B2288 - ISTITUZIONI DI MATEMATICA T-1

Learning outcomes

At the end of the course the student knows the methodological-operational aspects of mathematical analysis and some of its applications, with particular regard to the functions of one variable.

Course contents

WARNING: in order to optimally spread the study load between this course and the follow-up, Istituzioni di Matematica T-2, topic 8 has been reduced and point 9 has been added.

 

1. Fundamental concepts and tools:
   Sets, numerical sets, extremal points: supremum and infimum.
   
   
2. Functions:
   Definition of functions, domain, range, graphic; monotone functions, injective, surjective, bijective functions; function composition and inverse of a function. Elementary functions of real numbers.
   
   
3. Sequences and limits of sequences:
   Definition of sequence and of its limit, monotone sequences. Computation of limits. Some relevant limits. Comparison theorems.
   
   
4. Numerical series:
   Partial sums and convergin series. Necessary condition for convergence. Non-negative-terms series: convergence criteria. Variable-signed-terms series: absolute convergence and convergence. Geometric series.
   
   
5. Limits of functions of real variable:
   Cluster points. Definition of limit. Right and left limits. Asymptotes of functions.
   Computation of limits: fundamental limits, operations, limits of composition of functions.
   Continuous functions and points of discontinuity. Continuous functions on an interval: Bolzano’s theorem, intermediate values, Weierstrass’.
   
   
6. Derivatives of functions of real variable:
   Derivative of a function at a point: analytical and geometric interpretations. Derivative as a function and its computation: derivative of elementary functions, derivation rules. Differentiable functions and point of non-differentiability.
   Lagrange’s mean value theorem and its conseguences, monotonicity. Relative maxima and minimia and the Fermat’s theorem. De L’Hopital’s theorem.
   Higher-order derivatives: inflection points and concavity. Overview on Taylor’s series.
   
   
7. Anti-derivation and integrals:
   Indefinite integral and the inverse of derivation. Computation of indefinite integrals: integration by parts and by substitutions, and some rational functions.
   Definite integral of continuous real functions: properties and integral mean theorem. Fundamental theorem of calculus. Some applications in physics and geometry.
   
   
8. Analytic geometry on the plane and in the space:
   (FORMER VERSION, NO LONGER VALID)
   Points and vectors in the plane and in the space. Properties and operations between vectors. Lines and planes in the space. Distance, parallel lines and planes and intersections.
   (CURRENT REDUCED VERSION)
   Points on the plane: vectors and coordinates. Operations between vectors, computation of distance and angles. Lines on the plane: equations, parallel and crossing lines. Circles on the plane.
   
   
9. Complex numbers:
   Imaginary unit, definition and operations between complex numbers. Polar representation: powers and roots of complex numbers. Fundamental theorem of algebra (statement).
   

Readings/Bibliography

Theory: M. Bramanti, C. D. Pagani, S. Salsa: Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli.

Exercises: M. Boella: Analisi matematica 1 e algebra lineare. Eserciziario. Pearson

Teaching methods

Theoretical frontal lectures (about 4hrs per week).

Examples and exercises interactive sessions (about 1hr per week).

Assessment methods

The exam consists of a preliminar written test and an oral one.

The written examination targets the evaluation of the acquired knowledge in terms of problem solving. It is made of 5 to 6 exercises on the topics of the course. The completion of the test (minimum score: 15) is mandatory to attend the oral exam.

The oral test evaluates the acquisition of the theoretical contents of the course, including concepts, definitions and theorems. It can provide either a positive or negative score.

Six exam sessions will be scheduled, divided into three periods (January-February, June-July, September). The completion of the written test on the first session of each period allows to take the oral test in the same session or in the following one (within the same period).

Teaching tools

Virtuale platform for additional material

Office hours

See the website of Luca Ratti