## Learning objectives

Provide the basic tools of Mathematical Analysis

## Course unit content

<strong>First term.<br />

<br />

1. </strong><strong>Real numbers</strong>. <br />

<div align="""""justify""""">Axiomatic definition of real numbers, maximum, minimum, least upper and greatest lower bound; whole part and modulus of real numbers; powers, roots, nth roots of non-negative numbers; rational and irrational numbers; intervals, distance; neighbourhoods, cluster points, isolated points, interior points; closed sets, open sets, frontier. injective, surjective, and bijective functions, inverse function; graphs; real functions of a real variable, monotone functions, exponential and logarithmic functions; trigonometric functions.<br />

</div>

<br />

<strong>2. Functions.<br />

<br />

</strong>One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential and logarithmic functions; trigonometric functions.<br />

<br />

<strong>3. Limits. </strong><br />

<div align="""""justify""""">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; orders of infinitesimals and infinities, asymptotics.<br />

</div>

<br />

<strong>4. Continuous functions.</strong> <br />

<div align=""""""justify"""""">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; zero theorem; continuity and intervals; continuity and monotony; continuity of inverse functions; Weierstrass theorem.<br />

</div>

<br />

<strong>5. Differential calculus.</strong> <br />

<div align=""""""justify"""""">Difference quotients, derivatives, right and left derivatives; geometrical significance of the derivative; derivation rules: derivatives of the sum, product, quotient of two functions; derivatives of compound functions and inverse functions; derivatives of elementary functions; relative maximums and minimums; stationary points; relationship between monotony and 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, relationship between convexity and sign of the second derivative; Taylor's formula with Peano, Lagrange remainder; study of local maxima and minima via the study of higher order derivatives.<br />

<br />

<strong>6. Integrals.</strong> <br />

Partitions of an interval; upper and lower integral, integrable functions in an interval, integrability of continuous functions and monotone functions; geometrical interpretation of the integral; properties of integrals; mean of an integrable function; integrals on directed intervals; fundamental theorem of integral calculus; primitives, indefinite integrals; integration by parts and by substitution; integrals of rational functions. Taylor expansion with integral remainder.<br />

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<strong>Second term.</strong><br />

<br />

<strong>7. Complements of calculus.</strong><br />

Improper integrals. Convergence of the integral, absolute convergence, comparison tests. Uniformly continuous functions.<br />

<br />

<strong>8. Sequences. </strong><br />

<div align="""""justify""""">Mathematical induction; real and complex sequences; limit of a sequence; subsequences; Cauchy sequences; monotonic sequences; Neper's number; sequences defined by recurrence relation; upper and lower limits; Bolzano-Weierstrass theorem, compactness in the real line. <br />

</div>

<br />

<strong>9. Series. </strong><br />

Convergence criteria: comparison tests, ratio test, root test, integral test for positive valued series;absolute convergence; rearrangements; alternating series; examples: geometric series, harmonic series, power series.<br />

<br />

<strong>10. Complex numbers. </strong><br />

Definitions, operations, complex plane, polar form, root extraction.<br />

</div>

## Bibliography

E. Acerbi, G. Buttazzo: Primo corso di Analisi Matematica, Ed. Pitagora, 1997.<br />

E. Acerbi, G. Buttazzo: Analisi matematica ABC, Ed. Pitagora, 2000.<br />

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

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

## Teaching methods

The exam consists in a written part and an oral part in a different date.<br />

Two evaluations in itinere are fixed: one at the end of the first term and the second at the end of the second term. If both of them are positively marked, the student can skip the written part of the exam.