Unit information: Analysis 1 in 2014/15

Unit name	Analysis 1
Unit code	MATH11006
Credit points	20
Level of study	C/4
Teaching block(s)	Teaching Block 4 (weeks 1-24)
Unit director	Dr. McGillivray
Open unit status	Not open
Pre-requisites	Normally an A at A-level mathematics or equivalent.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Analysis introduces the style of logically precise formulation and reasoning that is characteristic of university-level mathematics; it studies the foundations of elementary calculus in this style. It starts from basic properties of the real numbers, and works up to a rigorous treatment of continuous and differentiable functions.

Aims:

The unit aims to provide some basic tools and concepts for mathematics at the undergraduate level, with particular emphasis on

fostering students' ability to think clearly and to appreciate the difference between a mathematically correct treatment and one that is merely heuristic; introducing rigorous mathematical treatments of some fundamental topics in mathematics.

Syllabus

Weeks 1-12

• Logical propositions and connectives.
• Sets; finite unions and intersections; differences.
• Ordered pairs; Cartesian products. Functions and their graphs.
• Very basic introduction to quantifiers; negating quantifiers.
• Injections, surjections, bijections. Invertible functions.
• Proof by Induction
• Rationals and reals; irrationality of square root of 2
• Definition of supremum and infimum with example.
• Completeness axiom and other axioms.
• Inequalities.
• Sequences and their limits.
• Theorems on limits of sums, products, quotients and compositions
• Series. Tests of convergence.

Weeks 13-24

• Limits of functions (epsilon-delta definition of limit).
• Theorems on limits of functions.
• Continuous functions. Definition and properties.
• Continuous functions on a closed interval.
• Intermediate Value Theorem; extremal values on closed intervals.
• Differentiation and its simple properties.
• Maxima and minima of functions.
• Rolle's Theorem; Mean Value Theorem and applications.
• Approximation by polynomials. Taylor's theorem.
• Inverse function; derivative of the inverse function.
• Series; alternating series; absolute convergence.
• Power series.
• The exponential and logarithmic functions.
• Trigonometric functions.
• Riemann integration in elementary terms.
• Fundamental Theorem of Calculus.

Relation to Other Units

The unit gives the foundations for all other units in the Mathematics Honours programmes.

Intended Learning Outcomes

At the end of the unit, the students should:

• be able to distinguish correct from incorrect and sloppy mathematical reasoning, be comfortable with "proofs by delta and epsilon",

• have a clear notion of the concept of limit as it is used in the context of sequences, series and functions,

• have a clear notion of the concepts of differentiation and integration.

Transferable Skills:

Clear logical thinking; clear mathematical writing; problem solving; the assimilation of abstract and novel ideas.

Teaching Information

Lectures supported by problem classes, homework problem sheets, and bi-weekly small-group tutorials.

Assessment Information

The final assessment mark for the unit is constructed from two unseen written examinations: a January mid-sessional examination (counting 10%) and a May/June examination (counting 90%). Calculators and notes are NOT permitted in these examinations.

The mid-sessional examination in January lasts one hour. There are two parts, A and B. Part A consists of 4 shorter questions, ALL of which will be used for assessment. Part B consists of three longer questions, of which the best TWO will be used for assessment. Part A contributes 40% of the overall mark for the paper and Part B contributes 60%. The summer examination in May/June lasts two-and-a-half hours. There are again two parts, A and B. Part A consists of 10 shorter questions, ALL of which will be used for assessment. Part B consists of five longer questions, of which the best FOUR will be used for assessment. Part A contributes 40% of the overall mark for the paper and Part B contributes 60%.

Reading and References

• C. W. Clark, Elementary mathematical analysis. Wadsworth Publishers of Canada, 1982

• J. M. Howie, Real Analysis. Springer-Verlag, 2001.

• S. Krantz, Real Analysis and foundations. CRC Press, 1991.

• I. Stewart and D. Tall, The Foundations of mathematics. Oxford University Press, 1977.

• D. J. Velleman, How to prove it. A structural approach. Cambridge University Press, 1994.