MA 2108, Fall 2005

Advanced Calculus II



Lecturer: J. Wu (e-mail: matwuj@nus.edu.sg )
Tutors: J. Wu (e-mail: matwuj@nus.edu.sg ) and Jelena Grbic (e-mail: matgj@nus.edu.sg).
Graders: Yang Jialiang (e-mail: g0306107@nus.edu.sg), Chen Yidi (e-mail: yidi@nus.edu.sg) .
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
Announcement
  • Important Notes on MA2108 and MA3110
  • Text Books:
    1. R. G. Bartle and D. R. Sherbert, Introduction to real analysis, 3rd edition, John Wiley, 2000. (Compulsory reading)

      Relevant Sections and Suggested Exercises from Bartle and Sherbert's book

    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 235 pages.
  • Packed lecture notes 85 pages
  • 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.
  • 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): .
  • Group 2 (ST2): .
  • Group 3 (ST3): .
  • Group 4 (ST4,STL): .
  • Group 5 (ST5): .
  • Group 6 (ST6): .
  • Group 7 (ST7):.
  • Group 8 (ST8): .
  • 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, 11 October, LT25, covering chapters 1 and 2.
  • Final Exam: 29/11/2005 (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%.
  • Rules for the Exams:
    1. NO Make-up exams.
    2. The final exam is a closed book exam, but you are allowed to bring along TWO help sheets.
    3. The midterm exam is a closed book exam, but you are allowed to bring along One help sheet.
    4. Definition of a help sheet: A help sheet is a piece of paper of size not larger than A4 (21 cm by 30 cm). Anything on the help sheet must be handwritten and may be written on both sides of the paper. The handwriting can be as big or as small as the candidate may desire. However, the help sheet must not contain any machine printed information of any kind (such as photocopy of a page from either a book or handwritten notes.)
  • Take-home Exams
  • Take-home Exam 1
    deadline: Tuesday, August 30
    The answers.
  • Take-home Exam 2
    deadline: Tuesday, September 27
    The answers.
  • Take-home Exam 3
    deadline: Tuesday, October 11
    The answers.
  • Take-home Exam 4
    deadline: Friday, October 28
  • 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
  • Practice Problems
  • Midterm Exam and the solutions
  • Past Midterms
  • Fall 2004 and the solutions
    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 2004 and the solutions
    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