Students must normally have A-level Mathematics, Grade A, or equivalent, in order to undertake this unit.
Aims & Learning Objectives:
Aims: To teach the basic notions of analytic geometry and the analysis of functions of a real variable at a level accessible to students with a good 'A' Level in Mathematics. At the end of the course the students should be ready to receive a first rigorous analysis course on these topics.
The students should be able to manipulate inequalities, classify conic sections, analyse and sketch functions defined by formulae, understand and formally manipulate the notions of limit, continuity and differentiability and compute derivatives and Taylor polynomials of functions.
Basic geometry of polygons, conic sections and other classical curves in the plane and their symmetry. Parametric representation of curves and surfaces.
Review of differentiation: product, quotient, function-of-a-function rules and Leibniz rule.
Maxima, minima, points of inflection, radius of curvature. Graphs as geometrical interpretation of functions. Monotone functions. Injectivity, surjectivity, bijectivity.
Inequalities. Arithmetic manipulation and geometric representation of inequalities.
Functions as formulae, natural domain, codomain, etc. Real valued functions and graphs.
Orders of magnitude. Taylor's Series and Taylor polynomials - the error term. Differentiation of Taylor series.
Taylor Series for exp, log, sin etc. Orders of growth.
Orthogonal and tangential curves.