Here are the 2011 exam solutions.
If you want to talk to me about the course at any time after Revision Week please e-mail me for an appointment.
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Lectures started on Monday 26th September and are:
| Time | Room |
|---|---|
| Monday 4-5 | Fogg Lecture Theatre (Biology) |
| Tuesday 10-11 | Fogg Lecture Theatre (Biology) |
| Thursday 1-2 | Law 210 |
All tutorials are on Mondays. They started on 3rd October, but we added a new one from 10th October. If you can, please come to the class corresponding to your surname:
| Time | Room | Surnames |
|---|---|---|
| 12-1 | Maths 103 (Maths Bldg) | A-G |
| 12-1 | BR 3.01 (Bancroft Road) | H-M |
| 2-3 | BR 3.01 (Bancroft Road) | N-Z |
The exercise sheets are an important part of this course. Each week you should attempt ALL the questions, and hand in your answer to the 'Feedback Question(s)' to the Orange Coursework Box on the ground floor of the Mathematics Building.
| Coursework Number | Date given out | Date Due | Solutions | Comments |
|---|---|---|---|---|
| Coursework 1 | 27th September | 2pm 4th October | Solutions 1 | Comments 1 |
| Coursework 2 | 4th October | 2pm 11th October | Solutions 2 | Comments 2 |
| Coursework 3 | 11th October | 2pm 18th October | Solutions 3 | Comments 3 |
| Coursework 4 | 18th October | 2pm 25th October | Solutions 4 | Comments 4 |
| Coursework 5 | 25th October | 2pm 1st November | Solutions 5 | Comments 5 |
| Coursework 6 | 1st November | 2pm 15th November | Solutions 6 | Comments 6 |
| Coursework 7 | 15th November | 2pm 22nd November | Solutions 7 | Comments 7 |
| Coursework 8 | 22nd November | 2pm 29th November | Solutions 8 | Comments 8 |
| Coursework 9 | 29th November | 2pm 6th December | Solutions 9 | Comments 9 |
| Coursework 10 | 6th December | No feedback questions | Solutions 10 |
Coursework 10 has more questions than usual but no feedback questions. There will be exercise classes as usual on Monday 12th December, at which you can discuss the questions on Coursework 10 with tutors and also get back your marked feedback questions from Courswork 9.
| Lecture | Topics | Scanned Notes |
|---|---|---|
| 1 | Introductory Lecture | Lecture 1 |
| 2 | Maximum of a set of real numbers; Demon games | Lecture 2 |
| 3 | Demon games continued; Negations | Lecture 3 |
| 4 | Upper bounds; least upper bound (supremum) | Lecture 4 |
| 5 | Definition of the real numbers; Consequences of the Completeness Axiom | Lecture 5 |
| 6 | Proof there is a real number with square 2 | Lecture 6 |
| 7 | Sequences: convergence to zero (definition) | Lecture 7 |
| 8 | Sequences: convergence to zero (examples) | Lecture 8 |
| 9 | Comparing sequences converging to zero | Lecture 9 |
| 10 | Constant multiples and sums of sequences converging to zero | Lecture 10 |
| 11 | Properties of modulus, and summary of results so far in this section | Lecture 11 |
| 12 | Products of sequences converging to zero | Lecture 12 |
| 13 | Sequences converging to real numbers: sums, products, quotients | Lecture 13 |
| 14 | Uniqueness of limits; behaviour of difference of consecutive terms | Lecture 14 |
| 15 | Proof that increasing sequences, bounded above, converge | Lecture 15 |
| 16 | Examples of bounded increasing sequences; definition of 'tends to infinity' | Lecture 16 |
| 17 | Increasing sequences converge or tend to infinity; Cauchy sequences (not for exam) | Lecture 17 |
| 18 | Series: definition of convergence | Lecture 18 |
| 19 | Basic theorems: geometric series, sums of series etc | Lecture 19 |
| 20 | Proof of Comparison Test; proof that absolute convergence implies convergence | Lecture 20 |
| 21 | Power series; exp(x), sin(x), cos(x). Radius of convergence | Lecture 21 |
| 22 | Radius of convergence (proof). Functions and continuity (definitions) | Lecture 22 |
| 23 | Continuous functions: examples | Lecture 23 |
| 24 | Sums, products and quotients of continuous functions | Lecture 24 |
| 25 | Compositions of continuous functions | Lecture 25 |
| 26 | Images of convergent sequences under continuous functions | Lecture 26 |
| 27 | The Intermediate Value Theorem | Lecture 27 |
| 28 | Applications of the IVT (Brouwer, Sarkovskii etc) | Lecture 28 |
| 29 | General form of IVT. Bolzano-Weierstrass Theorem | Lecture 29 |
| 30 | The Boundedness Principle | Lecture 30 |
| 31 | The Interval Theorem. Uniform convergence of sequences of functions | Lecture 31 |
| 32 | Differentiation | Lecture 32 |
2008 midterm. 2008 midterm solutions and Dr Walters' comments.
2009 midterm. 2009 midterm solutions and Dr Walters' comments.
2010 midterm. 2010 midterm solutions and Dr Walters' comments.
The test was on all the material in the first two sections of the course, 'Real Numbers', and 'Sequences', that is to say Lectures 1-17 inclusive and Coursework 1-6 inclusive - including the definition of when a sequence tends to infinity and the application of this definition to examples.
Here is the advice I put on this web-page before the test:
Here is a reasonable list of things that you should be able to do for the midterm test. When revising you should look at proofs and examples in your lecture notes as well as problems and solutions on the exercise sheets.
When you are asked in the test to prove something, if you prefer you may give a winning strategy for the corresponding demon game in place of a formal mathematical proof, provided you explain why your strategy always wins the game (in fact a winning demon game strategy, plus justification, is equivalent to a formal mathematical proof).
When you are asked to give a brief justification of a result, you may either quote results from the course by number (e.g. 'by Thm 16(iii) of the course') or by the statement of the result (e.g. 'because lim(x_n.y_n)=lim(x_n).lim(y_n), as proved in lectures').
There will be no exercise classes on Monday 7th November, but there will be a revision lecture on Tuesday 8th November 10.00-11.00a.m. in the Fogg Lecture Theatre. At this I will go through any topics and examples that you ask me to, and in particular I expect to go through the 2010 midterm test. The solutions to the 2010 midterm are now available above.
***** (Monday 14th November) HERE IS THE 2011 MIDTERM TEST AND HERE ARE THE 2011 MIDTERM SOLUTIONS AND COMMENTS. *****
Anyone who has not yet collected their marked paper from me at an exercise class or lecture can do so at one of my office hours.
I am not going to put 'model solutions' on the web for all these exams. Studying 'model solutions' for the exams of earlier years is not a useful way to prepare for an examination. A much better way is to try to work out your own answers (preferably under exam conditions): the way to understand mathematics is to do it. However here are the solutions to the 2010 exam to give you an idea of what is expected in your answers. Please do not look at them until you have attempted the 2010 exam.
Following the revision lecture on 25th April 2012 here are the solutions to 2011 exam. I am happy to mark your answers to the 2009 paper if you would like me to. E-mail me a scan of your answers and I will return it marked.
2. For statements with up to two quantifiers (e.g. one 'for all' and one 'there exists') formal proofs are expected and demon game proofs may lose you a mark or two.
3. For more complicated statements, a demon game proof could get full marks, but only if a watertight winning strategy (plus justification) is given.
It's very straightforward to convert a demon game winning strategy to a formal proof. Here is an instruction sheet From Demon Games to Formal Proofs written by Dr Mark Walters.
Whether or not you are taking Professor Vivaldi's course 'Mathematical Writing' you should read his web-book for the course, as it contains very good advice on how to write down mathematical reasoning clearly and precisely.
I hope to add further suggestions for additional reading as the semester progresses.
Last updated: 25th April 2012.
Prof Shaun Bullett.