This module provides a course on differential calculus and an introduction to integral calculus with applications for students likely specialize in mathematics, actuarial mathematics or statistics. The module builds on what the students learned at school but provides a greater depth of study and introduces new material and concepts.
1. Definition of function, domain and range; definition of injective, surjective and invertible function
2. Concept of limit of a functions; limit laws; calculation of limits via various standard techniques (e.g. recognise indeterminate forms and calculate indeterminate limits both by reducing them to determinate forms or by using de l’Hopital’s theorem)
3. Definition of continuous function and of differentiable function: state the definitions of continuous function and differentiable function and apply them to determine whether a function is continuous/differentiable
4. Mean value theorem
5. List and sketch standard functions (trigonometric, exponential, logarithms, polynomials, inverse trigonometric functions) and state their properties in terms of continuity, invertibility and differentiability
6. Determine the derivative of functions from first principles; list and apply rules of differentiation
7. Analyse functions and sketch their graphs: state and apply the definition of singular and critical points and their relation to global and local maxima and minima; determine global and local maxima and minima, find intercepts with x and y axis, intervals
8. Analyse and explain mathematical content clearly: independent learning of mathematical content and technical writing skills, as appropriate for first year level
By the end of the course, students should be able to do the following:
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SCQF Level: 7
Credits: 15