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Durham University

Department of Mathematical Sciences

Brush Up Your Skills (1H Support Classes)

Because of widening access, a broadening A-level syllabus and differences in the syllabuses of different boards, we facilitate revision and consolidation of the key skills required to embark on a mathematics degree through the Brush Up Your Skills course. The course covers material that most students will have seen at A-level, but as well as revision, the course is intended to cover any gaps there may be in any particular combination of A-level modules.

The course consists of 2 problems classes per week which complement the level 1 core modules. Attendance is not compulsory but is initially advised on the basis of a diagnostic test administered to all students at the beginning of the first term. The course is voluntary and does not form part of the degree, so students may attend only those sessions that deal with subjects where they feel weak. This facility is intended to help students take control of their own learning, recognize area of weakness and use the resources available to improve them. It is the first step on the road to becoming an independent learner.

More information, including the summer workbook, information about the diagnostic test, and timetabling, please refer to the BUYS home page.

Outline of Course

Aim: The Brush Up Your Skills course covers basic pre-calculus topics and broadly follows the Linear Algebra I and Calculus & Probability I syllabuses; most classes are led by questions posed by the students or suggested by the 1H lecturers so topics in other 1H courses (e.g. Analysis I and Programming & Dynamics I) are also addressed.

  • Basics: number systems, basic manipulation,quadratic equations, polynomials, partial fractions, linear and non-linear inequalities, exponents and logarithms, topics in discrete mathematics.
  • Functions: definition, domain and range, graphs, linear and quadratic functions, composition, inverse, modulus function, hyperbolic functions.
  • Coordinate Geometry: equations and properties of straightlines, general equation of circle, centre and radius, Cartesian and parametric equations of curves.
  • Trigonometry: trigonometric functions and identities, inverse trigonometric functions, solution of trigonometric equations.
  • Differentiation: definition and properties, interpretation as slope, chain rule, sum, product and quotient rules, simple functions defined implicitly or parametrically, maxima and minima, Taylor and Fourier series, differential equations.
  • Integration: basic definition, as inverse of differentiation,as area under curve, integration methods, definite integrals, multiple integration.
  • Vectors: definition, basic properties and operations, magnitude, dot and cross products, vectorial geometry.
  • Matrices: definition, basic properties and operations, inverse, determinants.
  • Probability: permutations and combinations, set theory, Venn diagrams, calculus of probabilities,random variables, discrete and continuous distributions, moments, inequalities, approximations, law of large numbers.

Reading List

  • Salas, S., Hille, E., Etgen, G., Calculus: One and Several Variables, J. Wiley & Sons, 10th ed., 2007
  • Anton, H., Elementary Linear Algebra, Wiley, 9th ed., 2005
  • DeGroot, M. H., Schervish, M. J., Probability and Statistics, Pearson, 3rd ed., 2003