# CALCULUS 1

CALCULUS 1 (UNIT 1) (9 credits) | CALCULUS 1 (UNIT 2) (6 credits) |

## Learning outcomes of the course unit

The course aims at providing students with the basic tools of Mathematical Analysis.

## Course contents summary

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.

Sequences and series.

Sequences of real numbers, 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. Series with nonnegative terms: three criteria for their convergence (convergence by comparison, root and ratio tests); absolutely convergent series; Leibniz criterion; Some examples: geometric series, telescoping series.

Continuous functions.

One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential, logarithmic and trigonometric 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; 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 intervals; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

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; 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.

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.

Improper integrals; convergence of the integral, absolute convergence,comparison tests. Integral test for positive valued series.

Asymptotic expansions

Landau symbols; Taylor's theorem; explicit formula of the remainder; Mac Laurin expansion of elementary functions; Taylor's series

Complements

Bolzano-Weirstrass theorem, compactness in the real line; Cauchy sequences; upper and lower limits; uniform continuity.

Complex numbers.

Definitions, operations, complex plain, polar form, root extraction.

Differential equations

Nomenclature: order, linear and nonlinear; first examples; solutions of linear first order equations; solution of separable differential equations; constant coefficients linear differential equations.

## Course contents

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.

Sequences and series.

Sequences of real numbers, 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. Series with nonnegative terms: three criteria for their convergence (convergence by comparison, root and ratio tests); absolutely convergent series; Leibniz criterion; Some examples: geometric series, telescoping series.

Continuous functions.

One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential, logarithmic and trigonometric 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; 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 intervals; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

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; 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.

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.

Improper integrals; convergence of the integral, absolute convergence,comparison tests. Integral test for positive valued series.

Asymptotic expansions

Landau symbols; Taylor's theorem; explicit formula of the remainder; Mac Laurin expansion of elementary functions; Taylor's series

Complements

Bolzano-Weirstrass theorem, compactness in the real line; Cauchy sequences; upper and lower limits; uniform continuity.

Complex numbers.

Definitions, operations, complex plain, polar form, root extraction.

Differential equations

Nomenclature: order, linear and nonlinear; first examples; solutions of linear first order equations; solution of separable differential equations; constant coefficients linear differential equations.

## Recommended readings

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

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

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

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

## Teaching methods

Lectures and classroom exercises. During the lectures the basic results of 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.

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

## Assessment methods and criteria

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

Some evaluations in itinere are fixed: if all of them are positively marked, the student are relieved from the written 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 the knowledge of the abstract results seen during the course, their proofs, and to evaluate the correct use of the mathematical terms.