Math Major Requirements and Recommendations

Math 4031, followed by either Math 4032 or Math 4035, satisfies the Advanced Calculus requirement for the Mathematics major with a mathematics concentration. It prepares students for graduate study of mathematics and its applications. The Department strongly recommends that Mathematics majors planning graduate study in Mathematics take all three Advanced Calculus courses: Math 4031, 4032, and 4035.

Prerequisites

Either MATH 2057 or 2058, and 2085 or 2090, or equivalents.

Attendance

Attendance is required and you will be responsible for classwork on examinations. Your presence and participation in class is an essential part of this course. Do not miss class without a valid excuse. When you are absent, you are missed. If you must miss class, please keep track of where we are in the syllabus online, and be sure to visit my daily office hours so that I can help you to keep up with the work you missed.
Statistics show a strongly postive correlation between regular attendance and test grades. Most LSU students are conscientious and sensible about coming to class unless there is a serious, excusable reason for not being able to do so. However, there is an unfortunate tendency for some students to become discouraged as the term progresses and to cease regular attendance. This happens despite the fact that a student who is feeling discouraged has an especially great need to be in class and to ask questions. I encourage every student to come to class even when the going gets tough.
I will require documentation always if you are absent from an hour test or from the Final Examination.

If you are unavoidably absent on a day when homework is due to be turned in for grading, email a pdf scan file or a clear photographic image of your homework solutions directly to the grader before class time.

Homework is required and will be part of your final grade

Problems, mainly proofs, will be assigned frequently: approximately 3 assignments every two weeks. The assignments are your main work in this course. You are encouraged to seek hints to help you get started with these problems! It is required to turn in every assignment! The key to learning to prove theorems lies in how you study Advanced Calculus. It is very important to understand thoroughly how and why the proofs presented in the book and in class work. Please read the Introduction to your textbook! We will go over every collected homework problem in class, to help you prepare for tests. At the end of the course, your homework average on a 10-point scale will be added to your Exam average to produce your final average. For example, if your average on the homework is 5 points out of 10, and you have an 85% exam average, your final average would be 90%. In this example the homework credit would raise your grade from B to A-. This is an increase of two grade levels on LSU's +/- grading system. Proofs assigned for homework are a very important learning experience. Some students try a shortcut - copying the correct proofs from the board after the homework has been graded, without turning in their own efforts. This tends to produce proofs on tests that are written by rote from memory, and these tend to be lacking in logic and thus incoherent. It results also in low grades on Part I of each test, because the student's own conceptual errors have not been turned in and thus have not been corrected. Remember that homework is required! In order to learn the logical structure of advanced calculus, one needs to follow a given set of definitions and theorems from start to finish. If you wish to use other definitions or theorems from a different book, you must also include a proof that the definition or theorem you have chosen is equivalent to the one we used in the course. This will require that you do much more work than is needed to follow the definitions you have been given in our course. There are unscrupulous businesses online that will sell you solutions to homework problems. If you were to avail yourself of such a service, then at best you would be cheating yourself out of this part of your education. The result will be an unacceptably low grade and very likely the need to repeat the course and pay tuition a second time for the same course. Moreover, I have seen some of these illegal and unauthorized solutions to problems in my book for sale online that were utterly wrong and must have been written by someone incompetent in mathematics. Buyer beware!! Your learning of Advanced Calculus will come only from your own work. There are no shortcuts. You need to turn in every assignment on time, come to class daily from the first day of the semester to the last, ask questions about everything you do not understand clearly, and ask questions about any errors indicated on your returned homework assignments.

When should you ask questions?

You should ask questions every time you do not understand something and also every time you are curious about something. Ask questions in class. Be aware that when I am writing on the whiteboard I have no way of seeing your raised hand. So speak up with your questions! Our class size is small so you should view our meetings as a two-way discussion and not a formal lecture. It is a good thing to speak up with your questions!
Ask questions after class. Ask questions in my office hours. Ask questions by email. Please consider this: I have been teaching as a University faculty member for more than half a century. So I ask you to consider that I just may be able to help you with whatever is causing you difficulty if you will permit me to do so. It is a pleasure for me to help each and every student, so please give me a chance to help you to the best of my knowledge and ability. Every single one of you is important to me.

Lateness and Classroom Conduct

Please try to arrive on time for class. But sometimes it may be unavoidable to be late. If you are late, please come right into the classroom, doing so as quietly as you are able so as not to disturb other students. You should have as much class time as possible, so please just come in--quietly--and join the class even if you are late. Also, if homework is due that day, remember to turn it in on paper to me. Class time is a time for work. So when class begins please turn your attention to the work of the class.

Tests

We plan to have 3 hour tests, and they will be closed-book tests. No notes, whether on paper or electronic, are allowed. No communication devices are allowed. Part I of each hour test will consist of a choice of 8 out of 12 short answer questions, and Part II will offer a choice of 2 out of 3 proofs. (The Final Exam will be equivalent to two hour tests.) The proofs will be modeled closely on the collected homework, and they are sometimes identical. The short questions will be small variations of homework problems---including those not collected---together with examples from the lectures and notes. Thus if you have been coming to every class and have done the homework conscientiously, you should be prepared well for all tests. If you must miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable.

Grades

We plan to have three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your test average TA will be the sum of your final exam grade and your hour test grades divided the maximum possible cumulative score, expressed as a percentage. Let HA denote your homework average on a 10 point scale. Your Final Average FA will be FA = TA + HA . (Alternatively, if it will benefit you, instead of adding your HA to your TA, we will replace your lowest test grade with your HA converted to a 100% scale. But experience shows most students benefit most from the calculation with TA + HA.) Thus 0<= FA <= 110. The minimum grade for each letter grade is as follows:
A+, 97
A, 93
A-, 90
B+,87
B, 83
B-, 80
C+, 77
C, 73
C-, 70
D+, 67
D, 63
D-, 60
F, below 60
You should save all your graded work for future study and in case you think your final grade is in error. Please take note: Number of Dropped Grades: 0 Curve: None.

General Advice

• Many students need help to learn how to write proofs. If you feel confused, it is important to see me for help as soon as possible. If you don't know how to start a homework problem, ask for a hint---either in class or in my office---or even by email. If you ask me a question about the homework, or if you email such a question to me, I may be able to think of a good hint and then I would email it to the whole class as a hint. I can guarantee you it is possible to learn to write sound proofs: Learning begins with your efforts and your persistence. If you are feeling 'lost' please see me privately in my Zoom office hour location so that I can help you. I am very happy to meet with each individual student to help.

• "Why is it important for me to be able to write proofs?" Read an opinion by a distinguished expert at this link!

• Attend class and ask questions. Non-attendance or lax attendance is usually the first sign of impending academic difficulty. Sometimes a student feels frustrated because of not understanding the classwork. If that is the case, it is necessary to ask more questions. Advanced calculus is a subject you can learn---but you must participate in this work.

• Assignments to be turned in by the start time of class. Do not turn it in later than that, because it is not fair to the graduate teaching assistant, who will be busy enough with the work of grading the assignments that are turned in by the proper time.

• LSU offers extensive academic support services to help students adjust to the demands of university studies: List of Frequently Used Services. HOWEVER, there are businesses both locally and online that sell their own instructional services to students for fees. LSU does not authorize or approve any such services and does not recognize their competence to serve your educational interests. Advertising for any such commerical service is prohibited in our classroom. If you see such advertising being circulated in our classroom please report it to me. And buyer beware!

• Caution about Online Sources: Many of us make use of online information occasionally. But students need to be aware that your homework must be logically consistent with the development of Advanced Calculus in our course. What is an exercise in one course may be a theorem in another course. And the order of logical development varies from one course to another. Also, remember that online sources, such as UTube, are not universities. They have no faculty and no academic accreditation. Also, proofs generated by artificial intelligence are apt to be completely lacking in logic. They are just chains of words that commonly follow one another somewhere in a data base. When it comes to learning advanced calculus, there are no shortcuts. We need to do our own work to cultivate a strong sense of logical reasoning.

Homework Assignments and Downloads

We will update the list of assignments and tests below as the semester progresses. You will know an assignment has been updated if a due-date appropriate to this semester appears in the left-hand column. However, sometimes we will assign a problem for a certain date and then postpone it because we don't cover as much as planned in class. So check regularly for updates as to what is due and when. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course!

Academic Honesty

The University has clear policies requiring academic honesty. If you get an idea from another book or an online source, or from talking with a friend, academic honesty requires that you acknowledge your sources openly. Above all, never copy directly from another person's written work as though it were your own. Remember that your own good name is irreplaceable. This is a sound principle which will serve you well throughout your life. Moreover, on a practical level, it is very foolish claim as your own an argument from a former student in this class or from a textbook. The arguments which are copied can be recognized very easily as not coming from the student, and often the precise source can be identified readily. This means that the honorable course of action is also the practical one.

Due Date

Assignments: Hand in problems in red bold face for grading. The problems in red bold face are required. Assignments must be written neatly so that the grader can read them. There is also a class of optional problems, called Bonus Problems, which are intended for those students who find the required homework easy and want to be seriously challenged. These are worth up to 20 extra homework points per problem. Bonus problems need to be emailed in pdf format directly to me on a separate sheet from the regular homework, clearly marked Bonus Problems at the top. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. Bonus problems must be handed in separately from the normal homework, and they will be graded more strictly for logical rigor than the required homework. Please read the Academic Honesty policy above!

January 17

January 19

Read Pp. xxi--5.

January 22

January 24

Hand in: 1.3, 1.7. Please remember to write neatly so the grader can read your work, and put your name on your paper so he can record the grades!

January 26

1. Use the result of homework problem 1.3 to prove that the real number 1 is positive. That is, prove that 1>0. (Hint: Use the Axiom of Trichotomy to see that there are only 3 potential cases for the number 1. Show that two of those cases lead to an impossibility, leaving only one possibility.)
2. Do problem 1.9.

January 29

Read from Example 1.3 through Definition 1.2.4. Then do: 1.12 -- 1.16, 1.18, 1.20--1.21, 1.23 -- 1.24, 1.26, 1.27.

January 31

1.19, 1.22. Downloadable Optional Bonus Problem B1 to hand in a week from today. Bonus Problems are intended for those students who find the required homework easy and want to be seriously challenged. These are worth up to 20 extra homework points per problem. Bonus problems need to be emailed directly to me, your teacher on a separate pdf file from the regular homework, clearly marked Bonus Problems at the top. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. Bonus problems must be submitted by email to me, at rich@math.lsu.edu, separately from the normal homework, and they will be graded more strictly for logical rigor than the required homework.

February 2

1.31, 1.33.

February 5

1.28, 1.32. (Hint for 1.32: Remember that the supremum and the infimum of a set might or might not be in the set. So don't make any unwarranted assumptions!) Also: 1.29, 1.30, 1.34--1.39.

February 7

1.41 --1.44.

February 9

1.47--1.50, 1.53--1.55.

February 14

1.51, 1.57. Also: 1.59--1.61, 1.63. (Hint for 1.51: Do not compute with (or even write) lim x_n or lim y_n until you have proven that these limits exist. Otherwise, you will be operating with undefined terms.) (Hints for 1.57: It may help to introduce the notations s_n(x) for the supremum of the nth tail T_n(x) of the x sequence, and similarly for the y sequence and the x+y sequence.) Please note that the Bolzano-Weierstrass theorem, the Nested Intervals theorem, and the Heine-Borel theorem will require your focused attention! Here is an optional bonus problem B2, 1.56, for those students who are looking for more challenging problems than those in the required homework. Email this to me, rich@math.lsu.edu, separately from the regular homework one week from the due date printed to the left of this box, if you choose to do it. Bonus problems will be graded more strictly for logical rigor than the required homework. Remember, bonus problems will be be emailed to me, and not handed in with the regular homework.

February 16

1.62, 1.64; Also 1.67-1.70. Please note that the Bolzano-Weierstrass theorem, the Nested Intervals theorem, and the Heine-Borel theorem will require your focused attention!

1.71, 1.72. Here is another optional bonus problem B3, for one week from the date to the left if you choose to do it. This question is similar to 1.70, but not identical! Suppose a_n is a strictly increasing sequence, b_n is a strictly decreasing sequence, and a_n < b_n for all n. Prove or Give a Counterexample: The intersection from 1 to infinity of (a_n,b_n) is nonempty.

February 21

1.76--1.80, 1.82, 1.85, 1.86. Be careful with terminology! If E is contained in the union U_{a in A}O_a of open sets O_a then it is the set {O_a | a is in A} that is an open cover of E. Do not confuse the open cover with its own union. The union of a family of sets is only one set!

February 23

1.83, 1.84. Be careful with terminology! If E is contained in the union U_{a in A}O_a of open sets O_a then it is the set {O_a | a is in A} that is an open cover of E. Do not confuse the open cover with its own union. The union of a family of sets is only one set!

February 26

February 28

First Hour Test. This test will cover all assigned work that was due before today.

February 29

Please download and read carefully the Hour Test #1, Spring 2024, Solution Sketches and Class Statistics.

March 4

1.87, 1.89, 1.90, 1.92, 1.94.

March 6

1.88, 1.91. Here are two more Bonus Problems, for those who seek more challenging exercises: B4 - 1.93, and B5-1.96. If you choose to do these, turn them in separately from the regular homework a week from today. After any bonus problems are graded and returned to you, please feel free to come to my office to ask for correct solutions.

March 8

2.1--2.5, 2.13--2.15. Here is another optional bonus problem, for a week from today if you choose to do it: B6-Problem 2.16.

March 18

2.6, 2.7 , 2.8; Be sure to read Cor. 2.1.1 and its proof to see how the sequential criterion for limits of functions is used. Read also Definition 2.1.3. (Hints: For 2.6, use the sequential criterion to prove the limit of the function does not exist. Do not use L'Hopital's Rule for 2.7(b)! Instead, draw a unit circle and find useful inequalities by comparing areas of triangles and a circular sector, expressible in terms of x, sin x, and tan x.)

March 20

2.19--2.20, 2.22-2.23, 2.25.

March 22

2.21, 2.24, 2.27. For 2.27, follow the sequence of steps given. This exercise is a theorem discovered by Cauchy. For optional bonus credit: B7-2.26 and B8-2.28. These two would be for one week from this date.

March 25

2.29--2.34, 2.40--2.41, 2.43--2.47.

March 27

Bring Questions to Review for the Second Hour Test today! This test will cover the assignments that were due after the first hour test.

March 29

April 1

Second Hour Test today!

April 2

April 3

2.35, 2.37, 2.42. In problem 2.42, remember to prove your conclusions for each of the three stated questions!

April 5

2.48, 2.50, 2.52-- 2.59.

April 8

2.49, 2.51. For 2.49, see Example 2.6.

April 10

April 12

2.64, 2.68 . In these exercises, you may use derivatives and L'Hospital's rule although they do not appear in this text until later.

3.1, 3.2, 3.4--3.9, 3.11, 3.12. These problems that are not to hand in but are important. Here is an optional bonus problem that would be due 1 week from today: B9- 3.14.

April 19

3.3, 3.10 For problem 3.3 use only the definition of the Riemann integral to show for each epsilon >0 there exists a delta >0 such that ||P||< delta implies that |P(f,{x_i bar}) - 0|< epsilon. For 3.10 you will benefit from the hint in the statement of the problem in the text.

3.18--3.23. Please read the statement of Theorem 3.24 and use it freely in the homework. It is a very powerful theorem.

3.24 .

Bring questions to review for the third hour test!

April 29

Third Hour Test today! This test will cover the work that was due since the second hour test.

May

3.25, 3.26, 3.30b. ((Hint for 3.25: In 3.25, the strictness of the inequality is the whole problem.) (Hint for 3.30b: Only part (b) is assigned, but here is a hint anyway for the unassigned part (a): In 3.30(a) one of the directions of implication has already been proven in the text. The other direction remains to be proven. (Hint: Use theorem 3.2.4, the variant of the Darboux Integrability Criterion.) Also: 3.27.

3.34 -- 3.37. Bring questions to review for the Final Exam! Also, please remember to fill out the end-of-course evaluation form that is available to you online! Your anonymous feedback is very important: It helps your teacher and the University to serve your needs to the best of our ability. Remember: Your opinion matters: You are the reason we are here! Thank you..
Don't forget to review from the beginning of the course! This 200-point exam will cover the whole course in a uniform manner, so remember to review from the beginning of the course. Your final grade for the course will be the larger of the following two: 1. The grade guaranteed by the formula provided higher on this page. 2. One letter below the final exam grade. Thus the final exam provides a safety net that supplements the calculations specified above. As an extreme example (that has not actually occurred in my experience) one might have a failing final average and an A on the final exam, resulting in a B for the course. But less extreme examples do in fact occur, showing the value of catching up by the end of the course.

Please download Final Exam, Spring 2023, Solution Sketches and Class Statistics.