MA 2108/MA2108S, Fall 2004

Advanced Calculus II



Lecturer: J. Wu (e-mail: matwuj@nus.edu.sg )
Tutors: J. Wu, D. Q. Zhang (e-mail: matzdq@nus.edu.sg ), Y. Yang (e-mail: matyangy@nus.edu.sg ), Pan Suqi (e-mail: g0203637@nus.edu.sg )
Graders: Yang Jialiang, Zhao Xinyuan, Wang Liping, Ayineedi Venkateswardu
Office: S14, 04-07, Faculty of Science, Lower Kent Ridge Road.
Office Hours: Friday 2-4pm or by making appointment through e-mail
e-mail: matwuj@nus.edu.sg
Phone: 6874-4940.
Problem-based learning of this module
Text Books:
  1. R. G. Bartle and D. R. Sherbert, Introduction to real analysis, 3rd edition, John Wiley, 2000. (Compulsory reading)
  2. W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. (Supplementary reading)
  3. Manfred Stoll, Introduction to real analysis, 2nd Edition, Addison Wesley Longman, Inc. press, 2001.
  4. James Stewart, Calculus, 4th Edition, Brooks/Cole Publishing Company press, 1999.
  • Lecture Notes, consisting of 89 pages. ( Note. There may be some changes of lecture notes in classes according to the situation.)
  • Un-packed lecture notes, 231 pages, which will be mainly talked in class.
  • Syllabus
    Completeness axiom of the real number system. Sequences, limits (epsilon-N definition), monotone convergence theorem, Cauchy's criterion for convergence, sup, inf, lim sup and lim inf of a sequence. Infinite series, Cauchy's criteria, absolute and conditional convergence. Tests for convergence. Power series and the radius of convergence. Review of the elementary functions and their properties via power series. Pointwise and uniform convergence of a sequence of functions, Weierstrass M-test. Integration and differentiation of a series of functions.
    Additonal topics of MA2108S: Conditions equivalent to the completeness axiom, rearrangement of series, trigonometric series.
  • Triangular Numbers
  • Hexagonal numbers
  • Convergence of a sequence
  • Lectures: Tuesday 10-12 LT25; Friday 10-12 LT 25.
  • Everybody must attend all lectures and should arrive on time. In case you missed the class due to sick, then you should submit your MC to me. If you missed some classes without sufficient reasons, you might get negative credit to your grade from the course.
  • Tutorials: (Online Registration through CORS)
  • Group 1 (ST1): TUESDAY 1500-1600 S9A-0102.
  • Group 2 (ST2): TUESDAY 1600-1700 S9A-0102.
  • Group 3 (ST3): WEDNESDAY 1100-1200 S9A-0102.
  • Group 4 (ST4): WEDNESDAY 1200-1300 S9A-0102.
  • Group 5 (ST5): THURSDAY 1100-1200 S14-0301.
  • Group 6 (ST6): THURSDAY 1200-1300 S14-0301.
  • Group 7 (ST7): FRIDAY 1200-1300 S13-0503.
  • Group 8 (ST8): FRIDAY 1300-1400 S13-0503.
  • Group 9 (ST9): SATURDAY 1100-1200 S13-0506.
  • Group 10 (ST10): SATURDAY 1200-1300 S13-0506.
  • Group 11 (long Tutorial, STL): FRIDAY 1600-1800 S13-0506.
  • Tutorial of MA 2108S: S9A-01-02
  • Wed 2-4 pm.
  • Everybody must attend tutorials and should arrive on time. In case you missed any tutorial due to sick, then you should submit your MC to me. If you missed some tutorials without sufficient reasons, you might get negative credit to your grade from the course.
  • Exams
  • Midterm: Tuesday, 5 October, 10-11:30 am, LT25, covering chapters 1 and 2.
  • Final Exam: 30/11/2004 (Tue) 1 pm (From the Examination Time-Table of NUS.)
  • CA, which includes the midterm test, take-home exams, participation in tutorials and classes, and etc, will be counted as 30% and the final will be counted as 70%.
  • Assessment for MA 2108S: CA 50%, which includes tutorial presentation and tests, and the final 50%.
  • Rules for the Exams:
    1. NO Make-up exams.
    2. During the final exam, a help sheet (two-sided up to A4 size) is allowable.
    3. During the midterm exam, a help sheet (two-sided up to HALF A4 size) is allowable.
  • Take-home Exams
    It is strictly NOT allowed to copy other people's solutions; but you are allowed to discuss the problems with your classmates.
  • Take-home Exam 1
    deadline: Monday, August 30
    Hand in your papers to: the Mailbox on the 1st floor of S14.
    The answers.
  • Take-home Exam 2
    deadline: Monday, September 13
    Hand in your papers to: the Mailbox on the 1st floor of S14.
    The answers.
  • Take-home Exam 3
    deadline: Monday, September 27.
    Hand in your papers to: the Mailbox on the 1st floor of S14.
    The answers.
  • Take-home Exam 4
    deadline: Monday, October 11.
    Hand in your papers to: the Mailbox on the 1st floor of S14.
    The answers.
  • Take-home Exam 5
    deadline: Monday, October 25.
    Hand in your papers to: the Mailbox on the 1st floor of S14
    The answers.
  • Take-home Exam 6
    deadline: Monday, November 8.
    Hand in your papers to: the Mailbox on the 1st floor of S14.
    The answers.
  • Tutorials
  • Tutorial 1 and the answers
  • Tutorial 2 and the answers
  • Tutorial 3 and the answers
  • Tutorial 4 and the answers
  • Tutorial 5 and the answers
  • Tutorial 6 and the answers
  • Tutorial 7 and the answers
  • Tutorial 8 and the answers
  • Tutorial 9 and the answers
  • Tutorial 10 and the answers
  • Tutorial 11 and the answers
  • Practice Problems
  • Midterm Exam Paper and the solutions
  • Past Midterms
  • Fall 2003 and the solutions
    Spring 2003 and the solutions
    Fall 2002 and the solutions
    Spring 2002 and the solutions
    Fall 2001 and the solutions
    Summer 2000 with suggested answers.
  • Past Finals
  • Fall 2003 and the solutions
    Spring 2003 and the solutions
    Fall 2002 and the solutions
    Spring 2002 and the solutions
    Fall 2001 and the answers
    Summer 2000 with the answers.
  • Computation of Pi using Taylor series
  • A little history about the number Pi:1500 years ago our ancestor already knew that Pi is between 3.1415926 and 3.1415927. This is really remarkable during the time when nobody knew any calculus. By using Taylor series together with computer, we obtain that Pi is about 3.14159264, a little more accurate than what our 1500 years old ancestor did.
  • Tsu Ch'ung Chi ֮(430-501)
    Links to Calculus at Other Universities
  • Calculus II at the University of Pennsylvania.
  • Calculus at Harvard
  • These pictures are created by Maple. Here are my Maple commands for the left picture: 1. with(plots,animate); 2. animate([e/(1+e*sin(125*t)),t,t=0..2*Pi],e=-2..2,coords=polar,view=[-4..4,-4..4],numpoints=150,frames=30);
    Have a fun with Math!

    Maple commands for the right picture above.
    More pictures.

    Feedback on matters related to the course will be welcome. Please send it to
    matwuj@nus.edu.sg