In this module we will study rigorously real functions and their properties, focussing in particular on continuity and differentiability. We will give a mathematical definition of limits at a point, continuity, the derivative and the Riemann integral. We will show how to derive rigorously many of the computational rules already used at A-level.

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.

Later in the module we will introduce the Taylor expansion, which allows us to approximate most mathematical functions by polynomials. We will then study the general properties of the Riemann integral in detail, followed by the demonstration of the techniques of integration.

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

Precursor Modules: MA1005 Foundations of Mathematics I

• 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;
• Understand analytic and geometrical properties of Riemann integral and compute the Riemann integral in several different contexts; compute improper integrals and integrals of discontinuous functions.

54 fifty-minute lectures

10 fifty-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 100 hours private study including preparation of worked solutions for tutorial classes.

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.

Transferable Skills:

The ability to think and argue clearly and precisely.

The ability to derive solutions of many problems starting from a few rigorous assumptions (Problem Solving)

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 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.  Section B has a choice of three from four equally weighted questions. 

• Functions and limits
  • Cluster points.
  • Definition of limits for functions (for bounded and unbounded functions in bounded and unbounded domains).
  • Sequential Criteria, algebra of limits and other properties.
  • One-side limits.
• Continuity
  • Definition of continuity.
  • Algebra of continuous functions. Continuity of elementary functions.
  • Behaviour of continuous functions over open and closed intervals. Intermediate value theorem.
  • The Weierstrass Extreme Value Theorem.
• Differentiation
  • Definition of 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.
  • Stationary points, local and global maximum/minimum points.
  • Theorems for differentiable functions. Mean Value Theorem.
  • Qualitative behaviour of functions using first derivatives: Increasing and decreasing functions.
  • Higher derivatives. Qualitative behaviour of functions using second derivatives. Stationary values and classification of local maxima and minima. Convexity, concavity, inflection points. Qualitative study for the graph of functions.
• Taylor expansion
  • Taylor’s Theorem with applications. Convergence of Taylor series using the remainder in the Taylor formula. Absolute ratio test and radius of convergence for power series.
•  Riemann integral
  • Definition of Riemann integral. Illustrative examples of the use of the definition. Geometric interpretation. Riemann integrability. Properties of the Riemann integral. Arc lengths.
  • Fundamental Theorem of Calculus and applications.
  • Integration by substitution. Integration by parts. Integration of rational functions and of rational functions of trigonometric functions.
  • Improper integrals. Integral test for convergence of series with non-negative terms.

Howie, J.M. 2000. Real analysis. Springer

Binmore, K.G. 1982. Mathematical analysis: a straightfoward approach. 2nd ed.  CUP

Courant, R. 1988. Differential and integral calculus. Wiley.

Clark, C.W. 1982. Elementary mathematical analysis. 2nd ed.  Brooks/Cole.