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MATH1051: Analysis I

Please ensure you check the module availability box for each module outline, as not all modules will run in each academic year. Each module description relates to the year indicated in the module availability box, and this may change from year to year, due to, for example: changing staff expertise, disciplinary developments, the requirements of external bodies and partners, and student feedback. Current modules are subject to change in light of the ongoing disruption caused by Covid-19.

Type Open
Level 1
Credits 20
Availability Available in 2023/24
Module Cap
Location Durham
Department Mathematical Sciences

Prerequisites

  • Normally grade A in A-Level Mathematics (or equivalent).

Corequisites

  • Calculus I (MATH1061) and Linear Algebra I(MATH1071) OR Calculus I (Maths Hons) (MATH1081) and Linear Algebra I (Maths Hons) MATH1091). Note that for some students, this module may be taken as Level 1 course in the second year, but that such students will havetaken Calculus I (MATH1061) and Linear Algebra I (MATH1071) OR Calculus I (Maths Hons) (MATH1081) and Linear Algebra I (Maths Hons) (MATH1091) in their first year.

Excluded Combinations of Modules

  • Maths for Engineers and Scientists (MATH1551), SingleMathematics 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 valuedfunctions.
  • Convergence of sequences: Examples, Basictheorems.
  • Bolzano-Weierstrass theorem.
  • Convergence of series: Examples, tests for convergence,absolute convergence, conditional convergence.
  • Limits and Continuity: Functions of a real and complexvariable.
  • Epsilon-delta definition of limit of afunction.
  • Continuity.
  • Basic theorems on continuity.
  • Intermediate Value theorem.
  • Differentiability: Definition.
  • Differentiability implies continuity.
  • Basic theorems on differentiability.
  • Proof of Rolle's theorem, Mean Valuetheorem.
  • Step & regulated functions.
  • Integration for regulated functions.
  • Fundamental theorem of calculus.
  • Basic theorems on integration.
  • Pointwise and uniform convergence.
  • Limit theorems.
  • Real (and complex) power series: Radius of convergence,Basic theorems.
  • Taylor series.

Learning Outcomes

Subject-specific Knowledge:

  • By the end of the module students will: be able to solve arange of predictable or less predictable problems inAnalysis.
  • have an awareness of the basic concepts of theoreticalmathematics in the field of Analysis.
  • have a broad knowledge and basic understanding of thesesubjects demonstrated through one or more of the following topicareas: Numbers, supremum, infimum.
  • Convergence of sequences and series.
  • Limits, continuity, differentiation, integration.
  • Real and complex power series.

Subject-specific Skills:

  • students will have basic mathematical skills in the followingareas: Spatial awareness, Abstract reasoning.

Key Skills:

  • students will have basic problem solving skills.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

  • Tutorials provide active engagement and feedback to thelearning process.
  • Lectures demonstrate what is required to be learned and theapplication of the theory to practical examples.
  • Weekly homework problems provide formative assessment to guidestudents in the correct development of their knowledge and skills. Theyare also an aid in developing students' awareness of standardsrequired.
  • The examination provides a final assessment of the achievementof the student.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures472 per week in term 1, 2 or 3 per week in term 2 alternating with Problems Classes and collection examination, 2 revision lectures in term 31 Hour47 
Tutorials14Weekly in weeks 2-10, fortnightly in weeks 13-19, and one in week 21.1 Hour14Yes
Problems Classes4Fortnightly in weeks 14-201 Hour4 
Preparation and Reading135 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 90%
ElementLength / DurationElement WeightingResit Opportunity
Written examination3 hours100Yes
Component: Continuous AssessmentComponent Weighting: 10%
ElementLength / DurationElement WeightingResit Opportunity
Weekly written assignments during the first 2 terms. Normally, each will consist of solving problems and will typically be one to two pages long. Students will have about one week to complete each assignment.100Yes

Formative Assessment

45 minute collection paper in the beginning of Epiphany term.

More information

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