3.2.3 FOUNDATION MATHEMATICS – MATH1641 (41 lectures)

Dr S. Borgan

This is a Mathematics course intended for students without an A level in mathematics, but whohave at least a grade C in GSCE or AS-level mathematics. The aim of the course is to bring thesestudents to A-level standard. For students successfully completing the module, the Department ofMathematical Sciences will issue a certificate stating that, in its opinion, the student has attainedA-level standard, with a grade corresponding to the mark obtained in the Foundation examination.

For students who reach a suitable standard in Foundation Mathematics, the Department will sup-port a request that they be permitted to register, in their second year, for level 1 mathematicsmodules normally requiring A-level mathematics as a prerequisite. In particular, such studentsmight be able to take the double module Core Mathematics A in their second year, which in turnwould enable them to take appropriate level 2 mathematics modules in their third year of study.

There will be three hours per week of lectures and tutorials and students will be required to submitcoursework throughout the year. It is very important that lectures and tutorials attended and thatcoursework is done conscientiously. The final mark for the module is based on a three-hour exam-ination (worth 90%) taken in May or early June and an assessment of coursework (worth 10%).There will also be a compulsory Collection examination in January.

Recommended Books

The notes given in lectures and tutorials should be self-contained but you may find it helpful toconsult the books below.

Dexter J. Booth, Foundation Mathematics, 2nd Edition, Addison-Wesley, 1993L. Bostock and S. Chandler, Core Mathematics for A-level, S. Thomas, 1990Anthony Croft and Robert Davidson, Foundation Maths, 3rd edition, Prentice Hall

Calculators

Approved electronic calculators are allowed in the examinations.

Outline of course Foundation Mathematics

Term 1 (20 lectures)

Differentiation: Functions, graphs and the gradient of a straight line. Derivative as the gradientof a curve. Differentiation from first principles. The rule d

dxxn = nxn−1. Differentiation of sumsand differences. The chain rule. Differentiation of products and quotients. Higher derivatives.Stationary points of a curve. Velocity and acceleration.

Elementary Integration: Integration as the reverse of differentiation:R

xn dx = xn+1

n+1 +c. Indefiniteintegrals. Area under a curve. Definite integrals.

Logarithmic and Exponential Functions: Definition of lnx as the integral of 1x . Derivative of

lnx. Graph of lnx. The rules ln(ab) = lna + lnb etc. Definition of e. Exponential function ex asthe inverse of lnx. Graph of ex. Derivative and integral of ex. The rules ea+b = ea eb etc. Examplesof exponential growth and decay.

Methods of Integration and Differential Equations: Integration by substitution. Integration byparts. First order separable differential equations with applications to chemical reactions.

Terms 2 & 3 (21 lectures)

Trigonometrical Functions: Radian measure. Sine, cosine and tangent of any angle. Properties ofthe trigonometrical functions. Identities: sin2 A + cos2 A = 1, addition formulae and double angleformulae. Polar coordinates. Examples of curves given in terms of polar coordinates. Differen-tiation and integration of the trigonometrical functions. Derivatives of the inverse trigonometricalfunctions.

Algebra: Algebraic manipulation. Solution of simultaneous linear equations in two unknowns.Matrix algebra: mainly 2x2 matrices, application to solution of simultaneous equations, additionand multiplication of matrices, identity matrix, transpose and symmetric matrices, Hermitian ma-trices, connection with geometry. Solution of quadratic equations with a brief treatment of complexnumbers and complex conjugates. Arithmetic and geometric series. Sum of an infinite geometricseries, idea of convergence of an infinite series. Binomial series of (x + y)n where n is rational,applications to numerical approximations.

Coordinate Geometry in the Plane: Cartesian coordinates in the plane. Distance between twopoints and Pythagoras’ theorem. Equation of a straight line. Intersection of two lines. Parallellines, perpendicular lines, perpendicular bisectors. Vectors in two and three dimensions, magnitudeof a vector, algebraic operations of vector addition and scalar multiplication and their geometricalinterpretations, position vectors, the distance between two points, vector equations of lines andplanes, the scalar product and the angle between two lines, connection with matrices.

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