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The course provides an introduction to differential and integral calculus for functions of one real variable; geometric and physical motivations and applications thereof will be stressed throughout.
Mathematical Analysis (advanced module): Syllabus (tentative) ?The course provides an introduction to differential and integral calculus for functions of one real variable; geometric and physical motivations and applications thereof will be stressed throughout.
Historical introduction. The notion of derivative and its geometrical and physical meaning.
Derivatives of higher order . Derivatives of elementary functions. Rules of differentiation.
Critical points. The theorems of Fermat, Rolle, Lagrange, Cauchy. de l'Hopital's rule. Convexity and second derivative. Taylor's formula. Applications to function plotting.
The definite (Riemann) integral. Integrability of continuous functions. The integral function. The mean value theorem for definite integrals. Indefinite integrals. The fundamental theorem of calculus. Integration techniques. Applications to areas and volumes. Cavalieri's principle. Numerical series. Improper integrals. Introduction to first order differential equations; geometrical meaning; elementary methods of integration (separation of variables, variation of parameters).
NOTES: 1. Some complementary lecture notes to the text below will be made available.
2. The programme is tentative and subject to slight changes.
Reference text (with exercises):
M.CONTI, D.L. FERRARIO, S.TERRACINI, G.VERZINI, Analisi Matematica -
Dal calcolo all'analisi, vol.1 Apogeo, Milano, 2006.
Assessment : written exam (with some "qualifying" questions therein, among others), followed by an oral exam (optional).
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