General description
This module continues the thread of MA12610 with optimisation of functions of several variables and Lagrange Multipliers. It also includes basic matrix techniques and simple differential and difference equations. Like MA12610, it will be illustrated with applications to Economics.

Aims
To introduce the basic Mathematical techniques required for degrees involving Economics or Accounting.

Learning outcomes
On completion of this module, a student should be able to:

• find the first and second order partial derivatives of a function of 2 or 3 variables;
• optimize a function of 2 variables using either the substitution method or Lagrange multipliers;
• perform basic matrix algebra, find the inverse of 2 by 2 or 3 by 3 matrices and use inverses to solve equations;
• evaluate 2 by 2 and 3 by 3 determinants and apply Cramer's rule to solve equations;
• reduce a matrix to echelon form, find its rank and solve associated systems of equations;
• integrate simple polynomial, rational and exponential functions;
• calculate the area under a curve;
• find the total cost function given any marginal cost function;
• find the total revenue function given any marginal revenue function;
• find the consumption and savings functions given either the marginal propensity to consume or the marginal propensity to save.
• solve first order linear difference / differential equations;
• solve differential equations by separation of variables;
• analyse the stability of the solutions of difference / differential equations.

Syllabus
1. FURTHER OPTIMIZATION: Constrained optimization in functions of one variable, the interpretation of Lagrange multipliers, unconstrained optimization in functions of more than one variable, constrained optimization in functions of more than one variable.
2. MATRIX ALGEBRA: Simultaneous linear equations, Gaussian reduction, laws of matrix algebra, identity and null matrices, matrix addition and subtraction, scalar multiplica-tion, matrix multiplication, matrix inversion, determinants, determinants and matrix non-singularity, higher-order determinants, matrix inversion using determinants, Cramer's Rule, matrix rank.
3. INTEGRATION: The notion of an integral. Area under a curve. Integration of simple power and exponential functions. Integration of marginal functions.
4. DYNAMICS: First-order differential equations, separation of variables, first-order difference equations, stability.