MATH 452-01 Advanced Calculus
Syllabus for Fall 2023

Mon & Fri, 8:00 – 8:50am
Wed, 8:00 – 9:50am

326 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 D2L Brightspace.

Prerequisite: MATH327: Foundations of Mathematics and MATH312: Multivariable Calculus

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------- 18.8%
                        Written Homework (scaled as needed)                    100 points------- 12.5%
                        Memory Quizzes (scaled as needed)                       150 points------- 18.8%
                        Writing Project (scaled as needed)                            50 points-------- 6.3%
                        Chapter Exams (50 pts each)                                   200 points------- 25.0%
                        Comprehensive Final (oral & written)                    150 points------- 18.8%
                                                                                                     --------
                                                                                                         800 points total

Grades:  A = 90% (720 pts), B = 75% (600 pts), C = 60% (480 pts), D = 50% (400 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 0 – 2 corresponding to their level of preparedness and ability to utilize feedback on the fly. When presenting a solution, you should be prepared to answer questions clarifying your work and/or implement feedback given by the instructor and/or classmates. 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 don’t understand the solution and maybe didn’t even create it yourself. 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 an Oral Homework Day. This work should be typed (preferably in LaTeX/Overleaf), 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.) Revisions are allowed within a week of receiving feedback.

http://37.media.tumblr.com/923ea3d8c6d2d43831998571b03b36f6/tumblr_mlyrq3ODiZ1rvnpe0o1_500.png

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. The second-best way to memorize these is to make flash cards and really pay attention to the details (“there exists” vs “for all”, “unique” or not, etc.).

Written Project:        A two-page mathematics narrative addressing your personal history with the subject of mathematics. Details in D2L. Important Dates: Best Version Due October 23, Revised Version Due November 27. 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.

Exams:     There will be in-class exams at the completion of each chapter. Exam dates are tentative (below) 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 4, 8:00am – 10:00am. The oral portion of the final exam will be made by appointment with the instructor near the end of the semester.

D2L Brightspace:      Many course materials can be found on D2L 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 October 19, 2023; subject to change)

 

Week Beginning

Monday

Wednesday (1st Hour)

Wednesday (2nd Hour)

Friday

8/21

Introductions

Pre-Test

Oral Homework

 

Written:

0.3
Sets, Induction, Functions, Cardinality

(0.3 #1, 4, 6ab, 8abc, 13, 19, 20)

Written HW Due

 

 1.1
Basic Properties

 

(1.1 #2, 4, 5, 6, 8, 9, 10)

8/28

Memory Quiz

 

1.2
Real Numbers

Oral Homework

 

Written: 0.3.4b, 1.1.4, 1.1.5, 1.1.9


1.2, cont.
Archimedean Property, Sup & Inf,

 

(1.2 #2, 4, 7, 9, 13)

Written HW Due

 

1.3
Absolute Value

1.4
Intervals, Uncountability

 

(1.3 #1, 2, 5)

9/4

NO CLASS
Labor Day

 

Memory Quiz

 

Oral Homework


1.4, cont.

1.5
Decimal Representation

 

(1.4 #6, 7)

Written HW Due

 

1.5, cont

9/11

Chapter 1 Exam

 

Go over Exam

Memory Quiz

 

2.1
Sequences and Limits

 

Exam Corrections Due

 

2.1, cont.
Monotone Sequences, Subsequences

2.2
Limit Facts

9/18

Memory Quiz

 

2.1, cont.

Monotone Convergence Theorem

 

(2.1 #4, 6, 8, 9, 12, 15, 16, 20)

Oral Homework

2.2
Algebraic Properties of Limits

Written HW Due

 

2.2, cont
Convergence Tests

 

(2.2 #3, 5, 7, 8, 9, 12)

9/25

Memory Quiz

 

2.3
Limsup & Liminf
Bolzano-Weierstrass

 

(2.3 #5, 9)

Oral Homework

2.4
Cauchy Sequences

 

(2.4 #1, 2, 4, 8)

Written HW Due

 

2.5
Series Definition and Properties

10/2

Memory Quiz

 

2.5, cont.

(2.5 #4, 9, 10, 11, and behavior of series Sum(1/(n^2+n)))

Oral Homework

3.1
Cluster points, limits of functions

Written HW Due

 

3.1, cont.
Sequential Limit

10/9

Chapter 2 Exam

Go over Exam

Memory Quiz

 

3.1, cont.

 

3.2
Continuous functions

 

(3.1 #1b,d, 2, 7, 8, 9, 12, 13)

 

(3.2 #2, 8, 9, 10, 15)

Exam Corrections Due

 

3.3
Min-Max and Intermediate Theorems

10/16

Memory Quiz

 

3.3, cont

3.4
Uniform continuity

 

(3.3 #2, 7, 10, 11, 13)

Oral Homework

3.4, cont.

(3.4 #2, 8, 11)

Written HW Due

 

4.1
Derivative definition

 

(4.1 #1, 2, 3, 5, 12)

10/23

Memory Quiz

 

4.2
Mean Value Theorem

Best Version of Narrative Due

 

Oral Homework (3.4 & 4.1)

4.2, cont.

(4.2 #1, 5, 8, 9, 12)

Written HW Due

 

5.1
Riemann Integral

10/30

Chapter 3 Exam

Memory Quiz

 

5.1, cont

 

5.2

Properties of the Integral

(5.1 #2, 5, 7, 10, 11b)

Go over Exam

Exam Corrections Due

 

Oral Homework (4.2 & 5.1)

11/6

Written HW Due

 

5.2, cont.

 

(5.2 #4, 5, 6, 7, 11, 12, 13, 16b)

Memory Quiz

 

5.3
Fundamental Theorem (Part 1)

5.3, cont
Fundamental Theorem (Part 2)

 

(5.3 #1, 2, 5, 7, 10, 11)

NO CLASS
Veterans’ Day

11/13

Oral Homework (5.2 & 5.3)

Written HW Due

 

Memory Quiz

TBD

Exam Review

11/20

Chapters 4 & 5 Exam

NO CLASS
Thanksgiving

11/27

Last Revised Version of Narrative Due

 

Go over Exam

Exam Corrections Due

 

Final Oral Exams

Final Oral Exams

Final Oral Exams


Final Exam (Written Portion)

Wednesday, December 4

8:00am – 10:00am

 

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.

The Standard Disclaimer applies.

© Eric Errthum, October 2023, all rights reserved.