PRINCIPLES OF MATHEMATICS
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- Versione italiana
- Academic year
- 2015/2016
- Teacher
- UMBERTO MASSARI
- Credits
- 12
- Didactic period
- Annualità Singola
- SSD
- MAT/05
Training objectives
- The aim of the course is to give some basic concepts of calculus and analytic tools to work with real numbers, complex numbers and real functions of one real variable.
The main knowledge provided by the course will be:
- the concepts of real number and real function of one real variable;
- the concepts of lower extreme (infimum) and upper extreme (supremum) for sets and functions;
- the induction principle;
- the concept of limit for a sequence of real numbers and for a function;
- some methods for the calculation of limits;
- the concept of continuity for a real function of one real variable;
- the concept of derivability for a real function and the methods for the calculation of derivatives;
- the concept of Riemann integrability for a real function;
- the main methods for the calculation of primitives and definite integrals;
- the concept of complex number, its properties and the operations in the set of complex numbers;
- the concept of ordinary differential equation (ODE);
- some methods for the solution of ODE of the first and second order.
The main skills that students should acquire (that is to say, the abilities to apply their knowledge) will be:
- to be able to check the properties of real numbers and determine specific sets of real numbers;
- to be able to determine the lower bound (infimum) and/or the upper bound (supremum) of sets and functions;
- to know how to apply the induction principle;
- to be able to verify the properties of real functions and recognize the main elementary functions (power, logarithm, exponential, trigonometric, ...);
- to be able to recognize the properties of numerical series and real functions;
- to be able to check by definition the limit of numerical sequences and real functions;
- to know how to calculate the limit of numerical sequences and real functions by various methods;
- to be able to carry out operations with complex numbers and use their representation properties;
- to be able to check the continuity properties of real functions of one real variable;
- to be able to determine the derivability of real functions of one real variable;
- to know how to compute derivatives of every order of derivable real functions of one real variable;
- to know how to study a real function of one real variable and draw an accurate qualitative plot;
- to be able to determine the Riemann integrability of real functions of one real variable;
- to know how to compute the primitives and the definite integral of Riemann-integrable real functions;
- to be able to solve simple first order ODEs (linear and nonlinear) and second order linear ODEs with constant coefficients. Prerequisites
- Just some basic knowledge of elementary calculus is needed, which is usually taught in high schools: sets, numbers, algebraic operations, representations in the Cartesian plane, coordinate systems, solution of first and second order algebraic equations, inequalities, radicals, operations with algebraic polynomials (sum, difference, multiplication, quotient), factorization of algebraic polynomials (relevant products, Ruffini rule, relevant polynomials).
Course programme
- The course is divided in two modules and lasts 96 hours of lessons. About half of total class time is devoted to examples and exercises.
Module 1 content (48 hours):
- main properties of real numbers, the function concept and some recalls about elementary functions (10);
- upper bound (supremum) and lower bound (infimum) of sets and functions (10);
- sequences and sequence limit (6);
- limits of functions (10);
- continuous functions and properties of continuous functions defined over intervals (7);
- complex numbers (5).
Module 2 content (48 hours):
- differential calculus for functions of one real variable: properties of derivable functions on an interval (8);
- applications to the study of the plot of a function (10);
- calculus: definition of Riemann-integrable functions and integral properties (16);
- first order linear differential equations and second order linear differential equations with constant coefficients (14). Didactic methods
- The course includes theoretical lectures, accompanied by blackboard exercises carried out by the teacher on all the topics.
Learning assessment procedures
- Purpose of the examination tests is to check whether the students achieved an adequate level of the course educational goals or not, with respect to both the knowledge and the skills.
The examination consists of a written test, aimed to assess the student's ability to solve problems and exercises, and of an oral test, aimed at evaluating the theoretical knowledge. The oral test may be optional, if the written test is marked sufficient.
The final grade of the examination is given by:
- one mark for the written test, possibly acquired through partial tests,
- one mark of a (possibly optional) oral test.
Written test.
The score for the written test is given by the mark of the test session, or by partial tests. In the case of partial tests, the final score of the written test is the arithmetic average of the score achieved in the partial test of module 1 and the final score achieved in the partial test of module 2, rounded up to the nearest integer. If that average is greater than or equal to 14, it will be the final score of the written test. Notice that this mark will NOT be greater than 24 in any case. If the average is less than 14, then the written test is not passed and the student have to repeat it at one of the next written test sessions (no additional partial tests are planned after the end of the lessons).
Oral test.
Depending on the final grade reported in the written test, the oral examination can be mandatory or optional. If the final score of the written test (determined by whether partial tests or session test) is between 14 and 17 (endpoints included), then the oral test is mandatory to pass the exam. If the final score of the written test is greater than or equal to 18 points, then it is up to the student to choose whether to give the oral test or not,
to try to improve the exam final score. Notice that the oral test is not necessarily ameliorative. In the case a student decides to give the oral test to improve his rating after the partial tests, the starting point for evaluation will be the average of the partial tests scores, even if it is larger than 24.
Notes
Those students having their partial tests scores greater than or equal to 18, can still participate to the session test: in this case, if the test is delivered to the teacher for correction, then this score DEFINITIVELY replaces the score achieved with the partial tests (it will NOT be kept the best of the two). If the student chooses to try the session test, it is on him to decide whether to hand over the task or not. Further specifications for additional special cases can be found on the course website.
Duration.
The final score of the written test, no matter how it is acquired, survives until the last examination session (January-February session) of the academic year where the course to which the test refers was held. Reference texts
- Main references:
- Teacher's notes;
- G. Buttazzo, G. Gambini, E. Santi: "Esercizi di Analisi Matematica I", Pitagora Editrice, Bologna, 1991.
Additional readings:
- F. Rosso, L. Fusi: "Matematica per le lauree triennali" (possibly, but not necessarily, with the volume "Esercizi di Matematica per le lauree triennali"), CEDAM, Trento, 2013;
- P. Marcellini, C. Sbordone: "Analisi Matematica I", Liguori Editore, Napoli, 1998;
- M. Bartsch, R. Dal Passo, L. Giacomelli, "Analisi matematica", seconda edizione, McGraw-Hill Italia, Milano, 2011.
