The profit-maximisation problem (for production of one good) is introduced as motivation for the development of general optimisation techniques. The general concept of a critical (or stationary) point is presented, together with the method for finding such points and classifying their nature in two different ways: by examining the sign of the derivative around the point and by using the second-derivative test. Optimisation on intervals and infinite intervals is then discussed (where the end-points must be taken into consideration). Additional economic and financial applications are given.

Matrices are introduced and it is explained how matrix addition, scalar multiplication and multiplication of two matrices works. As an example of matrix multiplication, it is demonstrated how investment portfolios can be modelled, and their returns in various states quantified by multiplication with a returns matrix. The concept of an arbitrage portfolio is explained.

This chapter introduces (with production function as an example) functions of more than one variable. Then we define partial and second partial derivatives and explain how to calculate them, and present the chain rule for partial differentiation.

Important special functions and their properties are described. In particular, the exponential function and its connection to continuous compounding is discussed, together with the logarithm and trigonometrical functions. It is explained how one can interpret not just integer, but rational and then irrational powers of positive real numbers. The derivatives of exponential, logarithmic and trigonometrical functions are studied.

The standard integrals of the previous chapter are of fairly limited use, so this chapter develops some much more widely applicable techniques. These are integration by substitution, integration by parts and integration by partial fractions.

The general matrix formulation of a system of linear equations is described. It is explained that a system may have no, one or infinitely many solutions. We begin to describe a general approach to solving such systems by performing row operations on the augmented matrix in order to reduce this to echelon form. This chapter gives examples in the case where the system has a unique solution. (The next chapter considers other cases.)

This is a discussion of sequences and first-order recurrence (or difference) equations and the behaviour of the solutions to such equations. It contains some economic applications.

We give a discussion of some basic mathematical concepts and terminologies connected with sets, numbers, functions and graphs.

We start by introducing the key ingredients in macroeconomic modelling: investment, production, income and consumption, and explain the corresponding equilibrium conditions. Modelling these quantities in discrete time, we describe the multiplier-accelerator model, a classic model of macroeconomic dynamics, and an example of a second-order recurrence equation. We then embark on describing how to solve linear constant-coefficient second-order recurrence equations in general. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. There is a general method for determining the solution of the homogeneous equation, involving the solution of a corresponding quadratic equation known as the auxiliary equation.

It is explained what is meant by a function defined implicitly and how the derivative of an implicitly defined function can be determined via partial differentiation. The general concept of the contour of a two-variable function is presented, together with the special case of this when the function is a production function, and the contours are known as isoquants. It is explained how the slopes of contours can be determined. Then, the concept of homogeneous functions and the connected economic interpretation of returns to scale are considered, along with Euler's Theorem and its economic interpretation in terms of marginal product of labour and marginal product of capital.

Basic concepts in finance are introduced and modelled via first-order recurrence equations. In particular, we discuss compound interest, present value and the present value of an annuity.

The concept of consumer surplus is introduced and this motivates the problem of determining the area under the graph of a function. We indicate the connection between this problem and anti-derivatives (or integrals), defining what we mean by a definite integral. We illustrate with some examples after developing a repertoire of standard integrals.

This chapter introduces the important idea of a vector through the example of bundles of goods. The dot product of two vectors is defined and it is shown how a budget constraint can be expressed in terms of dot product. It is explained how, in order to rank bundles according to a particular consumer's preference, we can use a utility function. Indifference curves are defined as the contours of the utility function. Linear and convex combinations and the concept of a convex set are explained. The utility maximisation problem -- to maximise utility subject to a budget constraint -- is explored and the relevance of convexity is emphasised.

Regarding the key components of macroeconomics as continuous (rather than discrete) and considering the corresponding dynamics leads to differential equations. In this chapter, we focus on first-order differential equations and consider two methods (applicable to certain types of equation): separation and the use of integrating factors. We look at economic applications to continuous-time price adjustment and continuous cash flows.

A mathematical discrete-time population model is presented, which leads to a system of two interlinked, or coupled, recurrence equations. We then turn to the general issue of how to solve such systems. One approach is to reduce the two coupled equations to a single second-order equation and solve using the techniques already developed, but there is another more sophisticated way. To this end, we introduce eigenvalues and eigenvectors, show how to find them and explain how they can be used to diagonalise a matrix.