This module introduces differential calculus and analysis. It develops problem solving skills and introduces concepts such as definition, lemma, theorem, proof and different methods of proof, including direct, contrapositive and induction.

• Basic properties of the real numbers: Important subsets (natural, integers, rationals), open and closed intervals, neighbourhoods, supremum, infinimum, boundedness, compactness. • Algebra of Complex numbers: modulus, phase, Argand diagrams, de Moivre's theorem and roots of complex numbers. • Real valued functions: Definition of function, properties of functions: one-to-one, onto, inverse function, composition of functions, parametric functions. • Limits and continuity: Definition of limit, limit theorems, limit points, definition and meaning of continuity, examples of discontinuous functions (e.g. Heaviside step function), Squeezing Theorem, Intermediate Value Theorem, Bisection Method. • The derivative and differentiation techniques: Differentiation from first principles, derivative of sums, products, quotients, inverse of a function, chain rule, smoothness of a function, Rolle's theorem, Mean Value Theorem. • Properties of transcendental functions: Including trigonometric, exponential logarithmic and hyperbolic functions; derivatives and inverse functions. • Applications of differentiation: Finding roots of equations (Newton's method), Indeterminate forms (L'Hopital's rule); implicit differentiation; optimisation applications, the Second Derivative Test. • Curve sketching: Domain and range, roots of equations, increasing and decreasing, maxima and minima, concavity, points of inflection, symmetry, asymptotes.

1. Give an account of the construction of the real number system, and in particular, give a coherent description of irrational numbers. 2. Understand properties of the complex number system, in particular the Argand diagram and polar form, and be able to find roots of complex numbers. 3. Use the formal definition of a (epsilon-delta) of a limit. 4. Decide whether a function is continuous via a limit argument. Apply the Intermediate Value theorem and the bisection method. 5. Evaluate limits and derivatives both from first principles and from a practical computational view. 6. Understand properties of inverse and transcendental functions. 7. Use Newton's method to find roots of nonlinear equations and use L'Hopital's rule to calculate limits of indeterminate forms. 8. Be able to graph polynomial functions and identify the locations of the intercepts, relative extrema and inflection points.

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Normal weekly lectures. Mid-term exam and final exam. Graduates will possess a firm grounding in the rigorous approach to calculus and have an appreciation of its many applications.