MA10207: Analysis 1
[Page last updated: 03 August 2022]
Academic Year:  2022/23 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  12 [equivalent to 24 CATS credits] 
Notional Study Hours:  240 
Level:  Certificate (FHEQ level 4) 
Period: 

Assessment Summary:  EX 100% 
Assessment Detail: 

Supplementary Assessment: 

Requisites:  While taking this module you must take MA10209 AND take MA10210 . Students must have a grade A in Alevel Mathematics or equivalent in order to take this unit. 
Learning Outcomes:  Learning Outcomes: After taking this unit, the students should be able to:
* State definitions and theorems... * Present proofs of key theorems... * Apply these definitions and theorems to a range of examples... * Construct their own proofs of simple unseen results... ...relating to the analysis of sequences and to functions of one real variable. 
Aims:  To define the notions of convergence and limit precisely, to give rigorous proofs of the principal theorems on the analysis of sequences and series, and to develop the theory of continuity, differentiation and integration for functions of one real variable. 
Skills:  Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials)

Content:  Logic, quantifiers. Real and complex numbers, order, absolute value, triangle and binomial inequalities.
* Sequences and limits, uniqueness, divergence, infinite limits, complex sequences. Examples: 1/n, a^{n}. Algebra of limits. Convergent sequences are bounded. Bounded monotone sequences converge. Subsequences, BolzanoWeierstrass Theorem. Cauchy sequences. * Convergence of series. Geometric series. Comparison and Ratio tests. Harmonic series. Absolute convergence. Leibniz's Theorem (alternating series). * Intervals, connectedness and sequential continuity. Nested intervals. * Decimal expansions. Sup and inf, limsup and liminf. * Power series, radius of convergence and sequential continuity, exponential and trigonometric functions, exp(x+y)=exp(x)exp(y), logarithms and powers. * Countability: uncountability of R, countability of Q. Density of Q in R. * Continuity and limits of functions. Inertia principle. Limits at infinity. Algebra of limits and continuous functions, polynomials. Composition of continuous functions. Relation to sequential continuity. Weierstrass's Theorem. Intermediate Value Theorem. Continuous inverse of strictly increasing function on interval. * Definition of derivative. Rules of derivation. Chain Rule. Inverse functions. * Rolle's Theorem. Mean Value Theorem. Sign of derivative; monotonicity. Sign of second derivative; maxima and minima, convexity. Cauchy Mean Value Theorem. L'Hopital's Rule. O and o notation. Taylor's Theorem with Lagrange remainder. * Integration on closed bounded intervals: Riemann sums, linearity, integrability of continuous functions, fundamental theorem of calculus, substitution, integration by parts. Integration for open and unbounded intervals, functions with singularities. 
Programme availability: 
MA10207 is Compulsory on the following programmes:Department of Computer Science

Notes:
