In this module we will study rigourously real functions and their properties, focussing in particular on continuity and differentiability. We will give a mathematical definition of limits at a point, continuity and derivative and show how to derive rigourously most of the computational rules already used in Calculus I.

Particular attention will be given to proving theorems for differentiable functions (as e.g. the Intermediate Value Theorem) and applications to maxima and minima, convexity and concavity. These tools can later be applied to qualitative study of functions and their graphs.

At the end of the module we will introduce the Taylor expansion, which allows to approximate most of mathematical functions by polynomials.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

Precursor Modules:  MA0123 Analysis I

• compute limits and derivatives of functions at a point, by using the rigourous mathematical definition;
• define continuity and differentiability for real-valued functions of a real variable;
• recognize the different kinds of discontinuity for functions;
• prove and use theorems for continuous functions and for differentiable functions, e.g. the Weierstrass Extreme Value Theorem and the Intermediate Value Theorem
• prove by only using the definitions algebra of limits and algebra of differentiation;
• describe increasing and decreasing functions in terms of their derivatives;
• find all local maximum-points and minimum-points of a given function and understand the role played by first and second derivatives;
• understand the relation between second derivatives and convexity of a function
• prove Taylor's Theorem and know the conditions under which a Taylor series is convergent;
• find the radius of convergence of a power series using the ratio test.

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills:

The ability to apply theorems from the course to the solution of problems requiring differential calculus.

The ability to construct rigorous mathematical arguments using only the properties available from the hypotheses, without importing extraneous material from one's intuition.

The ability to apply “epsilon-delta-definitions” in a wide range of problems.

Transferable Skills:

The ability to think and argue clearly and precisely.

The ability to derive solutions of many problems starting from a few rigourous assumptions.

A sound knowledge of differential calculus as required in most engineering, physical and financial applications.

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.  

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Continuity
  • Definition of continuity at a point and its illustration by simple examples. Sequences and continuity.
  • Algebra of continuous functions. Continuity of elementary functions. Behaviour of continuous functions over open and closed intervals. Intermediate value theorem. Piecewise continuity.

• Differentiation
  • Definition of a derivative. Illustrative examples of the use of the definition. Geometric interpretation. Continuity of a differentiable function.
  • Derivatives of elementary functions. Algebra of derivatives: sums, products and quotients. Chain rule.  Derivatives of inverse functions. Derivatives of parametric functions.
  • Qualitative behaviour of functions using first derivatives: Increasing and decreasing functions. Critical points. Local and global extreme values. Tangents and normals to curves. Mean value theorems.
  • Higher derivatives. Qualitative behaviour of functions using second derivatives. Stationary values and classification of local maxima and minima. Convexity, concavity, inflection points.

• Taylor and Maclaurin series
  • Taylor formula with remainder. Applications to simple numerical approximation. Convergence of Taylor series using the remainder in the Taylor formula. Absolute ratio test and radius of convergence for power series.

Please see Background Reading List for an indicative list.

Analysis 1, Tao, T., Hindustan Book Agency

Real Analysis, Howie, J.M., Springer

Calculus, Spivak, M., Benjamin Cummings

Mathematical Analysis, Binmore, K.G., Cambridge University Press

Numbers and Functions: Steps into Analysis, Burn, R.P., Cambridge University Press