Department of Mathematical Sciences, Unit Catalogue 2009/10 |
MA10207: Analysis 1 |
 Credits:
| 12 |
| Level:
| Certificate |
| Period: | Academic Year |
| Assessment:
| EX 100% |
| Supplementary Assessment: | MA10207 Mandatory extra work (where allowed by programme regulations) |
| Requisites:
| While taking this unit you must take MA10209 and take MA10210 |
| Description: | Students must have a grade A in A-level Mathematics or equivalent in order to take this unit. Aims: To define the notions of convergence, limit and continuity precisely and to give rigorous proofs of the principal theorems on the analysis of real sequences and real functions of a real variable. Learning Outcomes: After taking this unit, the students should be able to: * State definitions and theorems in real analysis. * Present proofs of the main theorems. * Apply these definitions and theorems to simple examples. * Construct their own proofs of simple unseen results. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials). Content: Quantifiers. Definitions; sequence, limit. Numbers, order, absolute value, triangle inequality, binomial inequality. Convergence, divergence, infinite limits. Examples: 1/n, an. Algebra of limits. Uniqueness of limits. Growth factor. Convergent sequences are bounded. Axiom: bounded monotone sequences converge. Sequence converging monotonically to root 2. Observations: roots generally not algebraically constructible, transcendental functions are defined as limits. Subsequences, Bolzano-Weierstrass Theorem. Cauchy sequences. Convergence of series. Geometric series. Comparison and Ratio tests. Harmonic series; condensation. Absolute and conditional convergence. Leibniz's Theorem (alternating series). Nested intervals. Application: uncountability of R. Countability of Q. Sup and inf via convergence of bounded monotonic sequences. Limsup and liminf. Existence of n-th roots, definition of rational powers. Infinite decimals. Continuity and limits of functions of real variables. Inertia principle. Algebra of limits, continuous functions, polynomials. Composition of continuous functions. Relate continuity to convergence. Weierstrass's Theorem. Intermediate Value Theorem. Continuous inverse of strictly increasing (continuous) function on interval. Functions discontinuous on R or Q. Definition of derivative. Rules of derivation. Chain Rule. Inverse functions. Rolle's Theorem. Mean Value Theorem. Uniform continuity. Uniqueness of solution f to f ' = g. Sign of derivative; monotonicity. Derivative vanishes at minimum. Cauchy Mean Value Theorem. L'Hopital's Rule. Limits at infinity. Taylor's Theorem with Lagrange remainder. O and o notation. Sign of second derivative; maxima and minima, convexity. Power series. Uniform Convergence. Differentiation of power series. Series definitions of exponential and trigonometric functions. ex+y=ex ey by differentiation. Uniqueness for Cauchy problem for y'' = - y by differentiating (y')2+y2, trigonometric addition formulae. Sketch extension to complex arguments and exp(iz)=cos(z)+isin(z). Logarithmic function. ab for positive a, real b. Logarithmic series on (-1,1) from geometric series. Binomial Theorem for real exponent by differentiating quotient of series by (1+x)p. Density of Q in R. Fundamentals of Rn; norm, inner product, Cauchy-Schwarz inequality, convergent sequences, Bolzano-Weierstrass, open sets. |