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The mathematics course will provide you with an in-depth introduction to pure mathematics.
If you decide to study mathematics on the UPCSE course, it is important that you are fully familiar with the topics listed in the “Essential Background Knowledge in Mathematics” section outlined below. This may involve the need for private study and/or revision over the summer vacation before the UPCSE course begins.
External examiners have stated that the standards
reached by students are very high. In Mathematics and in Physics, the
quality of work was in many cases in excess of high-grade A-level.
Essential Background Knowledge in Mathematics
You should be completely familiar with the arithmetic of the integers, of fractions and of real numbers, laws of indices for integer and rational exponents, and the manipulation of surds. You need to be able to solve linear equations in two variables, and should be familiar with geometric interpretations of these. You also need to be competent with manipulation of algebraic expressions and the solution of quadratic equations by completing the square, solution by discriminant, and the graph of a quadratic.
You should have knowledge of the elementary trigonometric functions of sin/cos/tan. You should be able to solve problems using trigonometry and be able to use the sine and cosine rules appropriately. You should be familiar with measuring angles in degrees and radians.
You should be familiar with Pythagoras’s Theorem, with the solution of problems involving ratio and proportion (including similar triangles) and links between length, area and volume of similar figures. You need also to be familiar with the perimeter and area of a circle, and the volume of a cone and a sphere, the properties of a circle, and co-ordinate geometry and elementary 2D vectors.
Key Topics in Mathematics
A general outline of the topics that will be covered is given below.
Algebraic identities, inequalities and functions; partial fractions; quadratic equations; logarithms; remainder theorem; Pascal’s triangle; arithmetic and geometric series and their sums to n terms and sum to infinity of the convergent geometric series.
Mappings; domains and ranges; exponential and log functions; inverse functions; representing a function as a curve; curve sketching and even/odd/periodic functions; finding zeros, asymptotes, symmetries, maxima and minima of function. The modulus function.
Trigonometric functions and identities
Sine and cosine rules; trigonometric functions, their relationships and identities; graphical representations; periodic properties and symmetries of trigonometric functions; solution to trigonometric equations; hyperbolic functions and their identities.
To cover both differentiation and integration. Work on differentiation will include geometrical interpretation, derivatives of standard functions, differentiation of the sum, product and quotient of functions, derivatives of simple functions defined implicitly or parametrically. The derivative of the composition of two functions and its applications, along with applications to gradients, tangents, normals, maxima and minima.
Work on integration will include geometric interpretation as area under a curve, the fundamental Theorem of Calculus. Also covered will be the integration of standard functions, techniques of integration, evaluation of definite integrals, evaluation of areas under a curve or between two curves, and numerical appropriations of definite integrals
Work on vectors will focus on 2D and 3D vectors, algebraic properties of addition, scalar multiplication and their geometrical properties; Distance between two points; equations of lines and planes; Direction Ratios and direction cosines; Scalar and vector products.
Imaginary numbers; algebraic properties of complex numbers; complex roots of quadratic equations; argand diagrams and modulus/argument form of complex numbers; cube and nth roots of unity; DeMoivre’s theorem; exponential form of complex numbers; relationships between hyperbolic, trigonometric and exponential functions.
Column vectors; general matrix arithmetic; transformations in 2D; determinants; inverse matrices; solution to simultaneous equations; basic gaussian elimination.
Numerical methods for solving integration problems: Trapezium and Simpson’s rule, small increments and rates of change; numerical solutions to algebraic functions: graphical methods, interval bisection.
First order differential equations and integrating factor; second order linear differential equations with constant coefficients, general solutions and particular integrals.
Page last modified on 18 sep 13 13:53 by Martin L White