Undergraduate Programme and Module Handbook 2017-2018 (archived)
Module MATH1051: Analysis I
Department: Mathematical Sciences
MATH1051:
Analysis I
Type |
Open |
Level |
1 |
Credits |
20 |
Availability |
Available in 2017/18 |
Module Cap |
|
Location |
Durham
|
Prerequisites
- Normally grade A in A-Level Mathematics (or
equivalent).
Corequisites
- Calculus and Probability I (MATH1061) and Linear Algebra I
(MATH1071) -- note that for some students, this module may be taken as
Level 1 course in the second year, but that such students will have
taken Calculus and Probability I (MATH1061) and Linear Algebra I
(MATH1071) in their first year.
Excluded Combination of Modules
- Maths for Engineers and Scientists (MATH1551), Single
Mathematics A (MATH1561), Single Mathematics B (MATH1571).
Aims
- To provide an understanding of the real and complex number systems,
and to develop calculus of functions of a single variable from basic
principles using rigorous methods.
Content
- Numbers: real and complex number systems.
- sup and inf of subsets of R and of real valued
functions.
- Convergence of sequences: Examples, Basic
theorems.
- Bolzano-Weierstrass theorem.
- Convergence of series: Examples, tests for convergence,
absolute convergence, conditional convergence.
- Limits and Continuity: Functions of a real and complex
variable.
- Epsilon-delta definition of limit of a
function.
- Continuity.
- Basic theorems.
- Intermediate Value theorem.
- Differentiability: Definition.
- Differentiability implies continuity.
- Basic theorems.
- Proof of Rolle's theorem, Mean Value
theorem.
- Integration: Discussion of Riemann sums.
- Fundamental theorem of calculus.
- Basic theorems.
- Issues of convergence.
- Real and complex power series: Radius of convergence,
Basic theorems.
- Taylor series.
Learning Outcomes
- By the end of the module students will: be able to solve a
range of predictable or less predictable problems in
Analysis.
- have an awareness of the basic concepts of theoretical
mathematics in the field of Analysis.
- have a broad knowledge and basic understanding of these
subjects demonstrated through one or more of the following topic
areas: Numbers, supremum, infimum.
- Convergence of sequences and series.
- Limits, continuity, differentiation, integration.
- Real and complex power series.
- students will have basic mathematical skills in the following
areas: Spatial awareness, Abstract reasoning.
- students will have basic problem solving skills.
Modes of Teaching, Learning and Assessment and how these contribute to
the learning outcomes of the module
- Lectures demonstrate what is required to be learned and the
application of the theory to practical examples.
- Tutorials provide active engagement and feedback to the
learning process.
- Weekly homework problems provide formative assessment to guide
students in the correct development of their knowledge and skills. They
are also an aid in developing students' awareness of standards
required.
- The examination provides a final assessment of the achievement
of the student.
Teaching Methods and Learning Hours
Activity |
Number |
Frequency |
Duration |
Total/Hours |
|
Lectures |
47 |
2 per week in term 1, 2 or 3 per week in term 2 alternating with Problems Classes and collection examination, 5 revision lectures in term 3 |
1 Hour |
47 |
|
Tutorials |
15 |
Weekly in weeks 2-10,20,21 and fortnightly in weeks 12-19. |
1 Hour |
15 |
■ |
Problems Classes |
4 |
Fortnightly in weeks 12-19 |
1 Hour |
4 |
|
Preparation and Reading |
|
|
|
134 |
|
Total |
|
|
|
200 |
|
Summative Assessment
Component: Examination |
Component Weighting: 100% |
Element |
Length / duration |
Element Weighting |
Resit Opportunity |
Written examination |
3 hours |
100% |
Yes |
- One written assignment each teaching week.
Normally it will consist of solving problems from a Problem Sheet and
typically will be about 2 pages long. Students will have about one week to
complete each assignment. - 45 minute collection paper in the first week
of Epiphany term.
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University