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Academic year
Didactic period
Primo semestre (primi anni)

Training objectives

The Mathematical Analysis I course is divided into two segments: a part of 9 credits, Theory and Exercises, indicated below for brevity with the letter A, and a part of 3 credits, Exercises with MatLab, indicated with M.

A: The main aim of the course consists in providing the students of the basic tools of differential and integral calculus for real functions of a real variable, both form a theoretical and a practical viewpoint. A software is also taught in the course, in order to make students capable of simple elaborations on computers of the main notions of the course.

Here follows what a student shall learn during the course. Basic notions on numerical sets, with particular reference to real numbers.
Numerical sequences and series, the limit notion.
Differential calculus for real functions of a real variable: derivatives and their computation, study of the graph of a function.
Integral calculus for real functions of a real variable: Riemann and generalized integrals.

Here follows what a student should be able to do after having followed the course. To argue by simple logic and deductive arguments, with particular reference to mathematics in general. To acquire a good level of computing derivatives, integrals and whatever in the course is related, basing on solving simple problems. To understand some simple mathematical proof of important results, by following in particular their logic development.

M: To use a specific software for solving simple problems treated in the course, in particular about differential and integral calculus.


A: The course does not require any particular knowledge different from those already studied in high school. However, a particular emphasis is given to the following topics:

Algebraic equations and systems.
Inequalities and systems of inequalities.
Elementary mathematical functions: power, exponential, logarithmic, trigonometric functions and their main properties.

M: No prior knowledge is required.

Course programme

The course consists of 120 hours of teaching, including exercises (30 hours) and computer elaborations (about 20 hours). Here follow the main topic of the course.

•Natural, integer, rational, real numbers. Maximum, minimum, least upper and lower bound of a set. The completeness property of the set of real numbers (5).
•Sequences. Limit of a sequence. Convergence of bounded sequences. Existence of the limit of a monotone sequence*. The geometric sequence. Fundamental operations with limits. Theorems about the sign, order and comparison of limits. The number e as limit of a sequence*. Order of divergence and vanishing of a sequence; asymptotic sequences. Stirling formula (10).
•Numerical series. Converging and undetermined series. The geometric series and its sum; the harmonic series, telescopic series. A necessary condition of convergence (vanishing of the general term). Convergence criteria for series with positive terms. The generalized harmonic series. Convergence criteria for series having sign changes: absolute convergence, Leibniz criterion (10).
•Functions of a real variable. Definition of a functions, domain, image, graph. Bounded functions, symmetric functions, monotone functions, periodic functions. Elementary functions. Operations with graphs. Composite functions, invertible and inverse functions and their graphs (15).
•Limits of functions and continuity. Limits of functions. Limits of functions and limits of sequences. Operations with limits of functions, theorems about the sign, order and comparison of limits. Change of variables in the limits of functions. Important limits. Continuous functions. Discontinuous functions. Asymptotes. Classes of continuous functions. Zeroes theorem*. Weierstrass theorem*. Intermediate values theorem. Study of a function (20).
•Differential calculus. The derivative of a function in a point, the derivative functions. Tangent line in a point. Higher order derivatives. Derivatives of elementary functions. Right and left derivative. Non differentiability points. Every differentiable function is continuous. Operations with derivatives: derivatives of sums, products, quotients and compositions of functions. Differentiability of the inverse function. Extremal points. Fermat and mean value theorem. Characterization of monotone functions by the sign of the derivative. Stationary and inflexion points. The second derivative: convexity and concavity. Convexity and concavity by chords and tangent lines. Study of maxima and minima by the second derivative. Hospital theorems. Taylor and MacLaurin polynomials (20).
•Integral calculus. Riemann integral for continuous functions and its geometric interpretation. Property of the integral: linearity, additivity with respect to the integration interval, monotonicity. The integral mean theorem and its geometric meaning. The primitive of a functions; characterization of primitives in an interval. First fundamental theorem of integral calculus. Primitives of elementary functions. The integral function: the second fundamental theorem of integral calculus. Techniques of integration; elliptic integrals. Generalized integrals (20).

•Exercises with MATLAB. Introduction to the program (cw, workspace), matrices and vectors, commands size e length. Sequences and series: script files, 2d graphics, the commands cumsum and cumprod, the for cycle and the conditional instruction if. Functions: logic and relational operators, graphs in logarithmic scales, the commands polival e polyfit. Functions: creation of a function, the while cycle, the fzero and fminbnd command. Derivatives: the command diff for computing numerical derivatives , the computation of symbolic derivatives. Integrals: numerical integration (command trapz), the integral functions, the command in. Introduction to the symbolic toolbox (20).

•All the statements lf the course are completed by a proof, excepted those denoted with a *

Didactic methods

The course is organized as follows.

A: Theoretical lessons in classroom about all topics of the course (Matlab excepted). The lessons will be in the presence; recordings of the lessons of the previous academic year will be available.

Exercises in classrooms: solution of exercises previously given as homework; students are strongly encouraged to try to solve them at home. Students are also pushed to collaborate among them.

M: Computer elaborations in computer science classrooms, with Matlab lessons. The lessons will be in the presence; recordings of the lessons of the previous academic year will be available.

Learning assessment procedures

To verify the level of achievement of learning objectives listed above we proceed as follows.

Two written theoretical partial tests will be carried out during the course, a mid (late October, early November) and one at the end (usually in the first days of January). They will focus solely on the program carried out during the first or the second part, respectively. A student is admitted to the second partial test only if he/she had a grade larger than or equal to 9/30 (every grade is expressed on a range of 30). Analogously, two partial tests of MatLab will be carried out during the course; a student is admitted to the second partial test of MatLab only if he/she had a grade larger than or equal to 2.5/5.

The written theoretical part consists of a written test, which lasts two hours. It typically consists of eight exercises (intermediate partial tests are carried out similarly). In this test it is not permitted to use notebooks, books, calculators, smart phones, pc's, tablets, etc. ...; students are advised to bring a photo ID. Students in the first year are admitted to the oral exam if they have in the written test a rating greater than or equal to 9/30; the sophomores and subsequent students are admitted regardless of grades received in the written test. A written test is only valid within the exam period (January-February, June-July and September) in which it is carried out; in case of more written tests is considered the highest rating. The written exam can be replaced by the two partial tests that will take place during the course, with the performance as above; the arithmetic average of these tests is taken as the grade of the written test. About MatLab, the tests consists of two exercises and lasts 20 minutes; a student has passed the MatLab test if the average of the two partial tests is larger or equal than 2.5/5.

The written theoretical tests are at disposal of students during the usual consultation hours or on the first oral examination following the corresponding written test.

The oral part consists of an oral examination that focuses mainly on theoretical topics of the course (knowledge of the major theoretical results, connections between different parts of programs, demonstrations of proven results in class) and, for the students that did not pass the MatLab test, of a 20 minutes test about MatLab. Students are strongly encouraged to show up with a computer on which they have installed the software. The final grade can be understood as an average among the written theoretical test, the oral part and the MatLab part.

The oral examination shall normally be supported within the same examination session of the written test; students of second and later years are allowed to take the oral exam even outside the normal examination sessions, provided that this takes place before the next session to the written test.

All exams (both written and oral) as well as the partial exams will usually be "in presence".

Reference texts

M. Bramanti, C. D. Pagani e S. Salsa: Analisi Matematica I, Zanichelli 2008.
S. Salsa e A. Squellati: Esercizi di Analisi Matematica 1, Zanichelli 2011.
G. Jensen: Using MatLab in Calculus, Prentice-Hall, 2000.

Further readings:
E. Giusti: Analisi Matematica I, Bollati-Boringhieri, 2002.
W. J. Palm III: MatLab - Un'introduzione per gli ingegneri, Mc-Graw-Hill, 2011.

The following texts are handouts provided by the teacher, all freely downloadable from the website of the university:

notes of the teacher on the course;
texts of exercises;
texts of exams resolved;
exam exercises related to MatLab with solutions;
general collection of exam papers in recent years;
notes on MatLab.