Learning objectives
The course provides the basic mathematical instruments for a solid comprehension of the other courses. <br />
Prerequisites
Basic knowledge of mathematics.
Course unit content
<p>Preliminary requirements: elementary algebra; trigonometry; analytic geometry; rational powers; exponential and logarithm; elementary functions. <br />
Logic and set theory; equivalence and ordering. <br />
Numerical sets: natural numbers and induction principle; combinatorial calculus and elementary probability; integers and rationals; real numbers and supremum; complex numbers and their n-th roots. <br />
Real functions: maximum and supremum; monotone, odd and even functions; powers; irrational functions; absolute value; trigonometric, exponential and hyperbolic functions; graphs of the elementary functions and geometric transformations of the same. <br />
Sequences: topology; limits and related theorems; monotonic sequences; Bolzano-Weierstrass and Cauchy theorems; basic examples; the Neper number “e”; recursive sequences; complex sequences. <br />
Continuous functions: limits of functions; continuity and properties of continuous functions (including intermediate values, Weierstrass theorem); uniform continuity and Heine-Cantor theorem; Lipschitz continuity; infinitesimals. <br />
Properties of differentiable functions (including Rolle, Lagrange, Hopital theorems); Taylor expansion (with Peano and Lagrange remainder); graphing a function. <br />
Indefinite and definite integral: definition and computation (straightforward, by parts, by change of variables); integral mean and fundamental theorems; Torricelli theorem; generalised integrals: definition and comparison principles. <br />
Numerical series: definition, convergence criteria, Leibniz and integral criteria. <br />
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Full programme
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Bibliography
Theory and basic exercises: <br />
E. ACERBI e G. BUTTAZZO: "Primo corso di Analisi matematica", Pitagora editore, Bologna, 1997 <br />
D. MUCCI: “Analisi matematica esercizi vol.1”, Pitagora editore, Bologna, 2004 <br />
examination exercises: <br />
A. COSCIA e A. DEFRANCESCHI: "Primo esame di Analisi matematica", Pitagora editore, Bologna, 1997 <br />
Teaching methods
Oral lectures. Exercitations in small groups
Assessment methods and criteria
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Other information
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