Outline Description of Module:

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 cluster points, limits, continuity, derivatives and the definite 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 continuous functions (e.g. the Intermediate Value Theorem and the Weierstrass Theorem), for differentiable functions (e.g. the Mean Value Theorem) and applications to maxima and minima, monotonicity, 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 definite 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 cluster points and limits;
• 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, the Intermediate Value Theorem and the Mean Value Theorem
• prove by only using the definitions algebra of limits, algebra of continuous functions 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;

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

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

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.

Abstract mathematical reasoning.

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.

• • Functions and limits
    • Cluster points.
    • Definition of limits for functions (including the case of infinite limits and limits at infinity).
    • Sequential Criteria, Squeeze Theorem, 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.
    • The 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.
    • Monotone functions
    • Higher derivatives. Stationary values and classification of local maxima and minima. Convexity, concavity, inflection points.
  • Taylor expansion
    • Taylor’s Theorem with applications.
  •  Definite integral
    • Definition of the definite integral. Illustrative examples of the use of the definition. Geometric interpretation.
    • Fundamental Theorem of Calculus and applications.
    • Integration by substitution. Integration by parts. Integration of rational functions and of rational functions of trigonometric functions.
    • Properties of the definite integral. Integration of piecewise functions
    • Improper integrals.

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