Department of Mathematics math252-10B

Elements of analysis

Course Web Site

Please return your script from Test 2 to me or Glenys so the marks can be recorded.

Today's lecture we will continue with the examination papers from 2008 and 2009 which are linked under their dates, and it time permits, Q3,5 from Test 2. My special office hours during the study break are given below.

Solutions for Assignment 3 and Assignment 4 are linked below. The questions and solutions for test 2 are also linked below. Please pay careful attention to questions 3,4,5.

Here is a list of theory topics which might be asked in the examination:

  1. All definitions and theorem statements,

  2. Archimedian property and every interval contains a rational number
  3. increasing bounded sequences converge
  4. limit of a constant times a function is a constant times the limit
  5. useful limits
  6. every subseqence of a convergent sequence converges to the same limit
  7. comparison test for series convergence
  8. the ratio test for series convergence
  9. n^a a_n -> 0 for a>1 implies convergence
  10. necessary condition for convergence a_n-> 0
  11. absolute convergence implies convergence

  12. limit of the sum of functions is the sum of the limits
  13. both side limits of a function exist with the same value implies the full limit exists
  14. sandwich theorem for limits of functions
  15. if as x->c f(x)->L and a_n->c then f(a_n)-> L also

  16. continuous functions on closed bounded intervals attain their bounds
  17. if the derivative exists at a local max or min it is zero
  18. Rolle's theorem and the mean value theorem
  19. Chain rule
  20. f:[a,b]-> R continuous implies uniformly continuous
  21. Application of Taylor's theorem to sufficient conditions for a local mininum,
  22. Big O notation,

  23. Characterization of Riemann integrable functions (Thm 50),
  24. If c>0 and f is Riemann integrable so is c.f,
  25. If f<= g and both are Riemann integrable then ...,
  26. Comparison of the integral and absolute value assuming |f| is Riemann integrable with f,
  27. if f is Riemann integrable on [a,c] and on [c,b] then ..
  28. continuity of the integral as a function of the upper limit and differentiabilty ..
  29. continous implies RI.
  30. uniform convergent limit of continuous functions is continuous.

Web Site:

The paper outline is linked here.

Lecturer for weeks 1-12:

Professor Kevin Broughan, Office G3.22, Tel 838-4423, email

Lecture Times:

  1. Monday 1.10-2pm in KG.01,
  2. Wednesday 12.00-12.50pm in SG.03,
  3. Thursday 10-10.50am in G3.33

Math Help:

Special times during the study break - see the notice boards level 3, block G I will have office hours in G3.22, 2-4pm, 26-28th October and 10-noon Friday 29th October.

Texts and Information Sources

Prescribed Text:

  1. Lecture Notes for Math252, Introduction to Real Analysis Lecture Notes, by Kevin Broughan: available on this web site- see the link below.
  2. Schaum's outlines Calculus by Ayres and Mendelson (A&M).

Recommended Reading:

  1. A first course in mathematical analysis by J. C. Burkill (CUP)
  2. Real Analysis, A First Course by R. A. Gordon (Addison-Wesley).

Lecture Note link (version 14 to page 101).

Extra figures:
  1. Limit of a sequence,
  2. A sequence with no limit,
  3. The limit is unique,
  4. The squeezing lemma.
  5. Better proof of Leibniz theorem.

Exercises and readings set in class 2010:

  1. Read A&M ch 1 p1-3, Do page 3 #2,6,7,8.
  2. Sequences: Read A&M p353-355.
  3. Limit of a sequence: Read A&M p355 #1,2. do p358 # 30-33, 35-38.
  4. Increasing sequences: Read A&M p355 # 3-9, do p357 # 12, 14, 15, 17, 21, 23, 25,26.
  5. Useful limits: Do A&M p357 # 15,16,18,19,22,29,34,39,40.
  6. Series of positive terms I: Read A&M p360-363, Do p364 #12(a)(b), 14, 16(a)(b), 17(a)(b), 18(a)(c)(d)(g).
  7. Series of positive terms II: Read A&M p366-368 (note the Limit comparison test at the bottom of page 367), and p368 #2-4,10. Do p371 # 12, 15,17,22,24,25,27,28.
  8. Absolute convergence, alternating series: Read A&M ch 45 p375-376, p377 #1-10, Do p380 #23-28,43.
  9. Limits of functions: Read A&M ch 7 p56-58 #1,2,3,9,10,14, Do p62 # 16(a)(b)(c)(l), 17(d), 18(a)(b).
  10. Continuity: Read A&M ch 8 p66-70 #1-3, Do page71 #1,8.
  11. Continuity - maximum and intermediate value theorems: Find the global maxima and minima for (1) 2x^2-3|x| on [-1,1], (2) |2x-1| + |3x-2|+x^2 on [-1,1], (3) (x-1)|x-2| on [0,3], (4) Use the Intermediate Value Theorem to show f(x)=x^3+x has a solution f(x)=1 for some 0<= x <= 2, and then show that there is only one solution.
  12. Chain rule, Rolle's theorem and the Mean Value Theorem: Read A&M ch10, ch13. p98-100, #1,2,3,10,11,12, Do p103#16,17,22,23(a)(iii),(vii).
  13. Taylor's theorem and its applications: Read A&M ch 47 p396-398,#1,2,3,4(a)(b)(c). Do p402 #12,13,14,15.
  14. Taylor's theorem and its applications II: Read A&M ch27 p222-226, Do p227 #10(a)-(h).
  15. Riemann integration I: Read A&M ch23 p190-196. Do p196 #1(a)(c),14, using the definition of the integral.
  16. Riemann integration II: Do A&M p197 # 9,10,12,13(c)(d), 14,15.
  17. Uniform continuity: show y=f(x)=x^2 (x squared) is uniformly continuous on [0,2]. Then show the sum of two uniformly continuous functions is also uniformly continuous. Show the product is not always: test g(x)=x*x on [0,infinity).
  18. Work through the details of showing piecewise continuous functions are Riemann integrable from the notes.
  19. Riemann integration IV: Let f(x)=1 when 0<= x <1, f(1)=2, f(x)=3 when x in (1,2]. Compute the area function A(x) and show explicitly it is continous on [0,2], but not differentiable at x=1. Then integrate A(t) on [0,x] for x in (0,2) to get a function B(x), and show that its integral is differentiable, even at x=1, and that B'(x)=A(x).
  20. Uniform convergence: Work through the details of examples in the notes and see if you can repeat them without consulting the text. Then try the exercises in the notes on uniform convergence.

Tutorial sheets for 2010:

Tests and Assignments for 2010:

Assignments should be handed in through the slot marked 252B under the Mathematics Office reception counter (G3.19).

Tests and Assignments for 2009:

Tests and Assignments from 2008:

Test and Assignments from 2007:

  • Assignment 1 questions, solutions.
  • Assignment 2 questions. solutions.
  • Test 1 questions, solutions.

    Dates and values of Tests, Assignments and the Final Examination:

  • Assignment 1: Out Wednesday 21st July, due Wednesday 28th July, value 5% of the overall mark.
  • Assignment 2: Out Wednesday 28th July, due Wednesday 4th August, value 5% of the overall mark.
  • Test 1: Thursday 19th August, 10-11am in G3.33, value 15%.
  • Assignment 3: Out Wednesday 15th September, due Thursday 23rd September, value 5% of the overall mark.
  • Assignment 4: Out Thursday 23rd September, due Thursday 30th September, value 5% of the overall mark.
  • Test 2: Thursday 7th October, 10am-11am in G3.33, value 15%.
  • Final Examination: in October/November after the study break, at a date to be announced, value 50%.

    Kevin Broughan

    14th October 2010