Analysis I MAT 320 MAT 640 Spring 2022

MAT320 Analysis I: 4 hours, 4 credits. Introduction to real analysis, the real number system, limits, continuity, differentiation, the mean value theorem, Taylor's theorems and applications. Riemann integration and improper integrals.

Prerequisite: Either Vector Calculus MAT226 or Departmental permission

Professor Sormani: google "Sormani Math Lehman” for my webpage

Contact: sormanic@gmail.com (do not call the office and leave messages)

Welcome to the Class Playlist

Class Meets: Asynchronously Online. As with any advanced 4 credit math course, you can expect to spend 12 hours per week on the course. Because the course is asynchronous, you will complete two lessons per week on your own schedule. For each lesson: mark a 3-hour slot on your weekly calendar to watch each lesson and start the homework, and mark an additional 3-hour slot for completing and fixing the homework. Keep in mind that there are 28 lessons including homework that must be completed before taking the final exam. Students who fall behind may risk failing the course. Students cannot request an INC unless they have passed three exams during the semester.

Expectations: This course involves writing proofs and all proofs must be completed in two column format with numbered statements and justifications as taught in this course. Students are expected to learn both the mathematics covered in class and the mathematics in other assigned reading. Completing homework is part of the learning experience. Students should review topics from prior courses as needed using old notes and books.

Classwork: Classwork is required as proof of attendance. All your notes for each lesson will be submitted in a googledoc entitled MAT320S22-LessonX-Lastname-Firstname with the lesson number replacing X and your last name and your first name. The googledoc will be shared with the professor sormanic@gmail.com as an editor. Note that you can easily do this on a smart phone using the googledoc app by copying your photos and then pasting into the googledoc.

Homework: In addition to the classwork, approximately four hours of homework will be assigned in each lesson that should be completed before starting the next lesson. Note that a single problem may take an hour. In the schedule below, homework is written within the lesson notes. Classwork and *ed homework must be submitted as proof of completion of each lesson in the same googledoc. After receiving feedback from the professor, you will often be required to resubmit some of the problems. This is part of the learning process. The resubmission goes at the end of the doc for that lesson.

Office Hours: To ask a question, take a photo of your question and put it in the googledoc for its lesson with QUESTION typed in next to the photo. Then share the doc with the professor writing "See my question inside this doc". The professor will respond within 48 hours. Questions about homework or classwork must include a photo of the statement of the problem and your initial attempt to solve it.

Exams: Exams will be given on Tuesdays or Wednesdays 8-10pm. Each exam may only be taken two days after all the lessons leading to the exam have been completed and submitted. There are no retakes, so do not take an exam unless you are both healthy and prepared. All students will have individual exams and must upload each part of their exam to their googledocs within 25 minutes.

Grading Policy: There are three exams (20% each) and a final (40%).

Materials, Resources and Accommodating Disabilities

Lecture Notes: Will be provided under each lesson

A Textbook: Mathematical Analysis: a Straightforward Approach by Binmore, 2nd Ed Cambridge University Press ISBN: 9780521288828 (not required)

Supplementary Notes:

Rules of Proofs and Proofs and Dominos by Prof Sormani

Calculus Textbook: You should have a copy of your calculus textbook in case you need to review something. If not you may use the MIT Free Calc Textbook

Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441. They also have a webpage:

https://www.lehman.edu/student-disability-services/

Equipment and Online Technical Assistance:

please contact the Lehman College IT center:

https://www.lehman.edu/itr/help-desk.php

Campus Access:

https://lehman.edu/coronavirus/

Counseling Services: Lehman has it available online:

https://www.lehman.edu/counseling-center/

Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let me know, and something will be arranged for you.

Names/Gender: We will use last names (surnames) in this class. You may call me Sormani or Professor or Dr. Sormani and may use any pronoun you wish when referring to me.

Respect: All students will treat each other with respect. Let me know if you have concerns.

Course Objectives

At the end of the course students should be able to:

1. find limits, sups and infs by applying theorems (as part of department objectives in math A, B & E)

2. prove that a sequence converges and a function is continuous at a point (as part of E, F & G)

3. write a proof by contradiction (as part of F & G)

4. state, apply and prove theorems related to Calculus including Riemann sums (as part of E)

5. write a proof by induction involving series (as part of F & G)

6. find Taylor series, prove convergence theorems and find radii of convergence (as part of B, E & F)

These objectives will be assessed on the final exam along with other important techniques.

Course Calendar (2 lessons per week starting Feb 1)

Each lesson below will be linked to a googledoc which has links to videos and to classnotes. Classwork must be submitted as well as *ed homework to prove a lesson is completed.

Lesson 1: Introduction to Sets, Direct Proofs, Inequalities, Cases, and Quantifiers (Feb 1-2)

Lesson 2: Bounds, Max/Min, and Proof by Contradiction (Feb 3-4)

Lesson 3: Sup and Inf with the Continuoum Property of the Reals (Feb 8-10)

Lesson 4: Sup and Inf with the Archimedian Property (Feb 11-12)

Lesson 5: Metric Spaces, Balls, Open Sets and Converging Sequences of Points (Feb 23 including redoing Part I)

Lesson 6: Converging and Diverging Sequences in the Real Line (Feb 25)

Submit Lessons 1-6 by 3pm on Sun Feb 27 to take the exam on time:

Lesson 7: Exam I on Sequences (Mar 2-3) New date!

will have 4 parts (25 minutes each)

Part I: an upper bound proof,

Part II: a sup/inf proof,

Part III: an epsilon-N convergence proof for a specific sequence,

Part IV: an epsilon-N convergence proof about pairs of sequences

Lesson 8: Convergence and the Sandwich Lemma (Mar 4-5)

Lesson 9: Watch one of these Inspiring Talks or take Exam I (Mar 7-9)

Lesson 10: Monotone Sequences (Mar 10-12)

Lesson 11: Subsequences, liminf, limsup, and Bolzano Weierstrass Theorem (Mar 13-15)

Lesson 12: Limits of Functions: lim as x to c of f(x) is L (Mar 16-18)

Lesson 13: Continuity of Functions and Existence of Maxima (Mar 19-20)

Lesson 14: Review for Exam II (Mar 21-22)

Lesson 15: Exam II on Sequences, Continuity, and Limits (Mar 23-24)

Exam II was given March 30, March 31, April 6 and April 7. The final opportunity to take Exam II is Thursday May 5. Students who wish to take Exam 2 must have resubmitted Lessons 8-13 by Saturday and do Lesson 14 by Monday. After this I will send everyone their Exam 2 grades and 0% for all students that never took it. Students who fail should withdraw.

Exam II will have 4 parts (25 minutes each):

Parts I-II: short questions about given sequences: bounded, increasing, decreasing, liminf, limsup, etc

Part III: an epsilon-delta continuity or limit proof for a specific function,

Part IV: an epsilon delta continuity or limit proof for a combination of functions

Lesson 16: Uniform Continuity and Equicontinuity (Mar 29-30)

Lesson 17: Uniform Convergence, C([0,1]), and the Arzela-Ascoli Theorem (Apr 5-6)

Lesson 18: Differentiation (Apr 7-8)

Lesson 19: Extrema, Mean Value Theorem, Continuity, and Inc/Dec (Apr 12-13)

Lesson 20: Riemann Sums and Integrals (Apr 13-14)

Spring Break: (April 15-22) All students who are behind schedule should catch up during the break completing all the above lessons.

Lesson 21: Riemann Integration and the Fundamental Thm of Calculus (Apr 23-24)

Lesson 22: Series (Apr 25-26)

Lesson 23: Convergence of Series (Apr 27-28)

Lesson 24: Convergence of Taylor Series and Review for Exam 3 (May 2-3)

Lesson 25: Exam III on the Proofs of Calculus I - II (May 4-5)

• Fri May 6 8-10pm

• Wed May 11 12-2pm

• Wed May 11 8-10pm

• Th May 12 8-10pm

If you do not take Exam 3 on these dates, but have passed Exam 1 and Exam 2, then you may request an incomplete and finish the course in June. If you are graduating or need a grade in May for some reason, you may take parts of Exam III that you have completed the lessons for and get a 0% on the other parts. See below:

Exam III will have 4 parts (25 minutes each):

Part 1: Prove that a sequence of functions is equicontinuous as in Lesson 16 (need Lesson 16 done to take this part)

Part 2: Prove that a sequence of functions converges uniformly to a limit function as in Lesson 17 (need Lesson 17 done to take this part)

Part 3: Find the Riemann Sum as in homework from Lesson 20 being sure to state how large N must be taken to have an error less than a given value (need Lessons 18-21 done to take this part)

Part 4: Find the Taylor Series as in Lesson 24 and check where it converges using a test as in Lesson 23. (need Lessons 22-24 done to take this part)

Lesson 26: Natural Log and L’hopital’s Rule (May 9-10)

Lesson 27: Limits and Improper Integrals (May 12-13)

Lesson 28: Limits and Review for the Final Exam (May 16-17)

Final Exam: The final will be given during finals week.

Dates for the Final Exam:

*Wed May 18 12-2pm

*Wed May 18 8-10pm

*Th May 19 8-10pm

*Sat May 21 8-10pm

*Mon May 23 12-2pm

*Mon May 23 8-10pm

You may submit Lessons 26-28 before or after the final. If you do not take the final on one of these dates, but have passed Exam 1 and Exam 2, then you will get an incomplete and finish the course in June.

Students will need to know the statements of all the important theorems and definitions we've learned this semester including sup, inf, bound, limit, bounded increasing sequences converge, sandwich lemma, subsequences of bounded sequences converge, Cauchy sequences, theorems about these, continuity, theorems about this, differentiation, mean value theorem, Rolle's theorem, Riemann integration of continuous functions, theorems about integration, improper integrals, series, convergence tests including comparison, ratio, root and alternating series tests, Taylor series, radius of convergence, uniform convergence. You will also need working knowledge of these concepts in the sense that you must be able to find the limit of various given sequences, the sup and inf of various sets in order to complete the proofs.

The final will have four parts (25 minutes each):

Part 1: write a complete epsilon-delta or epsilon-R proof of a limit of a function (or of continuity)

Part 2: write a complete proof that a sequence of functions converges uniformly

Part 3: prove an upper bound is a sup using proof by contradiction and the Archimedean Principle

Part 4: find a Riemann Sum approximating a Riemann Integral up to a given error and find the radius and interval of convergence for a given Power Series

What happens if you get an Incomplete in the course? This means you get a temporary grade of INC. You will complete all overdue lessons and take all missing exams in June. When you have completed your work, the grade of INC will be changed into the grade you earn in the course. Students are only able to do this if they have passed Exams I and II. Students who fail one these first two exams were advised to withdraw from the course to avoid the F.

What if I have not finished all my lessons and really need a grade before June? I can compute a grade based on work completed thus far, averaging in a 0% for incomplete work. You may take the parts of Exam III that you are have completed the lessons for and be given a 0% on the parts you did not yet prepare for. You may take the Final Exam. Students who do this are at risk of failing the course.