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Mathematics is one of the oldest and most universal means of creating, communicating, connecting and applying structural and quantitative ideas. The discipline of Mathematics allows the formulation and solution of real-world problems as well as the creation of new mathematical ideas, both as an intellectual end in itself, as well as a means to increase the success and generality of mathematical applications. This success can be measured by the quantum leap that occurs in the progress made in other traditional disciplines once mathematics is introduced to describe and analyse the problems studied. It is therefore essential that as many persons as possible be taught not only to be able to use mathematics, but also to understand it.
The syllabus aims to:
- provide understanding of mathematical concepts and structures, their development and the relationships between them;
- enable the development of skills in the use of mathematical and information, communication and technology (ICT) tools;
- develop an appreciation of the idea of mathematical proof, the internal logical coherence of Mathematics, and its consequent universal applicability;
- develop the ability to make connections between distinct concepts in Mathematics, and between mathematical ideas and those pertaining to other disciplines;
- develop a spirit of mathematical curiosity and creativity, as well as a sense of enjoyment;
- enable the analysis, abstraction and generalisation of mathematical ideas;
- develop in students the skills of recognising essential aspects of concrete, real-world problems, formulating these problems into relevant and solvable mathematical problems and mathematical modelling;
- develop the ability of students to carry out independent or group work on tasks involving mathematical modelling;
- integrate ICT tools and skills;
- provide students with access to more advanced courses in Mathematics and its applications at tertiary institutions.
Reasoning and Logic MODULE 1: BASIC ALGEBRA AND FUNCTIONSOn completion of this Module, students should: 1.develop the ability to construct simple proofs of mathematical assertions; 2.understand the concept of a function; 3.be confident in the manipulation of algebraic expressions and the solutions of equations and inequalities; 4.understand the properties and significance of the exponential and logarithm functions; 5.develop the ability to use concepts to model and solve real-world problems.
The Real Number System –ℝ
- perform binary operations;
- use the concepts of identity, closure, inverse, commutativity, associativity, distributivity addition, multiplication and other simple binary operations;
- perform operations involving surds;
- construct simple proofs, specifically direct proofs, or proof by the use of counter examples;
- use the summation notation;
- establish simple proofs by using the principle of mathematical induction.
Exponential and Logarithmic Functions
- define an exponential function;
- sketch the graph of y = ax ;
- define a logarithmic function as the inverse of an exponential function;
- define the exponential functions y = ex and its inverse y = ln x, wherelnx ≡ logex;
- use the fact that y = ln x x = ey ;
- simplify expressions by using laws of logarithms;
- use logarithms to solve equations of the form ax = b;
- solve problems involving changing of the base of a logarithm.
- define mathematically the terms: function, domain, range, one-to-one function (injective function), onto function (surjective function), many-to-one, one-to-one and onto function (bijective function), composition and inverse of functions;
- prove whether or not a given simple function is one-to-one or onto and if its inverse exists;
- use the fact that a function may be defined as a set of ordered pairs;
- use the fact that if g is the inverse function of f, thenf[g (x)] x, for all x, in the domain of g;
- illustrate by means of graphs, the relationship between the functiony = f (x) given in graphical form and y f (x)and the inverse of f (x), that is,y f-1(x).