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Module Specifications..

Current Academic Year 2023 - 2024

Please note that this information is subject to change.

Module Title Analysis 1
Module Code MS148
School School of Mathematical Sciences
Module Co-ordinatorSemester 1: Paul Razafimandimby
Semester 2: Paul Razafimandimby
Autumn: Paul Razafimandimby
Module TeachersPaul Razafimandimby
NFQ level 6 Credit Rating 10
Pre-requisite None
Co-requisite None
Compatibles None
Incompatibles None
Repeat examination
Description

This module ensures that students acquire a thorough comprehension of the fundamentals of real analysis, in particular, differential calculus for numerical functions. The module also aims to support students' development of logical, computational and mathematical problem solving and writing skills, and to prepare students for later courses in mathematical analysis

Learning Outcomes

1. Acquire and use the basics of mathematical logic and proofs, to formulate state and prove/disprove mathematical propositions
2. Define a set, know the different operations and sets and their properties. and make the differences between finite, countable and uncountable sets.
3. Define a relation and a function and use them to solve mathematical problems. Know the definitions of injection, surjection and bijection.
4. Describe the fundamental properties of the set of real numbers that are of the essence for the rigorous development of real analysis.
5. Know the formal definitions and properties of the limits of functions at a point and at infinity and compute the limits of a variety of functions.
6. Use the notion of limit to define the continuity of a function, determine the continuity of a variety of functions and apply theorems for continuous functions in a variety of settings.
7. Differentiate a variety of functions and apply derivatives in a variety of settings



Workload Full-time hours per semester
Type Hours Description
Lecture40Lectures on course contents
Tutorial20Weekly tutorials
Independent Study190No Description
Total Workload: 250

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Short introduction to logic
Statements, Connectives, truth table and logical equivalence, predicate, quantifiers and the De Morgan laws

Sets and Functions
* Sets: Definitions, Operations on sets and their properties, Venn Diagram. * Functions: relations, functions, injection/surjection/bijection * Examples of functions: Polynomials, Rationals, Power functions, Logarithmic and exponential functions. * Finite, countable and uncountable sets

Mathematical proofs
Direct proof, Proofs by contraposition, contradiction, construction/counterexample, by exhaustion, and by induction.

Real number system
Fields, Ordered, Well-Ordered and Complete sets Density of rationals and the Archimedean Principle.

Limits
Formal definition and uniqueness of limits, operations on limits, Comparison/monotonicity and Squeeze theorems.

Continuity
Continuity Definitions, Operations on continuous functions, Right- and left-continuity, discontinuities, Boundedness, Extreme value and intermediate value theorems.

Differentiability
Definitions, basic properties of differentiable functions, differentiation rules, Mean value theorem(s) and l'Hopital's rule and the Inverse Function Theorem.

Applications of derivatives
Applications of first order derivatives: monotonicity, critical points and local extrema. Higher order derivatives and their applications: Taylor's polynomials/theorem and concavity test.

Assessment Breakdown
Continuous Assessment25% Examination Weight75%
Course Work Breakdown
TypeDescription% of totalAssessment Date
In Class Test2 Class test or 2 Homeworks25%n/a
Reassessment Requirement Type
Resit arrangements are explained by the following categories;
1 = A resit is available for all components of the module
2 = No resit is available for 100% continuous assessment module
3 = No resit is available for the continuous assessment component
This module is category 1
Indicative Reading List

  • W. Rudin: 1976, Principles of Mathematical Analysis, 3rd, McGraw-Hill Book Co,
  • W. R. Wade: 2010, Introduction to Analysis, 4th, Pearson,
Other Resources

None
Programme or List of Programmes
ACMBSc in Actuarial Mathematics
CAFMCommon Entry, Actuarial, Financial Maths
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