Module Specifications..
Current Academic Year 2023 - 2024
Please note that this information is subject to change.
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Repeat examination |
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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 | |||||||||||||||||||||||||||||||||||||||
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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 | |||||||||||||||||||||||||||||||||||||||
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 |
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Indicative Content and Learning Activities
Short introduction to logicStatements, Connectives, truth table and logical equivalence, predicate, quantifiers and the De Morgan lawsSets 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 setsMathematical proofsDirect proof, Proofs by contraposition, contradiction, construction/counterexample, by exhaustion, and by induction.Real number systemFields, Ordered, Well-Ordered and Complete sets Density of rationals and the Archimedean Principle.LimitsFormal definition and uniqueness of limits, operations on limits, Comparison/monotonicity and Squeeze theorems.ContinuityContinuity Definitions, Operations on continuous functions, Right- and left-continuity, discontinuities, Boundedness, Extreme value and intermediate value theorems.DifferentiabilityDefinitions, basic properties of differentiable functions, differentiation rules, Mean value theorem(s) and l'Hopital's rule and the Inverse Function Theorem.Applications of derivativesApplications of first order derivatives: monotonicity, critical points and local extrema. Higher order derivatives and their applications: Taylor's polynomials/theorem and concavity test. | |||||||||||||||||||||||||||||||||||||||
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Indicative Reading List
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Other Resources None | |||||||||||||||||||||||||||||||||||||||
Programme or List of Programmes
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