Elements of Mathematical Analysis 1 O - Z

Aim of the course is to improve the knowledge of Calculus, learn the basic notions of Real Analysis, understand the concept of proof and use the common tools of Analysis. Students will also be prepared for future courses in Analysis.  In particular the course objectives are:

Knowledge and understanding: students will learn calculus for one real variable functions.

Applying knowledge and understanding: by means of simple mathematical models, students will focus on the central role of Mathematics within science and not only as an abstract topic. 

Making judgements: students will learn the concept of proof and basic techniques to prove a statement, formulating problems and solving them through rigorous reasonings.

Communication skills: students will learn to communicate with clarity and rigour, both in the oral and written analysis. Moreover, students will learn that using a properly structured language is the the key to clear scientific communication.

Learning skills: students will be stimulated to examine in depth some topics, thanks to individual activities or working in group.

The methods and concepts of the course will be taught through frontal lessons. Problems and results will be discussed in class, in order to deepen student’s understanding and acquire the method of logical reasoning. If needed, lessons will take place online, with the same aims and methods. For each theme of the course, exercises will be solved in class. Similar exercises will be assigned to the students to be solved by themselves through discussion in class, and further exercises of the same kind will be assigned as homework. Exercises could be either multiple choice or problems or thematic arguments, in order to underline the theoretical aspects of some of the taught concepts. Some of these exercises will be made available through studium or proving pdf files, in order to encourage individual efforts. Ample room will be given to theoretical questions focused on linking various mathematical concepts, and the construction and proper use of correct and rigorous language. Some space will be devoted to review past lessons with emphasis on the most delicate ideas and techniques taught.

Students with any form of cognitive difficulties are encouraged to discuss with the teacher alternative learning activities, or anyway refer to CInAP at DMI.

Students should have a drive through logical reasoning and already master the basics of numbers, operations, polynomials and their algebraic properties, inequalities of various types and their solutions. All these arguments will be anyway reviewed and recalled during the various “Corsi Zero” taught at the beginning of the academic year.

Attendance to the lessons is mandatory according. Students are also encouraged to attend supplementary lessons and tutoring activities.

The course will revolve on around 6 main themes:

1.     Numbers sets and functions (25 hours)

2.     Limits of sequences and functions (10 hours)

3.     Continuous functions and their properties (4 hours)

4.     Differential calculus (6 hours)

5.     Application of differential calculus (7 hours)

A detailed outline of the course will be given at the end of it.

A diary of the arguments taught will be provided on the Studium platform weekly. Any argument contained in the latter can be asked during the final exam.

For the prerequisites: S. Barbero - S. Mosconi - A. Portaluri, Precorso di Matematica, 1a ed., Pearson.

2. Teacher's notes on Studium

Learning level will be monitored in class through questions and open discussions. The final exam will be written, in one of the days scheduled by the department. It will last 90 minutes and be splitted in two parts, the Theorical (T) part and the Practical (E) part.

The T part will have two open questions, the E part will have two exercises.

Grades: The T part gives at most 10 points, the E part 20 maximum, for a total maximum of 30 points.

Points are given evaluating both the correctedness of the answers of clarity of arguments involved.

In order to pass the written part the students must earn at least 6 points in the T part and at least 12 point in the E part. Few days after the test, the results will be published and the students who passed the written part can, if they wish, ask for an oral test. The final rest can be the one obtained in the written part or, if an oral discussion is made, its final result. The latter, in this case, is independent of the result of the written part. Teacher may, if the written part is barely sufficient, ask for a further oral test.

The final score of the exam will be

·      Not approved: this means that the student failed to grasp the basic concepts and techniques taught in the course

·      18-23: meaning that the student developed minimal confidence in the basic concepts and techniques taught in the course, can solve basic problems and his/her ability to correctly convey mathematical concepts is sufficient but modest.

·      24-27: the student developed good confidence in the basic concepts and techniques taught in the course, can solve problems with good reliability and his/her ability to correctly convey mathematical concepts is good.

·       27-30: the student developed excellent confidence in the concepts and techniques taught in the course, can solve more advanced problems with good reliability, can effectively connect different mathematical ideas and his/her ability to correctly convey mathematical concepts is excellent.

In order to participate to the written part of the exam the student must make a reservation to the exam through SmartEdu. The final exam could also be conveyed via web if conditions require, in which case the test will be exclusively oral, involving both theory and exercises and lasting 20 minutes.

Theory part

·      The statement, through correct formal language, of a Theorem

·      The proof of a theorem or a proposition

·      The precise definition of a mathematical object

·      Exhibition of an example having certain required properties

·      Exhibition of a counterexample to a mathematical statement

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·      Finding the true statement among false ones.

Exercises

·      Determining extrema of a numerical set

·      Producing the graph of a given function

·      Determining the equation of the tangent line to a graph at one point

·      Computing a limit, or a family of parameter dependant limits

·      Verifying a certain asymptotic comparison

·      Computing the limit of a recursively defined sequence

·      Solving an equation over the complex field

·      Verifying an inequality through graphical method

·      Determining global and local extrema of a function

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·      Computing the derivative of an inverse function.