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Analysis 1 (MATH11006)

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Administrative Information

  1. Unit number and title: MATH11006 Analysis 1
  2. Level: C/4 (Honours)
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 07/08
  6. Unit Organiser: Misha Rudnev
  7. Lecturer: Michiel van den Berg (weeks 1-8), Misha Rudnev (weeks 9-24)
  8. Teaching block: 1 and 2
  9. Prerequisites: A at A-level mathematics or equivalent

Unit 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.

General Description of the Unit

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.

Relation to Other Units

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

Teaching Methods

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

Learning Objectives

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.

Assessment Methods

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%.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

The assessment mark is calculated as described in the Assessment section above. Details of the university's common criteria for the award of credit points are set out in the Regulations and Code of Practice for Taught Programmes at http://www.bristol.ac.uk/esu/assessment/codeonline.html

Note that for this unit:

  • first year students are expected to attend all the relevant tutorials,
  • all students are expected to hand in attempts to the weekly exercises set.

Transferable Skills

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

Texts

  1. C. W. Clark, Elementary mathematical analysis. Wadsworth Publishers of Canada, 1982
  2. J. M. Howie, Real Analysis. Springer-Verlag, 2001.
  3. S. Krantz, Real Analysis and foundations. CRC Press, 1991.
  4. I. Stewart and D. Tall, The Foundations of mathematics. Oxford University Press, 1977.
  5. D. J. Velleman, How to prove it. A structural approach. Cambridge University Press, 1994.

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.