Enrico Giusti, Elementi di Analisi Matematica, 2008, Bollati
Boringhieri
Suggested readings for the prerequisites:
Giuseppe Anichini, Antonio Carbone, Paolo Chiarelli, Giuseppe Conti, Precorso di matematica, 2010, Pearson Education, pag. 272, Euro 17,00 - ISBN 9788871925899
Roberto D'Ercole - Precorso di Matematica per Economia e Scienze, 2011, Pearson Education, pag.264 Euro 17,00 - ISBN 9788871926308
Paolo Boieri, Giuseppe Chiti, Precorso di matematica, 1994, Zanichelli, Euro 22,30 - ISBN 9788808091864
M.Castellani, F.Gozzi, M.Buscema, F.Lattanzi, L.Mazzoli, A.Veredice,
Precorso di matematica, 2007, Società editrice Esculapio, Pag. 205, e23,00
Texts for exercises:
Learning Objectives - Last names A-C
The students will learn the basic elements of the scientific method and of calculus: limits, continuity, and derivatives.
The aim of this course is to provide the necessary tools for developing and studying mathmatical methods using functions of a single variable.
The students will be able to develop a complicated logical-mathematical argument through elementary steps.
Students will be able to appreciate the importance of assumptions in theorems.
Students will be able to define and apply the basic ideas of calculus as limits and derivatives.
Students will be able to solve equations approximately, using appropriate algorithms.
Students will be ablve to descrive how the problems of calcolusm arise from economic, or social, or political, or environments phenomema.
Students will be able to model these problems in mathematical terms.
Students will be able to solve these problems and obtain results which will improve their understanding of reality.
Learning Objectives - Last names D-L
1. mathematical tools for the analysis of real functions of one real variable;
2. applications to simple economic problems.
Prerequisites - Last names D-L
1. Sets of numbers; Arithmetic.
The natural numbers. The integers. The rationals. An intuitive idea of
real numbers. Arithmetic operations and their properties. Percentages. Prime numbers. Factorization of a natural number. Greatest common divisor and least common multiple. Inequalities and manipulation of inequalities. Absolute value. Powers, roots and their properties. Simplifying expressions.
2. Basic algebra, equations, inequalities.
Polynomials. Special products. How to factor polynomial into irreducible
terms in simple cases. Polynomial identity. Ratios of polynomials. Identities
and equations. Solutions for an equation. Equations of degree one or degree
two. Solutions (roots) of a polynomial. Linear systems with two equations
and two unknowns. Equations containing ratios of polynomials. Equations
with radicals. Inequalities. Solving inequalities of degree one and degree two,
inequalities, with ratios of polynomials, with radicals.
4. Analytic geometry.
Cartesian coordinates in the plane. Pythagorean theorem, Distance between
two points. Equation of the line. Parallel lines and perpendicular lines. Equation of the parabola. Equatiion of a circle
Teaching Methods - Last names D-L
Class lectures. The course lenght is 12 weeks with three classes per week.
Further information - Last names D-L
Tutor junior.
Also for the academic year 2015-2016, it will be possible to attend review/exercise classes with some senior students.
Students can attend classes in any of the four offered courses, but they are invited to attend the course corresponding to the first letter of their last name.
Students have to take the exam with the professors teaching the course corresponding to the first letter of their last name.
The course has a Web page on the platform Moodle of e-learning, no password is needed. It is also possible to access as an host
For further ionformation on the preparation of the exam: see Regolamento on the Moodle page
Type of Assessment - Last names D-L
There will be a written exam. Reaching a passing grade on the written exam gives access to an oral exam. If this last is passed then the exam is succesfully concluded.
For further information on the exam rules, please look at the Regolamento on the Moodle page
Course program - Last names D-L
1. The role of mathematics in applications.
2. Sets theory.
3. Introduction to mathematical reasoning
4. Real numbers: algebraic and order properties. Geometric representation
of real numbers: the real line. Topology of the real line. Neighborhood
of a point. Interior points and accumulation points. Bounded sets, from
above/from below. Supremum and infimum of a set.
5. Functions. Domain, image and graph of a function. Injective and inverse functions. Composition of functions. Domain of the
composite function. The exponential function. The logarithmic function.
Other funtions: power, integer part, piecewise defined, Heaviside, sign, absolute value, sine, cosine, tangent.
6. Monotone functions. Strict monotonicity and invertibility. Concave and convez funztions. Functions
bounded above/below. Supremum and Infimum of a function; global maximum/minimum of a function, maximizer/minimizer
7. Limit of a function. Uniqueness of the limit. Sign invariance theorem.
Comparison or squeeze theorem. Left and right limits. Operations with
limits: multiplication by a number, sum, product, quotient. Limit of
composite functions (change of variables). Limits at infinity, infinite limits.
Infinities and Infinitesimals. Limits of rational functions, of logarithmic
functions, and of the exponential functions. Indeterminate forms. Special limits
8. Continuous functions. Definition and examples. Continuity of elementary
functions. Operations with functions and continuity. The intermediate
value theorem. Bisection method. Global and local maxima and minima and local minimizer/maximizer.
Weierstrass theorem.
9. Definition of derivative. Tangent line. Rules of differentiation for sum,
product, quotient and composition. Fermat Theorem, Rolle and Lagrande
Theorem, de l'Hospital Theorem. Sign of derivative and monotonicity.
Sign of second derivative and convexity. Sign of the
second derivative as a sufficient condition for local maxima and minima.
Taylor formula