This course aims to introduce gently the rigour of mathematical analysis to first year undergraduate students with some background of A-Level calculus. The concentration is on motivating results and concepts geometrically rather than on providing rigorous proofs. Concepts are defined carefully and results stated precisely, but illustrated by way of vivid, concrete examples.
|SEQUENCES. Definition (emphasis is on sequences of real numbers), examples (to include formulas and recursive definitions); arithmetic of sequences (adding and subtracting sequences etc); sequence types (monotone, alternating, constant, null sequences), examples and graphical illustrations; convergent sequences (introducing the concept of modulus and properties, then $\varepsilon$-definition of convergence, limits and uniqueness of limits), examples (including classical sequences like $(1+\frac 1n)^n$ ); bounded sequences, least upper bound, greatest lower bound; divergent sequences.||8|
|COORDINATE GEOMETRY. Geometric representation of numbers on the real line, inequalities and intervals; coordinates in the 2-dimensional plane; graphs (as subsets of $\Bbb R\times\Bbb R$), some special curves (the straight line, circle, parabola, the ellipse, hyperbola) and associated graphs; functions (emphasis is on real-valued functions): definition, domain, range, representation by arrow diagrams; arithmetic of functions; function types: even, odd, injective, surjective, bijective, identity function, inverse function; polar coordinates.||10|
|CONTINUITY AND SMOOTHNESS. Limits (at a point $x=x_0$), arithmetic with limits, special limits (e.g. $\lim_x\rightarrow 0\frac\sin xx$); continuity: definition using limits and using the $\varepsilon,\,\delta$-notation, examples and exercises, special continuous functions (eg polynomials, trigonometric functions, exponential and logarithmic functions); discontinuity and examples; monotone and bounded functions, existence of inverses, inverses of special functions; differentiability: definition (in terms of limit concept with geometrical representation as gradient) and evaluation for special functions, relationship with continuity (including why use the word "smoothness''), rules for differentiation (sums, products, quotients, chain rule); Rolle's theorem (without proof), the Mean Value theorem (without proof), higher order derivatives, L'Hospital's rule, detailed curve-sketching and Taylor's theorem.||15|
|INTEGRATION. Finding areas, volumes, distances travelled, etc.; the definite integral: definition from first principles using upper and lower sums; properties of the definite integral; the Fundamental Theorem of Calculus (geometrical proof); the indefinite integral as an "anti-derivative'', examples and a few integration formulae; some integration techniques (e.g. integration by parts, use of partial fractions, substitutions, inspection, numerical methods); analysis of the elementary functions and their inverses; applications of integration to include areas and volumes of revolution.||6|