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# Zvezdelina Entcheva Stankova

## Office:

713 Evans Hall

Berkeley, CA 94720-3840

Tel: 510-642-3768

Fax: 510-642-8204

Office Hours TTh 9:30-10:50am in Evans 713.

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 Thomas Brown, 965 Evans.

### 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:

HW3 Solutions.

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.

### Homework 3A, due Wed, Sept 17:

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.

Hints: #4 (arrange a,b,c in increasing order in your cases); #6 (x being odd and y being even is symmetric to another case); #8 (definitely try some small cases first before you find your example); #10 (Could they BOTH be perfect squares? Could (n+1)^2 – n^2=1? Why is this relevant in the problem?) #18 (place r on the real line between two consecutive integers, exactly one of which will be the required unique n in the problem; why?; what could go wrong if we allowed for r to be rational?) #22 (Do it as the hint suggests. And then apply the inequality from Example 14 to x^2 and 1/x^2 (instead of x and y) for a direct proof.) #30 (If they were such integers, how large can possibly be |y|?) #32 (The hint is giving you specific formulas for the integers x, y, and x. Substitute these formulas into the Pythagorean equation to show that they satisfy the equation. Do these formulas give infinitely many triplets of solutions? Why?) #34 (Adapt the proof that square root of 2 is irrational.) #36 (What kind of a number lies exactly in the middle between a rational and an irrational number? Use the average to get a formula for this middle number and show why it must be irrational by contradiction.) #42 (Direct tiling should do it. Try it.) #44 (What happens if you color the 5 x 5 board as a regular chessboard? Try finding an invariant.)

### Homework 2B, due Wed, Sept 10:

Read 1.6. Write #2,4,8,12,16,18,24,28. Challenge: #35*.

### Homework 2A, due Wed, Sept 10:

Note 1: The first quiz is on Wed, Sept. 3 and will be on the material from HW1. For the quizzes, you are allowed to have a cheat sheet containing material related to course. The cheat sheet can be only 1 page (one-sided!) of a regular 8”x11” sheet. It has to be hand-written by you (no zeroxing, copying, pasting, etc.)!

### 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.

Note 1: The implication "p --> q" can be read in a number of ways. One of the most confusing ways to read it is "p only if q", which means "if p is true then q must be true", i.e., "if p then q". I would suggest avoiding, whenever possible, the expression "only if" as it is doubly dangerous. Indeed, in everyday life it is used to mean "if and only if", while in math it is used only in one direction and it is often counterintuitive. For example, Problem #3 from 1.1 can be translated mathematically as: "If p then [q1 AND (not q2) AND (not q3)].", where p: "You graduate.", q1: "You fulfilled all requirements for your major.", q2: "You owe money to the university.", q3: "You owe a library book."

Note 2: The first quiz is on Wed, Sept. 3.

## Berkeley Math Circle

http://mathcircle.berkeley.edu

## Teaching: Calculus 1A

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

### Reviews for the Final Exam prepared by Students in 1A:

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.

### Mock-Final prepared by Zvezda:

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).

### Extra Office Hours between now and the final exam:

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.

### Questions on the final:

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.

## Competition for Best Review 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.

### MathSpace: A message from John Steel Undergrad. Vice-Chair

get more insight into what you're learning

work alone or connect with peers in a group

share tips for success

crazy math questions welcome!

Every Thursday from 5-7pm in 1015 Evans

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.

Midterm 1 Review Handout: PDF format

## SLC Drop-In Free Tutoring:

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.

### Homework assignments

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.

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

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:

## Final Exam in 16A: Wednesday, Dec 19, 2007; 12:30pm-3:30pm; 2050 Valley LSB.

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.

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.

## Notes after the Final Exam. Final Scores and Grades

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.

### All e-mails on questions addressed above shall be ignored.

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.

### Instructions for Midterm I

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

## Midterm I Review Handout

### Suggested problems for Review for Midterm I

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).

### Announcement:

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.

## Office hours on Tuesday, Feb 28

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.

## Correction on Solution of #32, HW6

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!)

## Midterm II Review Handout

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.

### As announced before, the quiz this week is moved for all sections to Friday April 7, and will be based on the Midterm 2.

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.

## Final Exam: Wednesday, May 17, 2006; 5:00pm-8:00pm; 1 Pimentel.

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."

### To clarify some questions regarding the lecture on non-homogeneous DE:

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.

### The material from last week of classes: 17.3 and 17.4, will also be on the final. The corresponding HW solutions will be distributed in sections sometime during the last week of classes.

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.

### Reviews for the Final Exam

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.

### Last Office Hours for this Semester

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.

## Notes after the Final Exam. Final Scores and Grades

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.

## Calculus 16A

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

## Teaching Spring'04: Calculus 1A

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

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

Homework Notes and Hints: Postscript format , PDF 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. -->