# CALCULUS 1

CALCULUS 1 PAR. 1 (6 credits) | CALCULUS 1 (UNIT 2) (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, R and C, 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.

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

None

## Course contents summary

The course aim 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.

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.

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. Uniformly continuous functions.

The concept of uniform continuity, main properties of uniformly continuous functions, Heine-Borel theorem, proof of the integrability of continuous functions over closed and bounded intervals.

11. Complements

Countable and noncountable sets, uncountability of real numbers.

## Recommended readings

Theory

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.

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.

Two evaluations in itinere are also fixed during the first half of the course: if the average of grades of these two tests is not less than 15 points, the student is allow to take the written part only on the topics of the second part of the course: if the average of the grade of this test and the average of the two evaluations in itinere of the first half of the course is not less than 15, the student is admitted to the the oral part of the exam.

The written part (or the evaluations in itinere) 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 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.