# CALCULUS 1

CALCULUS 1 (UNIT 2) (6 credits) | CALCULUS 1 (UNIT 1) (6 credits) |

## Learning outcomes of the course unit

Knowledge and understanding.

At the end of the lectures, students should have acquired knowledge and understanding of the numerical fields N, Q, Z and R, of the numerical sequences and series and of the differential and integral calculus for functions of one variable.

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.

Making judgements.

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

Students should be also able to produce precise mathematical arguments clearly identifying the assumptions and the conclusions.

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 numerical sequences, differential calculus for functions of one variable, 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 other activities of the degree course in Mathematics, which use notions from Mathematical Analysis.

## Prerequisites

Elementi di Matematica

## Course contents summary

The course aims at providing students with the fundamental notions of the numerical sets and with the fundamental concepts of infinitesimal and integral calculus for functions of one variable and of numerical sequences and series.

## Course contents

1. Real numbers.

Axiomatic definition of real numbers, maximum, minimum, least upper and greatest lower bound; integer part and modulus of real numbers; powers, roots, n-th roots of non-negative numbers; rational and irrational numbers and their density in the set of all the real numbers; intervals, distance; neighborhoods, accumulation points, isolated points, interior points; closed sets, open sets, frontier. The principle of induction.

2. Sequences of real numbers.

The concept of numerical sequence, convergent and divergent sequences, uniqueness of the limit; infinitesimal sequences; subsequences, a criterion for the non existence of the limit of a sequence; limit of the sum, product, quotient of sequences, permanence of the sign, comparison theorems; monotone sequences; the Nepero’s number; sequences defined by recurrence. Cauchy sequences; Bolzano-Weierstrass theorem.

3. Functions and limits.

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

4. Continuity.

The concept of continuous function, 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 intervals; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

5. Differential calculus.

Incremental ratio, derivatives, right and left derivatives; geometrical meaning 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; relative 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. Taylor expansions of smooth enough functions with Peano's and

Lagrange's form of remainder

6. Convexity.

Convex functions, monotonicity of incremental ratios, relation between convexity, first derivative and sign of the second derivative.

7. Riemann Integrals.

Partitions of an interval; Riemann sums; Riemann integral; integrability of monotonic functions and of continuous functions; integral mean; fundamental theorem of integral calculus; primitives; integration by parts; integration by substitution; integration of rational functions.

8. Series.

Definitions; convergent, divergent and undetermined series; series with positive terms, comparison, ratio and root tests; absolute convergence, Leibniz criterion; examples: geometric series, telescopic series, generalized harmonic series, alternating harmonic series.

9. Improper integrals.

Definition for bounded and unbounded intervals, convergence of the integral, absolute convergence, comparison tests. Integral test for positive valued series.

10. Ordinary differential equations.

Order of a differential equation, Cauchy problem, separation of variables, explicit solutions to homogeneous first order linear equations and second order linear equations with constant coefficients, variation of constants method for inhomogeneous equations.

11. Complements

Countable and noncountable sets, uncountability of real numbers. Power series and expansion in power series of smooth enough functions.

## Recommended readings

Theory

E. Acerbi, G. Buttazzo: Primo corso di Analisi Matematica, Pitagora Editore, 1997.

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

M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 1, Ed. Zanichelli, 2008.

M. Giaquinta, L. Modica, Analisi Matematica 1, vol. 1 & 2, Ed. Pitagora, 1998.

E. Giusti, Analisi matematica vol.1, Ed. Boringhieri, 2002

Exercises

Enrico Giusti "Esercizi e Complementi di Analisi matematica 1" Boringhieri

## Teaching methods

The course schedules 5 hours per week of lectures and classroom exercises. During the lectures the fundamental properties of the numerical sets will be illustrated and basic results of calculus, integration for functions of one variable will be analyzed and discussed. Students will be provided also with the basic results on sequences and series of real numbers. 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.

The didactic activities of the first half of the course are developped also with the help of a tablet PC which projects on a screen the notes the teacher is writing. A the end of each lesson, a pdf file with the notes of the lecture is uploaded on the elly website. Also a video is uploaded to elly which contains the screen shooting and the audio of the lectures.

The didactic activities of the second half of the course are developped in a traditional way using a blackboard.

## Assessment methods and criteria

The exam consists of a written part and an oral part in different dates.

An evaluation in itinere on the topics of the first part of the course is planned and is arranged with students in the last two weeks of the first semester

The written part (or the evaluation in itinere) is based on exercises (indicatively 3 or 4) 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 maximum score of the written part of the exam is 30. The written part is successful if the student reaches a score non inferior to 15. For students who have undergone the written test scheduled in the first semester, the written test will just focus on the program of the second part of the course and the vote of this will average with those of

the first two partial tests. For all the other students, it will cover the

whole program of the course.

The oral part is aimed at evaluating 1) the knowledge of the abstract results seen during the course and their proofs 2) the correct use of the mathematical terms, 3) the knowledge of those arguments which have not been included into the written test. The final vote will be given by a weighted average of the votes of the written and oral part of the exam.