AUTUMN SEMESTER:
Basic algebra (factorisation, partial fractions, roots of quadratics and other simple equations, linear simultaneous equations), geometry, trigonometry. Trigonometric identities and solutions of trigonometric equations.
Exponential and logarithmic functions and their properties. Odd, even and periodic functions. Concept of a function and inverse functions, trigonometric and inverse trigonometric functions, hyperbolic functions and their inverses, solution of trigonometric and hyperbolic equations.
Complex numbers: real and imaginary parts, polar and exponential form, Argand diagram, exp(jx)=cosx+jsinx, relationship between trigonometric and hyperbolic functions, De Moivre’s theorem and applications.
Concept of derivative and rules of differentiation for a function of one variable. Differentiation of trigonometric, exponential and logarithmic functions. Applications to gradients, tangents and normals, extreme points and curve sketching. Functions of several variables. The idea that the graph of z=f(x,y) is a surface. First and second order partial derivatives and their meanings as slopes in particular directions. The total differential and applications to errors and rates of change.
Arithmetic and geometric sequences and series.
Various techniques for the evaluation of limits.
Concept of indefinite integration as the inverse of differentiation and standard methods for integration such as substitution, integration by parts and integration of rational functions. Definite integration, areas under curves, use of recurrence relationships, numerical integration. Mean and rms values.
SPRING SEMESTER:
Fourier Series
Calculation of the Fourier Series of a periodic function. Representation of a Fourier Series in complex form.
Further integration
Hyperbolic function - definitions, identities and integrals. Conic sections - equations in implicit and parametric form. Integral requiring trigonometric or hyperbolic substitutions. Calculation of areas under curves given implicitly. Calculation of the length of a curve (given explicitly or as parametric equations)
Multiple Integration
Evaluation of multiple integrals with both constant and non-constant limits. Interpretation of the region of integration of a multiple integral and evaluation of multiple integrals by changing the order of integration and use of polar co-ordinates.
Power series, limits and approximations
Binomial expansion. Power series including Maclaurin and Taylor series expansions. The Newton-Raphson method. Calculation of approximations and limits using power series. Calculation of limits using L'Hôpital's Rule.
Ordinary Differential Equations
Classification of differential equations. First order differential equations with variables separable, and the use of an integrating factor. First and second order linear differential equations with constant coefficients (homogeneous and non-homogeneous where the RHS is an exponential, trigonometric or polynomial function).
