CALCULUS 1 (UNIT 2)
Learning outcomes of the course unit
Knowledge and understanding:
At the end of this course the student should know the essential definitions
and results in analysis in one variable, and he should be able to grasp how
these enter in the solution of problems.
Applying knowledge and understanding:
The student should be able to apply the forementioned notionsto solve medium
level problems, and to understand how they will be used in a more applied
The student should be able to evaluate coherence and correctness of the
results obtained by him or presented him.
The student should be able to communicate in a clear and precise way, also
in a context broader than mere calculus.
Course contents summary
Real analysis, functions of one variable, sequences, series
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.
Landau symbols; Taylor's theorem; explicit formula of the remainder; Mac Laurin expansion of elementary functions; Taylor's series
Bolzano-Weirstrass theorem, compactness in the real line; Cauchy sequences; upper and lower limits;
Definitions, operations, complex plain, polar form, root extraction.
Nomenclature: order, linear and nonlinear; first examples; solutions of linear first order equations; solution of separable differential equations; constant coefficients linear differential equations.
E. Acerbi, G. Buttazzo: Primo corso di Analisi Matematica, Ed. Pitagora, 1997.
E. Acerbi, G. Buttazzo: Analisi matematica ABC, Ed. Pitagora, 2000.
M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 1, Ed. Zanichelli, 2008.
M. Giaquinta, L. Modica, Analisi Matematica 1, vol. 1 & 2, Ed. Pitagora, 1998.
E. Giusti, Analisi matematica vol.1, Ed. Boringhieri, 2002
classroom lectures and classroom exercises
Assessment methods and criteria
The examination is both written and oral.
In the written part, the student will show his basis knowledge and his ability in solving some particular paroblem. In the oral part, the student will show his knowledge of the foundamental theorems of Mathematical Analisys 1. The oral exposition must be done using a proper mathematical formalism.