BSc Mathematics with an Integrated Foundation Year

The course unit aims to provide a basic course in various mathematical topics to students in Foundation Year.

On completion of the course the successful student will be able to:

- Calculate interpolating functions and values using linear and quadratic interpolation formulae and their inverses

- Select and apply numerical differentiation formulae to approximate derivatives, and discuss the accuracy of these approximations.

- Approximate integrals using the trapezium rule and Simpson's rule, and discuss the accuracy of these approximations.

- For each of circles centred on the origin, circles passing through the origin, straight lines, Archimedes and logarithmic spirals, ellipses, cardioids:
 --Given a curve represented by a written description, a polar-coordinate formula or a sketch, recall or derive the other two representations of the curve.
 --Recall how each of the parameters of the polar form of each curve relate to the characteristics of the plotted shape.
 --Scale and rotate the shape by multiplying the radius value and adding a constant to theta, respectively.
 --Convert between polar and Cartesian representations (points, lines, circles and ellipses only).

- Calculate the intersection points of polar curves, and use given intersection points to infer properties of the curves

- Recall the definitions of a root and the residual function.

- Apply the bisection method, the rule of false position and the Newton-Raphson method to solve scalar equations. For each method, explain the situations under which a root will be found and identify suitable initial values / intervals.

-Evaluate geometric integrals: the area inside a polar curve or between two polar curves; the volume of revolution of a curve (or between two curves) given in Cartesian form rotated about the x-axis; the volume of revolution of a curve rotated about the y-axis; the surface area of revolution of a curve given in Cartesian form about the x-axis; the arc-length of a curve given in either polar or Cartesian coordinates.

- Evaluate recurrence relations given a recurrence and initial term(s), and in simple cases evaluate later terms by identifying patterns in the sequence.

- Derive and apply reduction formulae to evaluate definite and indefinite integrals.

- Use the Mathematica language to achieve the above outcomes, including use of variables, functions and polar and Cartesian plots.

2: Numerical Interpolation: Linear Interpolation, Quadratic Interpolation.

1: Numerical Differentiation

2: Numerical Integration: The Trapezoidal Rule, Simpson's Rule.

3: Polar Coordinates: Polar coordinates of points. Polar coordinates of lines and curves. Points of intersection of polar curves

3: Numerical Solution of Equations: Bisection Method, Rule of False Position, Newton-Raphson method

3: Areas, lengths and volumes: Area inside a polar Curve, Volume of Solid of Revolution, Arc Length, Surface Area of Solid of Revolution

3: Recurrence Relations and Reduction Formulae: Recurrance relations, Reduction formulae [f(x)]^n with limits, reduction formulae x^n * f(x) with limits, reduction formulae without limits.

5: Mathematica: General introduction, application to topics in syllabus.

Coursework (Computerised Assignment) (week 7) Weighting within unit 10%

Mathematica Test (week 10 unless the project is submitted in week 10, in which case, the 0D2 coursework would be submitted in week 11.) Weighting within unit 10%

Examination (semester 2) Weighting within unit 80%

CROFT, A & DAVISON, R. 2010. Foundation Maths (5th ed.) Pearson Education, Harlow. (ISBN9780273730767)

BOSTOCK, L., & CHANDLER, S. 1981. Mathematics - the core course for A-level. Thornes, Cheltenham. (ISBN0859503062)

BOSTOCK, L., & CHANDLER, S. 1994. Core Maths for A-level (2nd ed.). Thornes, Cheltenham. (ISBN9780748717798)

STROUD, K.A, 2007. Engineering mathematics (6th ed.) Palgrave Macmillan, Baisingstoke. (ISBN9781403942463 / ISBN1403942463)

JAMES, G. 2001. Modern engineering mathematics (3rd ed.). Prentice Hall, Harlow. (ISBN0130183199 / ISBN9780130183194)

COOMBES, K. 1998. The Mathematica primer. Cambridge University Press, Cambridge. (ISBN0521631300 / ISBN0521637155  / ISBN9780521631303 / ISBN9780521637152)

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates