# CALCULUS 1

CALCULUS 1 (UNIT 1) (6 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

Classroom lectures and classroom exercises

## Assessment methods and criteria

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

Four evaluations in itinere are fixed: if all of them are positively marked, the student are relieved from the written part of the exam.