrium points in Predator-Prey Systems from
lecture today, April 20: We got 2 equilibrium points (1000,80) and
(0,0) for the first example where we used exponential growth for the
rabbits. (1000,80) cannot be called a "stable" equilibrium
here because one cannot get to it from anywhere else but from it (the
populations will follow the closed curve phase trajectories, which do
NOT go through (1000,80).) **

**On the other hand, in the second
example where we used logistic growth for the rabbits, we obtained 3
equilibrium points (1000,64), (5000,0) and (0,0). If we start on the
horizontral axis (A,0) with A>0 (i.e. we start with some rabbits
and no wolves), we will arrive at (5000,0). If we start on the
vertical axis (0,B) with B>=0 (i.e. no rabbits at all), the wolves
will die out and we will approach (0,0). Finally, starting from
anywhere in the plane (A,B) with A>0 and B>0 (i.e. some rabbits
and some wolves), we will approach (1000,64) following the
spiral-like trajectories. Because of this, we can call (1000,64) a
"stable" equilibrium, while (0,0) and (5000,0) can be
called "semi-stable" equilibria. **

**HW14. April 25. From 17.1. Homogeneous Second-Order Linear
DE with Constant Coefficients: #2-32. For extra challenge try
#33,34*. **

**April 27. From 17.2. Non-Homogeneous
Second-Order Linear DE with Constant Coefficients: #2-22,23,26,28.
**

**Please, do NOT e-mail me with
questions where Pimentel is: check the campus map. The exam will
start at 5:10pm sharp, so be seated in the auditorium at least 5
minutes early. **

**As the syllabus states: "I
shall not discuss bonus credit policy or grading policy with students
throughout the semester." "The professor will not answer
any math or grading policy questions on e-mail: professor's e-mail is
only for emergencies." "Administrative questions which are
addressed in this handout [the syllabus] or answered in lectures or
sessions will not be answered on e-mail or otherwise. For any missed
information: ask your classmates." **

**Both methods of undetermined
coefficients and of variation of parameters are regular
material and are fair topics for regular questions on the final exam.
You shall not be required to prove these methods, but you shall be
required to be able to state them and use them in problems. **

**HW15, May 2. From 17.3. Applications of Second-order DEs:
#2-12. For bonus: try #14-16. And for extra challenge, try #18. **

**May 4. From 17.4. Series Solutions of DEs:
#2-10. For extra challenge, try #12. From Review. Chapter 17. p.
1168: #2-18. For the final, you should also go over the Concept Check
and the True-False Quiz on pp. 1167-1168. **

**If everything goes according to plan, our
last lecture on May 9 will be a review for the Final. **

**Another review will be lead by Patrick Barrow
on Tues, May 16, 3pm-6pm in 10 Evans. The review is open to all
students. **

**Erik Closson: Sunday Dec. 19 at 12-2pm in 70
Evans, Monday Dec. 20 at 10:30am-12noon in 1039 Evans. **

**Dragos** **Ghioca: Saturday Dec. 18 at
10-12am in 230 D Stephens Hall. **

**Yonatan Harel: Tuesday Dec. 14 at 2-4pm. in
828 Evans. **

**Grigor** **Sarsyan: Sunday Dec. 19, at
2-4pm. in 824 Evans. **

**Zvezda: Monday Dec. 13 at 10-12noon, and
Monday Dec. 20 at 10am-12noon in 713 Evans. **

**Until posted here otherwise, the
remaining GSIs will hold their usual office hours during the week of
Dec. 13. **

**1. As a rule, the instructor and the GSIs do
NOT mail, e-mail or communicate in any way the final grades to the
students. The final grades are distributed to the students through
the regular university channels. There will be NO exceptions to this
rule regardless of how anxious you are to see your grade, regardless
of students' computer problems accessing bearfacts, regardless of
students' delinquency in paying bills and hence postponing the
posting of their grades on bearfacts, regardless of the fact that
students may be registered through UCB Extension and may receive
their grades later. Please, do NOT e-mail the instructor or the GSIs
asking for your grade: the GSIs are instructed not to communicate any
grades, and any such e-mails will be ignored by the instructor. **

**2. If a student wishes to dispute the final
grade, they have to talk directly to the instructor before June 1 or
after July 1 2006 (in June, the instructor will be out of the
country). However, the student must be aware that **

**(a) the final scores are calculated by the
percentages announced in the course syllabus, and they are calculated
for everyone in the same way: just like in the syllabus. Do NOT ask
for exceptions to "twitch your score a bit" so as to give
you a higher grade: this won't happen. The grading system is created
so that it is fair to everyone. And it will stay that way. **

**(b) a final grade case will be reopened only
if a student has a convincing reason that his/her final grade was
miscalculated. Final grade cases will NOT be reopened just to check
if the final grade was calculated correctly. The assumption is that
all final grades are calculated correctly, unless a convincing
evidence to the contrary is brought forward. **

**(c) a final grade case, once opened, will be
fully reviewed, and the final score and final grade will be fully
recalculated. This means that, if there is an error, the final grade
could go down as well as up. If the final grade calculation reduces
the grade, I will have no choice but to change the final grade to the
new lower grade and there will be no turning back. **

**2. If a student wishes to know facts about
the final exam (e.g. median, etc): this is internal information and
we do NOT communicate it to the students. **

**3. If a student wishes to view his/her final
exam: the final exams are out of our hands and into the hands of the
main office on the 9th floor in Evans Hall. The student must go there
and follow the rules for viewing the final exams within the allowed
viewing period. The student must also be aware that if he/she wishes
to contest the final exam score, the student CANNOT leave the main
office with the final exam under any circumstances: the student can
view the final only in the presence of the main office staff. Then,
if the student wants to contest the final exam score, the student
must come directly to me after the fall classes start, and I will
pull the final exam from the main office. A student CANNOT take the
exam from the office and bring it to me: if by any chance this
happens, the student will be disqualified from the right to contest
his final exam score. No exceptions. **

**4. If a student didn't take the final exam,
he/she gets an automatic F, as announced in the syllabus. If the
student has a documented medical or family emergency and wishes to
petition for an incomplete, all such petitions will be considered
before June 1 2006, and after July 1 2006. During June 2006, the
instructor will be out of the country. **

**TuTh** **2:00-3:30pm, 155 Dwinelle **

**Syllabus: Postscript
, pdf
**

**Midterm 1 Review Handout : Postscript
, pdf
**

**Midterms and Exams Instructions: Please, read before the exam.
Postscript
, pdf
**

**Midterm 2 Review Handout : Postscript
, pdf
**

**Final Exam Review Handout : Postscript
, pdf
**

**Midterm 2 Review Handout : Postscript
, pdf
(pictures may not appear in the pdf format) **

**Mock Midterm 2: Postscript
, pdf
**

**Final Exam Review Handout: Postscript
, pdf
**

**Mock Final: Postscript
, pdf
**

**TT 8:00-9:30am, 2050 Valley LSB **

**Course Syllabus, Homework Assignments and more: Postscript
format , PDF
format **

**Homework Notes and Hints: Postscript
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format **

**Midterm 1 Review Handout : Postscript
, PDF
**

**Midterms and Exams Instructions: Please, read before the exam.
Postscript
, PDF
**

**Midterm Extra Reviews: 3 GSIs, Arun Sharma, John Voight and
Michael West, are planning on having extra review sessions for the
midterm: **

**Monday evening (John Voight): 5:00-8:00 p.m.
at 105 North Gate Hall **

**Tuesday afternoon (Michael West): 4-5:30+pm,
105 Latimer. **

**Tuesday evening (Arun Sharma): 6:00-9:00pm,
101 Barker Hall. **

**When the rooms are assigned, the GSIs will post place of the
review sessions on their office doors and on my office door for
anyone who would like to attend. Everyone is welcome to attend the
review sessions: not just students belonging to the corresponding
GSIs' sections. **

**Midterm 2 Review Handout : Postscript
, PDF
(pictures may not appear in the PDF format) **

**Mock Midterm 2: Postscript
, PDF
**

**Midterm 2 Extra Reviews: **

**John Voight: Friday (April 2) from
5:00-7:00pm in 141 McCone. **

**Jeff Brown: Monday (April 5) 6:00-7:20pm in
60 Evans. **

**Arun** **Sharma: Monday (April 5)
7:00-10:00pm in 3113 Etcheverry. **

**Meghan Anderson: Tuesday (April 6)
4:00-5:30pm in 2040 VLSB. **

**Final Exam Review Handout: Postscript
, PDF
**

**Mock Final: Postscript
, PDF
**

**Final Extra Reviews: **

**John Voight: Thursday, May 13th, from 2:00 -
5:00 p.m. in 70 Evans. **

**Michael West: Thursday, May 20th, from
3:00-5:00 p.m. in 3 Evans. **

**Arun** **Sharma: Wednesday and Thursday,
May 19th-20th, 4:00-6:00 p.m. in 20 Barrows. **

**Jeff Brown: Thursday, May 20th, 5:00-6:30 pm,
in 60 Evans. **

**Meghan Anderson: Tuesday (April 6)
4:00-5:30pm in 2040 VLSB. -->
**