MATH 127B: REAL ANALYSIS

Winter 2021



FINAL EXAM

The above version corrects the original. In problem 1 the two '≤' were mistakenly '<' instead and in problem 3 'series' was written as 'sequence'.


All course meetings will be held using zoom.
See Canvas for zoom room numbers and passwords.

Lecturer: Eric Babson, babson@math.ucdavis.edu.
Lecture: Mondays, Wednesdays and Fridays at 9:00 am PST for sections A and 11:00 am PST for sections B in room 7150588313.
These will be recorded and linked below.
Office hours: for half an hour after each lecture.
These will normally not be recorded. Monday March 15 (Ides) 9-10 and 11-12.

Teaching Assistant: Neetal Neel, nneel@ucdavis.edu.
Sections A01 and A02: Tuesdays at 5:10 pm PST in 98515240131 and 6:10 pm PST in 9379860459.
Office hour: TBA

Teaching Assistant: Chutong Wu, ctwu@ucdavis.edu.
Sections B02 and B01: Thursdays at 4:10 and 5:10 pm PST
Office hour: TBA.

Texts: Introduction to Analysis by John K. Hunter and
Understanding Analysis (first or second edition) by Stephen Abbott.

Grading: There will be a total of 400 points possible with 250 points from your best 25 assignments (from among homework assigneents and quizzes) and 150 points from the midterm and final split either (50 + 100) or (75 + 75) whichever is better. For final grades a score of at least 360 yields an A+, at least 320 an A, at least 280 a B, at least 240 a C and at least 200 a D.

Honor Code: Please write and sign (or initial if typed) the following on each assignment and exam:
I pledge on my honor that I have not given or received any unauthorized assistance on this assignment/examination.

Participation: For each lecture a few people can be designated to either keep their cameras on or make a guess at any questions I pose. Each role is worth one extra credit point per lecture for a maximum of 5 points during the term. Please email me if you are willing to fulfill one of these roles during some lecture and I will suggest a date.

Homework: Most problems are from one of the two editions of Abbott's book. Those from the first edition will have a file linked. Those from the second will be denoted with ed2. The entire second edition is linked above. Please submit via Canvas or via the link below for your section.
You can type your solutions or write them out and send a picture. Please feel free to discuss the problems with others but if the work you turn in is not entirely your own please cite the group with which you worked.

Midterm: The midterm will cover chapters 8, 9, 10 and 13. One sheet of handwritten notes (both sides) will be allowed. It will be conducted via Zoom (see instructions) and you will scan and upload your solutions to Canvas. Here and here are a couple midterms from 2020 along with solutions and solutions.

Final: The final will be cumulative with more emphasis on the material not on the midterm and will not be proctored. It will be available at noon on Wednesday March 17 and due by noon on Friday March 19. Here are the spring 2020 final with answers and practice final with answers. You may use the course texts, notes and lectures for the final exam. You may not discuss the exam with others or use other outside materials.

Course Outline: The course covers formal definitions of derivatives and integrals along with spaces and sequences of functions with particular emphasis on power series as per the department syllabus.
This is a rough outline of when topics will be covered and will be edited as the term progresses.

Lecture Boardwork:9:00:Left Board Right Board 11:00:Left Board Right Board



DayDateTopicsHomework due: A1A2B1B2
MondayJan 48.1 Derivatives (av)(aud)
TuesdayJan 5A01 and A02 DiscussionQuiz 1
WednesdayJan 68.2 Properties (av)(aud) 5.2.7 ed2
ThursdayJan 7B01 and B02 DiscussionQuiz 1
FridayJan 88.3 Chain Rule (9:av)(9:aud) (11:av)(11:aud) 5.2.2 ed2, 5.2.6 ed2.
MondayJan 118.4 Extreme Values (11:av)(11:aud) 5.2.3 ed2.
TuesdayJan 12A01 and A02 DiscussionQuiz 2
WednesdayJan 138.5 Mean Value (9:av)(9:aud) (11:av)(11:aud) 5.2.5 ed2, 5.2.10 ed2.
ThursdayJan 14B01 and B02 DiscussionQuiz 2
FridayJan 158.6 Generalized Mean Value (11:av)(11:aud) 5.3.6 ed2.
MondayJan 18MLK Day: No Lecture
TuesdayJan 19A01 and A02 DiscussionQuiz 3
WednesdayJan 208: Darboux's Theorem (9:av)(9:aud) (11:av)(11:aud) 5.3.9 ed2, 5.3.10 ed2 (some texts may have g(x) printed wrong).
ThursdayJan 21B01 and B02 DiscussionQuiz 3
FridayJan 228: GMVT applications (9:av)(9:aud) (11:av)(11:aud) Darboux Problems.
MondayJan 259.1,2 Function Convergence (9:av)(9:aud) (11:av)(11:aud)5.3.3 ed2, 5.3.4 ed2, 5.3.7 ed2.
TuesdayJan 26A01 and A02 DiscussionQuiz 4
WednesdayJan 279.3 Cauchy (9:av)(9:aud) (11:av)(11:aud)6.2.1 ed2, 6.2.2 ed2, 6.2.9 ed2.
ThursdayJan 28B01 and B02 DiscussionQuiz 4
FridayJan 299.4 Uniform Properties (9:av)(9:aud)(9:av)(9:aud) (11:av)(11:aud)6.2.3 ed2, 6.2.5 ed2, 6.2.13 ed2.
MondayFeb 19.4.3 Differentiability (9:av)(9:aud) (11:av)(11:aud)6.2.6 ed2, 6.2.7 ed2, 6.2.10 ed2.
TuesdayFeb 2A01 and A02 DiscussionQuiz 5
WednesdayFeb 313 Banach Spaces (9:av)(9:aud)6.3.2 d2, 6.3.4 ed2, 6.3.6 ed2, 6.3.7 ed2.
ThursdayFeb 4B01 and B02 DiscussionQuiz 5
FridayFeb 513.6 Function Spaces (9:av)(9:aud) (11:av)(11:aud) Banach Problems
MondayFeb 810 Power Series (9:av)(9:aud) (11:av)(11:aud)6.4.2 ed2, 6.4.3 ed2, 6.4.4 ed2.
TuesdayFeb 9A01 and A02 DiscussionQuiz 6
WednesdayFeb 1010 Power Series (9:av)(9:aud) (11:av)(11:aud)6.5.2 ed2, 6.5.4 ed2, 6.5.8 ed2.
ThursdayFeb 11B01 and B02 DiscussionQuiz 6
FridayFeb 12Midterm AB5.3.8 ed2, 6.2.8 ed2, 6.4.8 ed2. (Feel free to turn in Wednesday).
MondayFeb 15Pres Day: No Lecture
TuesdayFeb 16A01 and A02 DiscussionQuiz 7
WednesdayFeb 1711.2 Riemann Integral (9:av)(9:aud) (11:av)(11:aud)
ThursdayFeb 18B01 and B02 DiscussionQuiz 7
FridayFeb 1911.3,11.4 Cauchy Criterion (9:av)(9:aud) (11:av)(11:aud) 7.2.2 ed2, 7.2.4 ed2, 7.2.5ed2.
MondayFeb 2211.5 Linearity (9:av)(9:aud) (11:av)(11:aud) 7.2.3 ed2, 7.2.7 ed2, 7.3.1 ed2.
TuesdayFeb 23A01 and A02 DiscussionQuiz 8
WednesdayFeb 2411.5, 13.2 L1 Norm (9:av)(9:aud) (11:av)(11:aud) 7.3.2 ed2, 7.3.3 ed2, 7.3.4 ed2.
ThursdayFeb 25B01 and B02 DiscussionQuiz 8
FridayFeb 2611.6, 11.8 Integrability (9:av)(9:aud) (11:av)(11:aud) 7.3.5 ed2, 7.3.6 ed2, 7.3.7 ed2.
MondayMar 112.1 Fundamental Theorem (9:av)(9:aud) (11:av)(11:aud) 7.4.3 ed2, 7.4.5 ed2, 7.4.6 ed2.
TuesdayMar 2A01 and A02 DiscussionQuiz 9
WednesdayMar 312.2 Weak Derivatives (9:av)(9:aud) (11:av)(11:aud) 7.5.1 ed2, 7.5.2 ed2, 7.5.3 ed2, 7.5.4 ed2.
ThursdayMar 4B01 and B02 DiscussionQuiz 9
FridayMar 512.3 Limits (9:av)(9:aud) (11:av)(11:aud) 7.5.6 ed2, 7.5.7 ed7, 7.5.10 ed2.
MondayMar 812.4, 12.5 Improper (9:av)(9:aud) (11:av)(11:aud) 7.4.7 ed2, 7.4.8 ed2, 7.4.11 ed2.
TuesdayMar 9A01 and A02 DiscussionQuiz 10
WednesdayMar 1012.7 Taylor's Theorem (9:av)(9:aud) (11:av)(11:aud) 8.4.9 ed2, 8.4.10 ed2.
ThursdayMar 11B01 and B02 DiscussionQuiz 10
FridayMar 12Review (9:av)(9:aud) (11:av)(11:aud) 8.3.9 ed2, 8.3.10 ed2, 8.3.11 ed2.
MarFinalCumulative