###### Main content

This entry is from Winter semester 2016/17 and might be obsolete. No current equivalent could be found.

# Analysis I (incl. Foundations of Mathematics) (dt. Analysis I mit Grundlagen der Mathematik)

 Level, degree of commitment Basic module, compulsory elective module Forms of teaching and learning,workload Lecture Grundlagen der Mathematics (2 SWS) lecture Analysis I (4 SWS) central recitation class (2 SWS) recitation class (2 SWS), 450 hours (150 h attendance, 300 h private study) Credit points,formal requirements 15 CP Course requirement(s): Written examination Examination type: Successful completion of at least 50 percent of the points from the weekly exercises. Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for the degree program B.Sc. Mathematics. Duration,frequency One semester, each summer semester Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Thomas Bauer, Prof. Dr. Pablo Ramacher

## Contents

Fundamentals of mathematics:

• elementary set theory
• natural and whole numbers, complete induction, rational numbers
• mappings, functions, relations
• elementary propositional logic and its application in mathematical proofs
• real numbers, inequalities (Bernoulli etc.), complex numbers
• groups, solids.

Analysis:

• Consequences: Limits, monotonicity, convergence criteria
• Series: limits, absolute convergence, convergence criteria, rearrangement of series
• Continuity and limits of functions: Concepts, equivalent formulations, properties of continuous functions on compact or connected sets (intermediate value theorem), uniform continuity and Heine's theorem.
• Important functions of analysis and their properties: exponential function and the number e, sine and cosine, logarithm
• Differentiability: terms, continuous differentiability, mean value theorem of differential calculus, Rolle's theorem, monotonicity, local extrema, l'Hopital's rule
• Sequences and series of functions: uniform convergence, continuity and differentiability, power series, Taylor's formula
• Integration theory: definition of the integral, criteria for integrability, primitive function, main theorem of integral and differential calculus, indefinite integrals and their calculation (partial integration, substitution), improper integrals, Fubini's theorem if applicable, Cavalieri principle (this complex of topics can alternatively be treated by the instructor in Analysis II)

## Qualification Goals

The students

• should learn the basics of mathematical reasoning and argumentation,
• understand the basic principles of calculus of a variable and can use them for the analytical treatment of geometrically, scientifically or technically motivated problems,
• master the basic concepts and techniques of calculus, especially approximations and limit transitions,
• use mathematical working methods on concrete problems, they can distinguish between mathematical intuition and formal precision and use and relate both components to each other,
• recognize the close connections between different mathematical areas by means of the linear structures within calculus as an example,
• improve their oral communication skills in the exercises by practicing free speech in front of an audience and by actively participating in the discussion.

None

## Applicability

Module imported from B.Sc. Mathematics.

It can be attended at FB12 in study program(s)

• B.Sc. Mathematics
• B.Sc. Business Mathematics

When studying B.Sc. Business Mathematics, this module can be attended in the study area Basics of Mathematics.

## Recommended Reading

• Forster, O.: Analysis 1 und Analysis 2, Vieweg-Verlag.
• Heuser, H.: Lehrbuch der Analysis, Teil 1 und Teil 2, Teubner-Verlag.
• Rudin, W.: Analysis, Oldenbourg-Verlag.

## Please note:

This page describes a module according to the latest valid module guide in Winter semester 2016/17. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

• Winter 2016/17
• Summer 2018 (no corresponding element)
• Winter 2018/19 (no corresponding element)
• Winter 2019/20 (no corresponding element)
• Winter 2020/21 (no corresponding element)
• Summer 2021 (no corresponding element)
• Winter 2021/22 (no corresponding element)
• Winter 2022/23 (no corresponding element)
• Winter 2023/24 (no corresponding element)

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.