The other half of this course, on Complex Analysis, will be given by Dr C. Walkden.

The official source of material for this course is the syllabus page, but here I will give more details of the course and its examination procedures. This page is a place of resources for students on the course.

To give you a brief idea of what you can expect to be able to do at the end of the course here are the Intended Learning Outcomes:

• State the definition of limit of a function; calculate the limit for simple functions;
prove and apply the Rules for Limits to calculations for more complicated functions,
• State the definition of continuity; prove that simple functions are continuous at given
points; prove and apply the Rules for Continuous functions to more complicated
functions,
• State the definition of differentiable; prove that simple functions are differentiable and
calculate their derivatives at given points; prove and apply the Rules for Derivatives
to more complicated functions,
• Prove and apply the Intermediate Value Theorem; Inverse function Theorem; various
results on the composition of functions; various mean value theorems,
• Calculate Taylor polynomials; state Taylors Theorem with an error term; derive
bounds on the error terms; prove that Taylor series for a function converge to that
function,
• State the definition of the Riemann integral; calculate the Riemann integral for various
functions.

## 48.3%

The final Unit Survey response rate is 48.3%. So no new coffee mug for Charles and me (we needed 60% or more for that).

Alas, 48.3% means that 167 of the 323 students in the class did not completed the survey. But I suspect that most of the students who attended lectures completed it.

Best of luck to all in the exam, though with sufficient revision luck will not be needed. And anyway, there is no such thing as luck.

## Exam, Solutions & Feedback

A lot of students did well on my questions, which was gratifying to see. On the attached you will find the exam solutions along with some comments on the commonly seen errors.  But, for many who did not do well, the main error was not attending lectures; not learning the proofs; not attempting the problem sheets and not attending the tutorials.

For example, students had a problem with

• the limit in A1iii, though a similar limit is seen in Qu 26 on Problem Sheet 1,
• deducing the Mean Value Theorem from Rolle’s Theorem; this is Theorem 3.2.8 and the proof was given in lectures,
• the inequality in A2iii; this is Question 15 on Problem Sheet 3  and the method was discussed in lectures and a tutorial,
• the Taylor polynomial in A3, due to errors when differentiating (in turn due to lack of practice),
• remembering the definition of the Upper and Lower sums in A4; these are given in the lectures and illustrated with diagrams to help with remembering which definition has a lub and which a glb,
• the calculation of the Lower Sum in A4, due to simple errors with the algebraic manipulation of the geometric series (again due to lack of practice).

Also, in almost all cases where students had answered more than the required number of questions they would have been better served by searching for a mark or two extra in the correct number of questions. Especially because so many marks were lost due to arithmetic and algebraic errors which could have been caught and corrected after a careful reading.

## The MATH20101 Timetable

For both Real and Complex analysis you will get
• 2 Lectures per week. The timetable shows all the lectures concerned with the Real Analysis half of the course.
• 1 feedback tutorial per week. The timetable shows all five real analysis classes, you will have been assigned to one of these classes.
9.00 10.00 11.00 12.00 1.00 2.00 3.00 4.00
Monday   Tutorial
G.205
Dr. C. Dean

Tuesday   Tutorial
Roscoe 1.010
Dr. M. Coleman
Tutorial
G.207
Dr. C. Dean
Tutorial
Simon 2.61
Dr. M. Coleman

Wednesday Lecture
Simon E

Thursday     Lecture
Roscoe A

Friday           Tutorial
G.207
Dr. C. Dean

You are, of course, welcome to make an appointment by email and come an see me in my office if you should have any problems with the course.

## Feedback Tutorials

When? The Feedback tutorials start in week 2. You will be assigned one of the five periods each week. It is important you go to your alloted tutorial, you will be recorded as not attending a class if you go to one you are not assigned.

What happens in a tutorial?
The Feedback tutorials will cover those Examples stated in lectures without solution.
In weeks 3, 5, 8 and 10 the tutorials will feature a Kahoot Quiz, a fun way of me seeing how well you are doing, and identifying any misconceptions the class might be having.
Otherwise, the tutorials are an opportunity for the tutor and a P.G. helper to go around the room and offer individual help. This is only of benefit if you have attempted questions before the tutorial.

## Before you start the course.

You need to recall some results and techniques from courses in your First Year.

For instance you will need to know the triangle inequality, |a+b||a|+|b| for all real numbers a,b. Perhaps you can use this to show that |a-b| ||a|-|b||?

For Trigonometric Functions you will not need to know more than sinθ and cosθ can be defined as ratios of lengths of sides in a right angled triangle. From such a definition it is simple to deduce –1 ≤ sinθ ≤ 1, –1 ≤ cosθ ≤ 1 and sin2θ + cos2θ = 1 for all θ, the latter just being a restatement of Archimede's Theorem that the square of the hypotenuse is equal to the sum of the squares on the other two sides.

You will need to know what is meant by the greatest lower bound and least upper bound for a set S. Then you will need know the Completeness Property of R, that a non-empty set bounded above has a least upper bound, and a non-empty set bounded below has a greatest lower bound.

For Sequences of Real Numbers you will not need to know much more than the definition of the limit of a sequence, the Sum, Product and Quotient Rules for limits along with the particular example that limn → ∞ xn/n! =0 for all real x.

For Finite Series you will need to know the formula for ∑i=1nik for k=1, 2 and 3, along with the formula for i=1nxi valid for x ≠1.

For Infinite Series of real numbers a1+ a2+ a3+ ... you will need to know what is meant by saying that it converges, namely that the sequence of partial sums, s1 = a1, s2 = a1+ a2, s3 = a1+ a2+ a3 , ... , sn =i=1nai, ... converges.

For Power Series you will need to know what is meant by the radius of convergence. In particular you should know the power series 1+ x + x2/2! + x3/3! + ... + xn/n! +... for the exponential function, ex, and that it has an infinite radius of convergence, i.e. converges for all real x.

In Differentiation you should know the derivatives of standard functions, i.e. the exponential function ex, the hyperbolic functions, sinhx etc., the trigonometric functions, sinθ, etc., and the logarithm, lnx. The point of this course is to give justifications for what you already know for differentiation. Similarly, we will give proofs of results you should already be familiar with, namely the Product, Quotient and Chain Rules. And by "know" I mean "be able to use" and the only way you can use these results effectively is to practice on many examples.

Here are four sheets of questions, they are not for revision after the course but to help preparation before.

## Question Sheets

There are four Question Sheets mirroring the four parts of the course; Limits, Continuity, Differentiation and Integration.
I do not give weekly sheets. You can compare the Question Sheets with the material for that week in the lecture notes to see what questions you can attempt.

## Lecture Notes

The course naturally falls into four parts of Limits, Continuity, Differentiation and Integration. After each part I will make available notes on that section. These notes will state carefully all definitions and Theorems. Hopefully the notes will also give motivation for the results. But the notes will not contain proofs of all the Theorems nor solutions to all Examples. There has to be some reason for attendance at lectures.

Notes Contents
Notes Part 1.1a Notes Part 1.1b
Part 1.1 Appendix
1.1 Limits limit of a real-valued function at a finite point, one-sided limits, limits of a function at infinity. Divergence.
Notes Part 1.2a Notes Part 1.2b
Part 1.2 Appendix
1.2 Limits Limits Rules, including the Sum, Product and Quotient rules and the Sandwich Rule.
Notes Part 1.3
Part 1.3 Appendix
1.3 Special Limits. The exponential and trigonometric functions.
Notes Part 2.1
Part 2.1 Appendix
2.1 Continuous functions Continuous functions and their properties. Definition. Continuity Rules, including the Sum, Product and Quotient rules and the Composite rule.
Notes Part 2.2
Part 2.2 Appendix
2.2 Continuous functions Properties of continuous functions, including the Intermediate Value Theorem, and the Boundedness Theorem.
Notes Part 2.3
Part 2.3 Appendix
2.3 Continuous functions Monotonic functions, the Inverse Function Theorem.
Notes Part 3.1a Notes Part 3.1b
Part 3.1 Appendix
3.1 Differentiation Definition. Rules for differentiation, including the Sum, Product and Quotient rules, the Chain Rule and the Inverse Rule.
Part 3.2 Appendix 3.2 Differentiation Derivative Results including: Rolle's Theorem, the Mean Value Theorem, Cauchy's Mean Value Theorem and L'Hôpital's Rule.
Part 3.3 Appendix 3.3 Differentiation Taylor Polynomials and Taylor's Theorem with Cauchy's and Lagrange's forms of the error. Taylor Series and a number of Standard series.

Part 4 Appendix
4 Integration Definition and examples. Fundamental Theorem of Integration.

Not given in 2016-17: How to use the Fundamental Theorem of Calculus; an example of a sequence of Riemann integrable functions whose limit is not Riemann integrable; if f is monotonic then f is Riemann integrable; Sum, Product and Quotient Rules for integration; Integration by parts; Integration by Substitution; Improper Integrals.

## Past exam papers

The solutions here are for the Real Analysis questions in the exam paper.
I must stress that you cannot pass the exam simply by doing past papers. You can get practice of how long the questions are, learn how each part of a question depends on previous parts of that question, and get a feel of what types of problems you may be required to solve, definitions of give and Theorems to prove. But to pass exams you have to learn, understand and remember the material in the course. For this, nothing is better than attending the lectures along with reading the notes on this site.