## MA10207: Analysis 1

[Page last updated: 20 April 2021]

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%
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: While taking this module you must take MA10209 AND take MA10210 . Students must have a grade A in A-level Mathematics or equivalent in order to take this unit.
Description: 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.

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.

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, an. Algebra of limits. Convergent sequences are bounded. Bounded monotone sequences converge. Subsequences, Bolzano-Weierstrass 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
• USCM-AFB20 : BSc(Hons) Computer Science and Mathematics (Year 1)
• USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 1)
• USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 1)
• USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 1)
• USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 1)
• USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 1)
Department of Economics
• UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 1)
• UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 1)
• UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 1)
• UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 1)
Department of Mathematical Sciences
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 1)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 1)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 1)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 1)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 1)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 1)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 1)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 1)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 1)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 1)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 1)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 1)
Department of Physics
• USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 1)
• USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 1)
• USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 1)
• USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 1)
• USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 1)
• USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 1)

 Notes: This unit catalogue is applicable for the 2021/22 academic year only. Students continuing their studies into 2022/23 and beyond should not assume that this unit will be available in future years in the format displayed here for 2021/22. Programmes and units are subject to change in accordance with normal University procedures. Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.Find out more about these and other important University terms and conditions here.