German original

# Analysis I (dt. Analysis I)

 Level, degree of commitment in original study programme Basic module, required module Forms of teaching and learning,workload Lecture (4 SWS), recitation class (2 SWS), Werkstatt (2 SWS), 270 hours (120 h attendance, 150 h private study) Credit points,formal requirements 9 CP Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written examination Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for study course B.Sc. Mathematics. Original study programme B.Sc. Mathematik / Mathematik Basismodule 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

• Sequences: Limits, monotony, convergence criteria
• Series: limits, absolute convergence, convergence criteria, reordering of series
• Continuity and limits of functions: Terms, equivalent formulations, properties of continuous functions on compact or connected spaces (mean value theorem), uniform continuity and theorem of Heine
• Important functions of calculus and their properties: exponential function and the number e, sine and cosine, logarithm
• Differentiability: concepts, continuous differentiability, mean value theorem of differential calculus, Rolle's theorem, monotony, local extrema, l'Hopital's rule
• Function sequences and series: uniform convergence, continuity and differentiability, power series, Taylor formula
• Integration theory: definition of the integral, criteria for integrability, primitives, main theorem of calculus, indefinite integrals and their calculation (partial integration, substitution), improper integrals, possibly theorem of Fubini, Cavalieri principle (these topics can be treated by the lecturer alternatively in Analysis II)

## Qualification Goals

The students

• understand the basic principles of calculus in one variable and can use them for the analytic treatment of geometric, scientific or technical problems,
• are proficient in the basic concepts and techniques of analysis, in particular approximations and limits,
• use mathematical working methods on concrete questions, they can distinguish between mathematical intuition and formal precision and use and relate both components to each other,
• recognize by means of the linear structures within calculus the close connections between different mathematical areas,
• improve their oral communication skills in the exercises by practicing free speech in front of an audience and by actively participating in the discussion.

## Prerequisites

None.

• 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.