Today we will continue with revision going through the 2008 examination paper and any theory topics requested. Office hours will be announced. The 2008 paper is linked here. A list of theory topics which might be included is given below.

Test 2 has been marked and will be handed back Wednesday. Worked solutions are linked below.

Here is a list of topics for revision:

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. continuous functions on closed bounded intervals attain their bounds
16. if the derivative exists at a local max or min it is zero
17. f:[a,b]-> R continuous implies uniformly continuous

18. Rolle's theorem
19. Chain rule
20. Application of Taylor's theorem to sufficient conditions for a local mininum,

21. Characterization of Riemann integrable functions,
22. If c>0 and f is Riemann integrable so is c.f,
23. If f<= g and both are Riemann integrable then ...,
24. The integral and absolute value assuming |f| is Riemann integrable with f,
25. if f is Riemann integrable on [a,c] and on [c,b] then ..
26. continuity of the integral as a function of the upper limit and differentiabilty ..
27. continous implies RI.

    limit |a_n/a_(n+1)| exists implies its value is that of the radius of convergence for the power series sum a_n x^n

28. convergence of the exponential series,
29. proof that exp'(x)=exp(x),
30. derivation and convergence of the series for ln(1+x) on [0,1].
31. derivation and convergence for arctan on (-1,1).
32. uniform convergent limit of continuous functions is continuous.

Web Site: http://www.math.waikato.ac.nz/~kab

Paper outline is here.

Lecturer for weeks 1-12:

Associate Professor Kevin Broughan, Office G3.22, Tel 838-4423, email kab@waikato.ac.nz

Office Hours: Normally Thursday 3-5pm in G3.22 during the teaching weeks 2-12.

Lecture Times:

1. Monday 1.10pm in KG.01,
2. Wednesday 12.00 noon in S 1.03, (Tutorial/Workshop)
3. Thursday 2.10pm in S1.03

Math Help:

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 links below.
2. Thomas' Calculus, 11th Edition.

Recommended Reading:

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

Lecture note Links:

1. Inequalities and absolute value: page 1, 2, 3, 4, 5.
2. The real number line R is complete: page 6, 7, 8, 9, 10,
3. Limit of a sequence, limit theorem: page 12, 13, 14, 15, 16, 17.
4. Subsequences, monotonic sequences, useful limits: page 18, 19, 20, 21, 22, 23. 24, 25, 26.
5. Series of positive terms I, D'Alembert's tests, Cauchy's test: page 27, 28, 29, 30, 31, 32, 33.
6. Series of positive terms II, Integral test, conditional and absolute convergence: page 34, 35, 36, 37.
7. Limits of functions, continuity: page 38, 39, 40, 41, 42.
8. Limit theorem for functions: page 43, 44, 45, 46.
9. One sided and infinite limits, l'Ho^pital's rule: page 47, 48, 49, 50, 51, 52.
10. Continuity - maximum and intermediate value theorems: page 53, 54, 55, 56, 57.
11. The derivative, maxima and minima: 58. 59, 60.
12. Chain rule, Rolle's and Mean Value theorems: page 61, 62, 63, 64, 65.
13. Taylor's theorem and its applications I: page 38, 39, 40, 41, 42,
14. Taylor's theorem and its applications II: 43, 44, 45, 46.
15. Riemann integration I - definition of the integral: page 59, 60, 61, 62.
16. Riemann integration II - properties: page 63, 64, 65, 66.
17. Uniform continuity: page 7, 8, 9, 10, 11, 12, 13, 14.
18. Riemann integration III - continuous and piecewize continuous functions: page 67, 68, 69, 70, 71, 72, 73.
19. Riemann integration IV - integral as a function of the upper limit: page 74,
20. Uniform convergence of sequences of functions: 75, 76, 77, 78, 79.
21. Exponential and log functions: page 80, 81, 82, 83, 84.
22. Power Series: page 85, 86, 87, 88, 89, 90, 91.
23. Sin and cos: page 92, 93, 94.
24. Binomial theorem: 95, 96, 97.
25. ArcTan function: 98, 99.

Exercises and readings set in class 2009:

1. Read Thomas' Calculus Section 1.1, Do page 7 #3, 9,11,15,19,21, 31,43,45.
2. Sequences: Thomas' Calculus (T) Sect 11.1 p747. Do p757 #13, 15, 23,25,29,35,47.
3. Limit of a sequence: Read T p751-756, Do p757 # 33, 53,55, 57, 61, 69,91, 97, 103.
4. Subsequences, useful limits etc: same reading and exercises as last lecture.
5. Series of positive terms I: Read T p761-769. Do p769 # 1,5,9,13, 19, 23, 35,41.
6. Series of positive terms II: Read T p772-775, p777-780, p781-785 and try some exercises from each of the related sections, e.g. p775 # 9,25, p781 #1,3,7, p786 # 1,3,5,23.
7. Limits of functions, continuity: T page 91-97, Do T p96 # 7,11, 17,27,29,31,39,47.
8. Limit theorem for functions: Read T page 97-98 Appendix A.2 p AP-4, Do T p100 # 51,53,57,59.
9. One sided and infinite limits: Read T sections 2.4, 2.5 Do p112 #17,51,57,70.
10. Continuity - maximum and intermediate value theorems: Read T p124-132. Do: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].
11. The derivative, maxima and minima. See the previous reading and exercises.
12. Chain rule, Rolle's theorem and the Mean Value Theorem: Read T p255-260, Do p260 #1,3,5,9,11(a).
13. Taylor's theorem and its applications I: Read T p811-819, Do p819 # 23, 27, 39, 41, 43.
14. Taylor's theorem and its applications II: Read T p805-810, Do p810 # 1,7,21,27,31.
15. Riemann integration I: Read T Sections 5.2,5.2,5.3.
16. Riemann integration II: Do T p354 # 77-80.
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 1, 2 and 3 on page 75 and see if you can repeat them without consulting the text. Then try example 1 on p79.
21. Exponential and log functions: Read T sections 7.2 and 7.3 and make a list of the properties of exp and ln that we did not demonstrate in the lecture. Attempt to unify the series approach with the integral definition of ln and of exp as its inverse.
22. Power series: lecture notes page 87 # 1,2,4,6,7.

Workshop/Tutorial sheets for 2009:

• Sheet 1: inequalities and absolute value,
• Sheet 2: limits of sequences,
• Sheet 3: definition of the limit of a sequence,
• Sheet 4: convergence of series.
• Workshop 5: going over Test 1.
• Sheet 6: Taylor expansions.
• Sheet 7: the Riemann integral,
• Sheet 8: uniform convergence of sequences of functions,

Tests and Assignments for 2009:

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

• Assignment 1 questions, solutions,
• Assignment 2 questions, solutions.
• Test 1 questions, solutions.
• Assignment 3 questions, solutions.
• Assignment 4 questions, solutions,
• Test 2 questions, solutions.

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