MATH 452-01
Advanced Calculus
Syllabus for Fall 2018
Mon & Wed, 2:00 – 2:50pm
Fri, 2:00 – 3:50pm
327 Gildemeister Hall
Instructor: Dr. Eric Errthum Winona Email Username: eerrthum Office: 205 Gildemeister Hall Office Hours: See homepage. Or by appointment on any day.
Text: “Basic Analysis: Introduction to Real
Analysis” by Jiˇrí Lebl
A pdf version of this text can be found on Brightspace.
Prerequisite: MATH327: Foundations of Mathematics and MATH312: Multivariable Calculus
Course Website: http://course1.winona.edu/eerrthum/math452
About This Course: This course is designed to rigorously prove all the relevant concepts typically presented in single-variable Calculus as well as provide a solid foundation on which to begin a study in the field of Analysis.
Nobody needs proofs in order to get something done, but understanding the theorems and their proofs will make anyone think more deeply about the domain, become better at problem solving, and feel more comfortable with the meaning - and limitations - of the practical procedures.
-- Alon Amit (http://qr.ae/TUNYYw)
Expectations: Students who complete this course with a passing grade are expected to be able to demonstrate the following skills: (i) Mastery of prerequisite material (conceptual calculus, basic proof writing, etc.), (ii) Memorize and use new definitions, (iii) Write analysis proofs, (iv) Explain the relevant conceptual ideas from analysis, (v) Communicate mathematical reasoning clearly in both an oral and written format.
Seems to me the goal of undergraduate math is to provide experience in concept formation (finding meaning in abstract definitions), a wide variety of examples of structures, relationships, and approaches to proof, and generally develop mathematical thinking. The actual content is less important than suitability of the topic for elementary development.
Remember that traditional undergraduate math is not a professional degree in that it is not intended to provide information needed for specific jobs. Not even graduate study. The expectation is that beginning graduate students should be able to pick basic stuff up quickly, not that they already know it. Non-academic employers of people with math degrees have the same expectation.
Questions about whether specific content is really needed downstream miss the point.
-- Frank Quinn (https://mathoverflow.net/q/25916)
Grading: Oral
Homework (scaled as needed) 150
points 17.6%
Written
Homework (scaled as needed) 100
points 11.8%
Memory Quizzes
(scaled as needed) 150
points 17.6%
Relevancy Projects
(scaled as needed) 50
points 5.9%
Chapter Exams (5 @
50 points) 250
points 29.4%
Comprehensive
Final (oral & written) 150
points 17.6%
--------
850
points total
Grades: A = 90% (765 pts), B = 75% (638 pts), C = 60% (510 pts), D = 50% (425 pts). There will be no curving of individual assessments.
Homework: Homework will be completed in two ways:
Oral Homework: According to the approximate schedule below, on Oral Homework days students will be chosen randomly to present individual solutions to problems from the homework assigned since the last homework day. For each problem presented, students will be graded a 0, 1, or 2 out of 2 corresponding to their level of preparedness (not necessarily correctness). When presenting a solution, you should be prepared to answer questions clarifying your work. It is not acceptable to write out a whole solution, but then when asked about a particular step to say “I don’t know.” To me this indicates you copied the homework from someone else without understanding it. If you miss an Oral Homework day, you will need to present on problems during office hours to make up the lost points.
Written Homework: The written portion of the homework is due the period after a Homework Day. This work should be typed (preferably in LaTeX), stapled, and presented in order. Each problem will be graded 0 – 2 on its level of mathematical correctness and 0 – 2 on how well-written it is. The written homework problems will be a subset of the oral homework problems. (If you don’t know about LaTeX, start here.)
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Don't just read it; fight it! Ask your own questions, look for your own
examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? ---
Paul R. Halmos |
Memory Quizzes: Once a week there will be a short quiz over the newly defined terms and simple proofs from the book. The best way to memorize these is to actually understand them. Mathematics is not a series of incantations.
Relevancy Projects: Students should pick a topic that illustrates the intersection
of this course with their interests in Math, MathEd, Statistics, or some other
field. Some example topics are below. Remember to make proper citations in your
paper as you learned in ENG111 or MATH/STAT/DSCI395. If you don’t know how, ask
the instructor.
Important Dates: Project Topic Okayed
by October 15, Best Version Due November 19, Revised Version Due December 7.
Note: Your best version will be
graded as if it is your final version. If you are satisfied with your grade on
this version, you will not be required to turn in a revised version.
· Give an exposition comparing the three types of continuity covered in our text and how they relate. Research the concepts of bounded variation and absolute continuity. Describe how they relate to the types above. Give examples (with proof) of functions that fall sharply into each type.
· Present, in one place, the full proof of the Mean Value Theorem. Comment on the parts that are easy and the parts that are technically difficult, i.e. discuss what parts of the proof could be accessible to a Calc I class.
· Give an exposition on the error bounds for the Midpoint and Trapezoid Rules. Give a rigorous proof of at least one of them, and give an example of a function that (almost) attains the bound.
· Give an exposition on Levy’s Continuity Theorem (with proof and examples) for someone who has never taken a Mathematical Statistics course, but has taken this course.
· Give an exposition on the Strong Law of Large Numbers, starting with pertinent definitions and working up to explaining “almost sure” convergence and the conditions under which it could occur.
· Give an exposition on the role of the absolute value function throughout analysis. What are the important properties it satisfies, how would analysis be different if a different function with those (or stronger) properties was used in place of the absolute value, what role does it play in important definitions and could you work around it, etc.
· Give an exposition on the Borel-Cantelli Lemmas, covering the necessary background material, giving proofs, and nontrivial examples.
· A topic of your choice with instructor’s approval.
Exams: There will be in-class exams at the completion of each chapter.
Exam dates are tentative until officially announced in class. The final exam
will consist of a comprehensive written final exam and an individual oral exam.
The written portion of the final exam is scheduled for Wednesday, December 12, 10:30am – 12:30am. The oral portion of the
final exam will be made by appointment with the instructor near the end of the
semester.
Brightspace: Many course materials can be found on Brightspace including homework problems, study materials, and approximate grades. If at any point during the semester you would like to know your exact grade, please email the instructor.
Late/Missed Work: Late textbook activities, homework or missed exams will result in a score of zero. Make-up exams before the time of the normal exam will be given at the discretion of the instructor. If you miss class, it is your responsibility to obtain notes and assignments from fellow students. If you have an unavoidable absence, please inform the instructor beforehand.
Study Groups: Students are allowed to form study groups for the course. However, students are strongly encouraged to work on the homework individually first. All students must put homework solutions into their own words. Copy and pasted typed homework (even with minor changes) will be considered cheating. Do your own written homework.
Academic Dishonesty: Any type of academic dishonesty (cheating, copying, discussing confidential oral-exam problems with other students, using a solutions manual to do homework, finding solutions online, etc.) will result in failure and will be reported to school authorities. If you are having trouble with an assignment, please see the instructor first.
Note: This syllabus is subject to change if deemed necessary by the instructor.
Tentative
Schedule of Events – Math 452
(as of November 18, 2018; subject to change)
Note: This is very tentative and will depend on our pace. Here is what is for
sure:
1) Each chapter will have an exam in the period after the last oral homework in
that chapter.
2) We must cover Section 5.3 at the very least.
3) The Relevancy deadlines and Final exam time is fixed.
|
Week Beginning |
Monday |
Wednesday |
Friday (1st Hour) |
Friday (2nd Hour) |
|
8/27 |
Introductions Pre-Test |
0.3 (0.3 #1, 4, 6ab, 8abc, 13, 19, 20) |
Oral Homework Written: 0.3 #4a1, 6a, 13 |
1.1 (1.1 #2, 4, 5, 6,
8, 9, 10) |
|
9/3 |
NO CLASS |
Written HW Due 1.2 |
Memory Quiz & Written: 1.1 #2, 4, 9 |
1.2, cont. (1.2 #2, 4, 7, 9, 13)
|
|
9/10 |
Written HW Due 1.3 1.4 (1.3 #1, 2, 5) |
1.4, cont. 1.5 (1.4 #6, 7) |
Memory Quiz & |
1.5, cont |
|
9/17 |
Written HW Due 2.1 |
Chapter 1 Exam |
Go over Exam |
2.1, cont. 2.2 |
|
9/24 |
2.1, cont. Monotone Convergence Theorem (2.1 #4, 6, 8, 9, 12, 15, 16, 20) |
2.2 |
Memory Quiz & Written: 2.1 #6, 12, 20 |
2.2, cont (2.2 #3, 5, 7, 8,
9, 12) |
|
10/1 |
Written HW Due 2.3 (2.3 #5, 9) |
2.4 (2.4 #1, 2, 4, 8) |
Memory Quiz & Written: 2.2 #9, 12(a), 2.3 #9 |
2.5 |
|
10/8 |
Written HW Due 2.5, cont. |
TBD |
Memory Quiz & |
3.1 |
|
10/15 |
Relevancy Projects Topics Okayed Written HW Due 3.1, cont. |
Chapter 2 Exam |
Go over Exam |
Out Early |
|
10/22 |
3.1, cont. 3.2 (3.1 #1b,d, 2, 7, 8, 9, 12, 13) (3.2 #2, 8, 9, 10, 15) |
3.3 |
Memory Quiz & (3.1 #8, 3.2 #10, 15) |
3.3, cont 3.4 (3.3 #2, 7, 10, 11, 13) |
|
10/29 |
Written HW Due 3.4, cont. |
Memory Quiz & |
Written HW Due 4.1 (4.1 #1, 2, 3, 5, 12) |
4.2 |
|
11/5 |
Chapter 3 Exam |
Go over Exam |
Memory Quiz & |
5.1 |
|
11/12 |
NO CLASS |
5.1, cont 5.2 Properties of the
Integral |
Memory Quiz & |
5.2, cont. (5.2 #4, 5, 6, 7,
11, 12, 13, 16b) |
|
11/19 |
Relevancy Projects Rough Draft Due 5.3 |
NO CLASS |
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11/26 |
5.3, cont (5.3 #1, 2, 5, 7, 10, 11) |
Memory Quiz & |
Written HW Due TBD |
TBD |
|
12/3 |
Chapters 4 & 5 Exam |
Go over Exam |
Final Oral Exams |
Final Oral Exams |
Final Exam (Written Portion)
Wednesday, December 12
10:30am – 12:30pm
Commitment to
Inclusive Excellence
WSU recognizes that our individual differences can deepen our understanding of one another and the world around us, rather than divide us. In this class, people of all ethnicities, genders and gender identities, religions, ages, sexual orientations, disabilities, socioeconomic backgrounds, regions, and nationalities are strongly encouraged to share their rich array of perspectives and experiences. If you feel your differences may in some way isolate you from WSU’s community or if you have a need for any specific accommodations, please speak with the instructor early in the semester about your concerns and what we can do together to help you become an active and engaged member of our class and community.
If you or a friend has been a victim of sexual assault, dating violence, domestic violence, or stalking, you can talk to a trained, confidential advocate by calling 507-457-5610. See the Sexual Violence page for more information about your rights and resources.
Campus
Resources
- Inclusion and Diversity Office, Kryzsko Commons Room 236, 457.5595
- KEAP Diversity Resource Center, Kryzsko Commons Room 162, 457.2286
- Access Services (Services for Students with Disabilities), Maxwell 314, 457.5878
- KEAP Diversity Resource Center Faculty Liaison, Professor Chuck Ripley, Minné Hall 307, 457.5445, wripley@winona.edu
- Affirmative Action/Title IX Office, Lori Mikl, Somsen 202, 457.2766, lmikl@winona.edu
- Warrior Success Center, Maxwell 314, 457.5878
- Advising Services, Maxwell 314, 457.5878
- Tutoring Services, Krueger Library 220, 457.5680
- Writing Center, Minné Hall 348, 457.5505
- TRIO Student Support Services, Krueger Library 219, 457.5465
- Counseling and Wellness Services, Integrated Wellness Complex 222, 457.5330
The Standard Disclaimer
applies.
© Eric Errthum, November 2018, all rights reserved.