Mathematics for business degrees, tutored in English

Linear algebra: matrices, determinants and systems

Matrix operations, the determinant and what a zero determinant tells you, the inverse, and rank. Then solving systems of linear equations with Gaussian elimination and Cramer's rule, and the Rouche-Frobenius theorem for classifying a system as consistent with a unique solution, consistent with infinitely many, or inconsistent. We finish with eigenvalues, eigenvectors and diagonalisation, which come back later in the dynamic models.

Calculus in one and several variables

Limits, continuity and differentiation, then the applications the module actually examines: elasticity, marginal analysis and curve sketching. In several variables, partial derivatives, the gradient, the chain rule, and total differentials for approximating a change. We cover homogeneous functions and Euler's theorem, because they appear in the production function questions, and integration for the consumer and producer surplus problems.

Unconstrained optimisation

First-order conditions to find the critical points, then second-order conditions to classify them. In one variable that is the second derivative; in several it is the Hessian matrix and the sign of its leading principal minors. We work carefully on the classification step, because finding the point is usually the easy half and losing the marks on whether it is a maximum, a minimum or a saddle point is avoidable.

Constrained optimisation: Lagrange and Kuhn-Tucker

The Lagrangian method for equality constraints, the first-order conditions, and the economic meaning of the multiplier as the shadow price of relaxing the constraint by one unit. Then the bordered Hessian for the second-order condition. For inequality constraints we cover the Kuhn-Tucker conditions and complementary slackness, which is the part people find strangest and which becomes clear once you work three or four cases.

Differential and difference equations

First-order linear differential equations and separable equations, with the applications to continuous growth models. Difference equations for discrete-time models, solving the homogeneous and particular parts. The two have different stability criteria and confusing them is a common source of lost marks: a difference equation is stable when the modulus of the root is below one, whereas a differential equation is stable when the real part of the root is negative.

How the lessons work

One to one, by video call, on a shared whiteboard where we work the problems out together. There is no fixed timetable and no minimum number of sessions. Send me your notes and past papers first: maths notation varies more between courses than any other subject, and we use yours.