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MA2509: ANALYSIS II (2020-2021)

Last modified: 13 Aug 2020 11:50


Course Overview

Analysis provides the rigorous, foundational underpinnings of calculus. This course builds on the foundations in Analysis I, and explores the notions of differential calculus, Riemann integrability, sequences of functions, and power series.

The techniques of careful rigorous argument seen in Analysis I will be further developed. Such techniques will be applied to solve problems that would otherwise be inaccessible. As in Analysis I, the emphasis of this course is on valid mathematical proofs and correct reasoning.

Course Details

Study Type Undergraduate Level 2
Session Second Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus Aberdeen Sustained Study No
Co-ordinators
  • Professor Benjamin Martin

Qualification Prerequisites

  • Programme Level 2

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

- Differentiation of functions of one variable: basic definitions and properties, chain rule, basic results on differentiable functions, Rolle's Theorem, Mean Value Theorem.

- Riemann integrability: Riemann sums, basic properties, the Fundamental Theorem of Calculus, improper integrals - Sequences of functions: pointwise convergence, uniform convergence, properties of limits of functions, series of functions

- Power series: convergence, continuity, differentiability, integrability, Taylor series

 

Syllabus

  • Differentiation: Definitions and properties, Standard rules for differentiation, Extrema, Mean value theorem, Monotonicity and Convexity.
  • Riemann integrability and the Riemann integrals.
  • Integrability of continuous functions; characterisations of integrability; properties of the integral.
  • Sequences and series of functions:
  • Pointwise and uniform convergence; examples of pointwise converging sequences with bad  behaviour (regarding continuity, differentiation, integration);
  • Theorems about good behaviour under uniform convergence; Weierstrass' M-test;
  • Dominated Convergence Theorem; pointwise limits of continuous functions.
  • Taylor series. Computing radius of convergence; uniform convergence of power series; Lagrange's form of the remainder.

 

Course Aims

To further develop understanding of the concepts, techniques, and tools of calculus. Calculus is the mathematical study of variation. This course emphasises differential and integral calculus, sequences and series of functions.

 

Learning Objectives

By the end of this course the student should:

  • be able to state the main definitions and theorems of the course;
  • be able to prove most results from the course;
  • be familiar with the concept of differentiability and understand theorems about differentiable functions;
  • understand Riemann integration and theorems about the Riemann integral;
  • be able to apply techniques for showing integrability or non-integrability of functions;
  • be able to distinguish between pointwise and uniform convergence of sequences of functions;
  • be able to compute Taylor series, compute the interval of convergence of power series, and use Taylor's theorem to estimate functions by polynomials.

Contact Teaching Time

Information on contact teaching time is available from the course guide.

Teaching Breakdown

  • 2 Lectures during University weeks 24 - 34
  • 1 Tutorial during University weeks 25 - 34

More Information about Week Numbers


In light of Covid-19 and the move to blended learning delivery the assessment information advertised for courses may be subject to change. All updates for first-half session courses will be actioned no later than 1700 (GMT) on 18 September 2020. All updates for second half-session courses will be actioned in advance of second half-session teaching starting. Please check back regularly for updates.

Summative Assessments

Homework

Assessment Type Summative Weighting 5
Assessment Weeks Feedback Weeks

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Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Homework

Assessment Type Summative Weighting 5
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Homework

Assessment Type Summative Weighting 5
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Exam

Assessment Type Summative Weighting 80
Assessment Weeks Feedback Weeks

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Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Homework

Assessment Type Summative Weighting 5
Assessment Weeks Feedback Weeks

Look up Week Numbers

Feedback
Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Formative Assessment

There are no assessments for this course.

Resit Assessments

Best of written exam (100%) or written exam (80%) with carried forward in-course assessment (20%)

Assessment Type Summative Weighting
Assessment Weeks Feedback Weeks

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Learning Outcomes
Knowledge LevelThinking SkillOutcome
Sorry, we don't have this information available just now. Please check the course guide on MyAberdeen or with the Course Coordinator

Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
FactualUnderstandbe able to state the main definitions and theorems of the course;
FactualApplyBe able to prove most results from the course;
ConceptualUnderstandbe familiar with the concept of differentiability and understand theorems about differentiable functions;
FactualUnderstandunderstand Riemann integration and theorems about the Riemann integral;
ConceptualApplyBe able to apply techniques for showing integrability or non-integrability of functions;
ConceptualApplyBe able to distinguish between pointwise and uniform convergence of sequences of functions
FactualAnalyseBe able to compute Taylor series, compute the interval of convergence of power series, and use Taylor's theorem to estimate functions by polynomials.

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