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This subject develops fundamental concepts and principles in mathematical analysis. Students should gain skills in the practical techniques of differential calculus, integral calculus and infinite series, and study selected applications of these techniques in mathematical modelling.
Topics covered rigorous discussion of limits of sequences and of real-valued functions, continuity and differentiability; Mean Value Theorem and applications; Taylor polynomials; Riemann integration, techniques of integration and applications, improper integrals; infinite series, with applications to power series representations of elementary functions and their generation by Taylor series and to the representation of periodic functions by Fourier series; first order differential equations, second order linear differential equations with constant coefficients and selected applications.
Intended learning outcomes
On completion of this subject, students should be able to:
- Explain the properties of sequences of real numbers;
- Identify the properties of a function of a real variable, such as limits, continuity and differentiability;
- Explain when proper and improper Riemann integrals exist and be able to use standard techniques to evaluate them;
- Determine the convergence and divergence of infinite series and to represent functions by Taylor series and Fourier series;
- Solve first and second order ordinary differential equations, and use these equations to model some simple physical systems;
- Describe simple rigorous proofs of fundamental results in real analysis.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team; and
- time management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 6 December 2023