# MATHEMATICS 1 AND EXERCISES

## Learning outcomes of the course unit

Knowledge and understanding.

At the end of the lectures, students should have acquired knowledge and understanding of the basic theory of numerical sequences and series, of the differential calculus and integration for functions of one variable, of ordinary differential equations, and of linear algebra (where particular attention is paid to the methods to solve linear systems and to diagonalize square matrices).

Applying knowledge and understanding.

By means of the classroom exercises students learn how to apply the theoretical knowledges to solve concrete problems, such as optimization problems for functions depending on one variable, models from physics and chemistry which lead to solving ordinary

differential equation or to solving particular integrals for functions of one variable.

Making judgements.

Students must be able to evaluate coherence and correctness of results obtained by themselves or by others.

Communication skills.

Students must be able to communicate in a clear and precise way mathematical statements in the field of study, also in a context broader than mere calculus. Through the front lectures and the assistance of the teacher, the students acquire the specific and appropriate scientific vocabulary.

Learning skills.

The student who has attended the course is able to deepen autonomously his/her knowledge of calculus for functions of one variable, of numerical sequences and series, of the Riemann integral for functions depending on one variable, of linear algebra, starting from the basic and fundamental knowledges provided by the course. He/She will be also able to consult specialized textbook, even outside the topics illustrated during the lectures. This to facilitate the learning of the subjects illustrated in the second mathematical course which students should attend in the second semester, as well as the future entry in the labour market or a second-level study in a field which requires good mathematical skills.

## Course contents summary

The lectures aim at providing students with fundamental concepts of infinitesimal and integral calculus for functions of one variable, of numerical sequences and series, of Linear Algebra (paying particular attention to the theory of linear systems and to the problem of diagonalize matrices) and of the theory or ordinary differential equations which can be solved in a elementary way.

## Course contents

1. Real numbers.

Maximum, minimum, least upper and greatest lower bound; integer part and modulus of real numbers; powers, roots, nth roots of non-negative numbers; rational and irrational numbers; intervals, distance. Complex numbers. The principle of induction.

2. An overview of linear algebra.

Vector spaces, linearly independent vecors, basis; matrix, determinant; linear operators; systems of linear equations. Lines and planes in the space.

3. Functions.

One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential and logarithmic functions; trigonometric functions.

4. Sequences and series.

Limits of sequences. Series with positive terms; criteria for their convergence.

5. Limits.

Limits of functions with real values, uniqueness of the limit, limits of the restrictions; limit of the sum, product, quotient of two functions; permanence of the sign, comparison theorems; right and left limit; limits of monotone functions.

6. Continuous functions.

Continuity of real functions of a real variable, restrictions of continuous functions, composition of continuous functions; sum, product, quotient of continuous functions; examples of continuous functions; discontinuity, examples of discontinuous functions; zeroes of continuous functions defined in an interval; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

7. Differential calculus.

Incremental ratio, derivatives, right and left derivatives; geometrical significance of the derivative; derivation rules: derivatives of the sum, product, quotient of two functions; derivatives of composite functions and inverse functions; derivatives of elementary functions; delative maxima and minima; stationary points; connection between the monotonicity and the sign of the derivative; Rolle's theorem and Lagrange's theorem and their geometrical interpretation, Cauchy's theorem and de l'Hopital's theorem; convex functions, derivatives of convex functions, connection between the convexity and the sign of the second order derivative; study of local maxima and minima via the study of the derivatives.

8. Integrals.

Partitions of an interval; upper and lower integral, Integrability of continuous functions; geometrical interpretation of the integral; properties of integrals; mean of an integrable function; integrals on oriented intervals; fundamental theorem of integral calculus; primitives, indefinite integrals; integration by parts and by substitution; integrals of rational functions.

9. Ordinary differential equations.

Separable differential equations; first-order linear differential equations with variable coefficients; linear differential equations of order n with constant coefficients.

## Recommended readings

M. Bramanti, C.D. Pagani, S. Salsa, Matematica: calcolo infinitesimale e algebra lineare. Seconda edizione. Zanichelli, 2004

## Teaching methods

The course schedules 6 hours of frontal lectures and classroom exercises per week plus two additional hours where other exercises are discussed and solve.

During the lectures the basic results of the calculus for functions of one variable will be analyzed and discussed. Students will be provided also with the basic results on sequences of real numbers and numerical series and with the basic knowledges of ordinary differential equations and linear algebra.

The classroom exercises aim at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.

Most of the didactic activities are developped with the help of a tablet PC which projects on a screen the notes the teacher is writing. At the end of each lesson, a pdf file with the note is created and uploaded on the elly website to make it accessible to students.

## Assessment methods and criteria

The exam consists of two parts: a written part and an oral part in different dates. The written part, which lasts approximately 2 hours, is based on some exercises and it is aimed at evaluating the skills of the student in applying the abstract results proposed during the course to some concrete situations. The oral part is aimed at evaluating the knowledge of the abstract results seen during the course and their proofs, and the knowledge of the topics which have not been included into the written test.

The maximum score of the written part of the exam is 30. If the score of the written part is less than 15, the exam is failed, otherwise students should take the oral part of the exam.

If interested, students may take two tests "in itinere": one around the end of november and the other one at the end of the lectures. The tests consists of exercises on the topics illustrated in first part and in the second part of lectures, respectively, plus (possibly) a theoretical question. The maximum score of the tests is 30 points. If the average of the two tests is not less than 15 points, the student just need to take the oral part of the exam.