# MATHEMATICAL ANALYSIS

## Learning outcomes of the course unit

At the end of the course the student should have acquired basic knowledge and skills in differential and integral calculus; he should also be able to apply them to concrete problems and to easily handle them in relation to other areas of knowledge. In particular, the student should be able:

1) to be familiar with the structure of the sets of real and rational numbers and with basic concepts of the differential-integral calculus for functions of one real variable (limits, derivatives, definite and indefinite integrals). He should understand and be acquainted with concepts related to sequences and series (convergence criteria and convergence speed) (Knowledge and understanding)

2) to apply the theoretical knowledge acquired to solve simple, concrete problems (limits, integrals) with the help of class exercises. Subsequently, be able to link different concepts in order to solve more complex tasks (draw an approximate graph of a function, for instance) (Applying knowledge and understanding)

3) to assess the consistency and accuracy of the results obtained and analyze the appropriate strategies to solve exercises. (Making judgments)

4) to use a formally correct language allowing to communicate in a clear and precise way the content of the program of the course. Frontal lessons and continuous interaction with the teacher will encourage the student to acquire a specific and appropriate scientific vocabulary (Communication skills)

5) to go deeper into the matter, starting from the basics provided in the course, in order to properly and effectively handle further mathematical tools and concepts. These will be important in the rest of the undergraduate courses, in later training courses, but also in view of future work experiences where mathematical elements and abilities will be required. (Learning skills)

## Prerequisites

None

## Course contents summary

Limits for sequences. Differential and integral calculus for real functions of one real variable

## Course contents

Sets and numbers: Basic set theory, operations between sets. Number systems: N, Z, Q, R, C. Representation of real numbers on a line; maximum, minimum, supremum, infimum of a set of real numbers; integer part and absolute value of real numbers; powers and roots. Complex numbers in various forms.

Functions: injective, surjective and bijective functions. Composition of functions; inverse function. Graphs. Real functions of one real variable. Monotonic functions. Powers with real exponent. Exponential and logarithmic functions. Angles; trigonometrical functions. Cardinality.

Sequences and series: Limits of sequences, Big-O and little-o notation. Numeric series and convergence criteria.

Limits and continuity. Limits of real functions of one real variable; properties. Continuity of real functions of one real variable; properties of continuous functions.

Differential calculus. Derivative and its geometric interpretation. Derivation rules (sum, product, ratio, inverse); chain rule; derivatives of the elementary functions. Relative and absolute maxima and minima; stationary points; monotony and the sign of the derivative. Main theorems (Fermat, Rolle, Lagrange aka mean-value, De l'Hopital); higher order derivatives; Taylor series development. Graphs.

Integral calculus: Primitive of a function defined in an interval; indefinite integrals. Geometric interpretation. Main properties. Fundamental theorem of the integral calculus. Integration techniques: by parts, by substitution; integration of rational functions.

## Recommended readings

M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, Mc Graw-Hill

E. Acerbi, G. Buttazzo: Analisi matematica ABC, Ed. Pitagora, 2000.

## Teaching methods

The course includes 8 or 9 hours of teaching per week, of which about 2 (or more) will be dedicated to exercises. Lessons will be traditionally held at the blackboard and the topics will be introduced starting from examples, whenever possible. Discussion with students will always be important, mainly in order to discover any preexisting gap and readily try to fix them. The course will give particular emphasis to applications and calculus aspects, not overlooking a rigorous approach to the theoretical aspects. To this end, exercises, where students will learn how to apply theoretical arguments to the resolution of a specific problem, will be taken in high consideration. A file will be uploaded to the elly portal weekly and it will contain various exercises; we shall require, as part of the individual study, the resolution of those exercises by the student. At least two hours per week will be devoted to the detailed resolution of some exercises from the previous week's deliveries.

## Assessment methods and criteria

The actual method of examination may depend on the pandemic. The information herein is not final and may be updated later accordingly.

About 8 weeks into the course, students will be asked to pass a test (questions concerning equations, inequalities, direct and inverse image of functions, ...) . Students that pass this test will have a small bonus at the end of the final exam.

The final exam is a written test (6 possibilities until september 2022); students will be asked to solve three exercises (computation of limits, integrals, plot of the graph of a fgiven function). Before the written test, there will be a quiz with questions on derivatives and integrals. In case of need, the tests will be taken at distance, with minor modifications of the above rules.

The examination will be done in two phases. The first part consists is the evaluation of a written test, done with the help of notes and books but without electronic devices, where the student must demonstrate to be able to apply the knowledge solving exercises; those will be on the model of the ones proposed in class. The exam is passed if the student reaches a score of at least 16; the maximum grade of the test is 30. The solution of the test will be uploaded to the elly portal as soon as the test will be concluded; the test will be corrected the same day and the results made available soon as possible. The success in the first part allows the student to take the second part of the exam, which will mainly focus on the theoretical part of the course (definitions, theorems, proofs). Such questions will be given in written form and the answers will be discussed. The final vote will be given by a weighted average of the two votes. The purpose of this type of performance assessment is to attempt to reliably assess the level of achievement of learning outcomes expected described above, in particular points 1) to 4).