MA 2108, Spring 2002

Advanced Calculus II



Lecturer: J. Wu
Tutors: J. Wu
Office: S14, 04-07, Faculty of Science, Lower Kent Ridge Road.
e-mail: matwuj@nus.edu.sg
Phone: 874-4940.


Text Books:
  1. William R. Parzynski and Philip W. Zipse, Introduction to mathematical analysis, International Edition 1987, McGraw-Hill Book Company Press.
  2. G. B. Thomas, Jr. and Ross L. Finney, Calculus and analytic geometry, 9th Edition, International Student Edition, Addison-Wesley Longman Inc. Press, 1996.
  3. M. Braun, Differential equations and their applications, 3rd Edition, Applied Mathematical Sciences V. 15, Springer-Verlag, 1986.


  • Course Outline
  • Lecture Notes
  • Supplement to Section 3.9
  • Supplement to Section 3.10


  • Lectures: Monday, Thursday, 4:00-5:30 pm, LT26.

  • 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:
  • Group 1: Tuesday 11-12, S13 0501
  • Group 2: Tuesday 12-1, S13 0501
  • Group 3: Tuesday 1-2, S13 0501
  • Group 4: Wednesday 4-5, S13 0502
  • Group 5: Friday 4-5, S13 0506
  • Group 6: Friday 10-12, S13 04-04

  • Everybody must attend all tutorials and should arrive on time. In case you missed the tutorial due to sick, then you should submit your MC to your tutor. If you missed some tutorials without sufficient reasons, you might get negative credit to your grade from the course.


  • Exams
  • Midterm: (Tentative time) Monday, March 18, 4:00-5:30, LT26,
  • Final Exam: Monday, April 22, EV.
  • CA, which includes the midterm test, participation in tutorials and classes, and etc, will be counted as 20% and the final will be counted as 80%.
  • Rules for the Exams:
    1. NO Make-up exams.
    2. During the final exam, a note sheet (two-sided up to A4 size) is allowable.
    3. During the midterm exam, a note sheet (two-sided up to HALF A4 size) is allowable.


  • 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
  • Practice problems for Chapter One and the answers
  • Practice Exam for the Midterm and the solutions
  • If you like to practice more, here is another one with suggested answers.
  • For convergence/divergence of series, there are a lot of questions in one of our text books (Thomas-Finney's book "Calculus and analytic geometry"). So you can find some questions there for practice.
  • Solution of the Midterm Exam
  • Practice exam for the final and the answers
  • If you like to practice more, here is another one with the answers.
  • Solution of the Final Exam


  • 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