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Before you start the course.

From results you will have seen before, 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 Archimedes' 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 lim_{n → ∞} xⁿ/n! =0 for all real x.

For Finite Series you will need to know the formula for ∑_i=1ⁿi^k 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 , ... , s_n =∑_i=1ⁿa_i, ... 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, e^x, 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 e^x, 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 ten Question Sheets. You only learn Mathematics by doing Mathematics so try to do all the questions, preferably before the tutorials.

Limits	Continuity	Differentiation	Integration
Question Sheet 1 Question Sheet 2 Question Sheet 3	Question Sheet 4 Question Sheet 5	Question Sheet 6 Question Sheet 7 Question Sheet 8 Question Sheet 9	Question Sheet 10

Solution Sheets

Limits	Continuity	Differentiation	Integration
Solution Sheet 1 Solution Sheet 2 Solution Sheet 3	Solution Sheet 4 Questions 1-7 Solution Sheet 4 Questions 8-10 Solution Sheet 5 Solution Sheet 5 Questions 5-10	Solution Sheet 6 Solution Sheet 7 Solution Sheet 8 Solution Sheet 9	Solution Sheet 10

Lecture Notes

The course naturally falls into four parts of Limits, Continuity, Differentiation and Integration. I expect you to read the notes and though the appendices are not compulsory they will illuminate and help you understand the compulsory notes.

Remember, video recording are not an alternative to reading the notes, instead they will help you understand the notes.

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.
Notes Part 1.2a Notes Part 1.2b Part 1.2 Appendix	1.2 Limits Divergence. 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.
Notes Part 3.2 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.
Notes Part 3.3 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.
Notes Part 4 Part 4 Appendix	4 Integration Definition and examples. Fundamental Theorem of Integration.

Recommended Texts

• Mary Hart, Guide to Analysis, Macmillan Mathematical Guides, Palgrave Macmillan; second edition 2001,
• R Haggarty, Fundamentals of Mathematical Analysis, Addison-Wesley, second edition, 1993.