Function theory: limits, continuity, implicitely defined functions, review of

inverse function theory.

Differentiation: Definition of the derivative, examples; the derivative of a

sum, difference, product, and quotient of two functions; derivatives of

polynomials, the trigonometric functions, logs, exponentials, and the

inverse trigonometric functions; higher-order derivatives, first-order

separable differential equations.

Applications of the derivative: Local maxima and minima, the second-

derivative test; global maxima and minima; maximization on a closed

interval; curve sketching.

The definite integral: Definition of  the integral, examples; The

Fundamental Theorem of Calculus; antiderivatives; u-du substitutions;

integration by parts; changes of variable for the definite integral.

                 

Applications of the integral: Volumes by cross sections and cylindrical

shells; arc-length; surface areas of revolution.

Sequences and series: criteria for convergence; techniques of integration, the Fundamental Theorem of Calculus; properties of differentiable functions; Taylor series; ordinary differential equations; an introduction to partial derivatives; parametric representation of curves.

Elements of set theory: elements of proof theory, relations and functions; Groups, including fine permutation groups; Rings and the Euclidean algorithm; homomorphisms; fields.

Sequences: Convergence, limit theorems, monotone sequences, Cauchy sequences.

 

Continuity: limits and limit laws; continuity; the intermediate value  theorem; uniform continuity.

 

Differentiability: The derivative and its properties; Rolle's theorem, the Mean-Value theorem.

 

Integration: Introduction to the theory of Riemann integral; Riemann sums; the Fundamental Theorem of Calculus; improper integrals;  functions defined by integrals.

 

Series: Comparison, ratio, root, etc., tests; absolute convergence;  alternating series; Cauchy criterion for convergence.

 

 Series of functions: Uniform convergence of sequences and series of functions; convergence of power series; Abel's and Weierstrass's tests; functions defined by power series; Taylor series.

Ordinary linear differential equations: existence and uniqueness theorems (no proofs), Wronskians; solution in series for first and second order non-singular and regular singular equations; methods of Frobenius.

Fourier series: two-dimensional separable linear partial differential equations; solutions by separation of variables and Fourier series.

Functions of a single complex variable: continuity, differentiability, Cauchy-Riemann equations; analyticity, power series; Cauchy's Theorem and applications to evaluation of integrals.

Differential equations and classifications - first order differential equations; the existence and uniqueness theorem; second and higher order linear differential equations; power series solutions; Legendre

 polynomials, Bessel functions; numerical methods.

Survival distributions and life tables, utility theory, life insurance, life annuities, commutation functions, net premiums and premium reserves, introduction to multiple life functions.

Linear programming and duality; mathematical modelling, mathematical structure of the primalprogramme; the simplex tableau and simplex techniques, dual linear programmes; complimentary slackness,  the duality theorem; networks; computations involving computers and software; sensitivity analysis.
       

Multiple life functions, multiple decrement model; insurance models  including expenses; nonforfeiture, benefits and dividends; valuation theory for pension plans.

Review of Macroeconomics; Characteristics of the various types of  investments used to fund financial security programmes; traditional  techniques of financial analysis used in selecting and managing

 nvestment portfolios.

 The course builds on the material in courses MS28D and MS38H,  introducing further tools and techniques of asset/liability management, general product design, as well as issues of pricing and valuation and  asset management.

Study is continued on the applied aspects of MATH2150/M25B such as  analysis of variance, regression analysis, design of experiments and categorical data analysis, time series analysis, stochastic processes and decision theory.

Syllabus:   Projections in R^n and C^n; the adjoint of a matrix; special classes of matrices (Hermitian, positive definite, normal and unitary); polynomials of matrices; the Jordan canonical form; the singular value decomposition.

Metric spaces, examples; continuity; completeness; topological spaces;  Hausdorff spaces; connectedness.

Partial differential equations and classifications; well-posed problems,  classical solutions, initial and boundary value problems; first order linear  and quasi-linear partial differential equations; method of characteristics; conservation laws; classification of general second order operators; the wave equation, D’Alembert method of solution; Laplace’s equation, the maximum principle; the heat equation; separation of variables; boundary  value problems and Sturm-Liouville theory.