# FUNDAMENTALS OF MATHEMATICS (004AR)

#### A.Y. 2019 / 2020

Professor
Period
First semester
Credits
8
Duration/Length
64
Type of Learning Activity
Core subjects
Study Path
[PDS0-2018 - Ord. 2018] common
Syllabus
Teaching language

Italian

Learning objectives

The course aims to introduce students to the fundamental techniques of mathematical reasoning; constitute objectives in the course of study:
•D1 Knowledge of the basic notions and tools of mathematical analysis and analytical geometry at a post-secondary level, characterized by the use of advanced textbooks;
• D2 Competence in using these tools for the application to particular cases, concrete or abstract, to the study of the graphs of simple functions.
• D3 Ability to draw conclusions from the collected data, to apply the tools to the study of experimental phenomena to correctly interpret the data and to solve more complex problems;
• D4 Communication skills: students must communicate the topics studied with a correct formal mathematical language, using the appropriate logical connectives without ambiguity.
• D5 Students will be able to read some mathematical passages autonomously from the texts from which to obtain the necessary data.

Prerequisites

Prerequisites are established with a placement test that will be administered in the first day.

Contents

SET THEORY: Sets; union, intersection, difference; subsets; powerset; complementary to a set; Cartesian product.
LOGICA: propositions; conjunction, disjunction, negation, implication and property; logical equivalence; truth tables; de Morgan's laws. Logical Predicates, quantifiers and basic properties. theorems; assumptions; thesis; proof by contradiction.
NUMERICAL SETS: The numbers: N,Z, Q, R. Real numbers axioms.
GEOMETRY IN THE PLANE AND SPACE.Coordinate Cartesian plane. Points and vectors. Distance in metric space; Euclidean distance on the line, plane and space. Absolute value. Density of Q in R. D Equation of circumference: from the point of the point to the equation; equation of a spherical shell. The conic as a locus; parable with axis parallel to the y axis equation , ellipse with fires on the x axis.
LINEAR ALGEBRA Vector space; sum of vectors, produced of a vector and a scalar, dot product of vectors in the plane and space. Condition of perpendicularity between vectors. Linear subspace; definition of linearly independent and dependent vectors; vector space generator system; base of a vector space; base of R2 and R3; a condition of parallelism between vectors. Parametric equation of the line, passage from parametric to Cartesian, Cartesian equation, equation in explicit form. perpendicularity conditions. Point-to-Point Distance. Plane equation known a normal vector, plane equation note a point and two vectors, plan equation known three points; special plans; equation of a line; directors parameters; fractional equation; straight as a two-planes intersection; relationship between plane and straight coefficients; distance of a point from one plane.
FUNCTIONS. Domain, range, function; gtsph of a function; vertical translation, exponential, logarithm of a function known to the graph. upper and lower extremities, maximum and minimum of a subset of R and of a function. Open and closed sets, intersections, accumulation points and isolated points, definitions, and examples. Limit definition. Limit operations, polynomial limits, and polynomial quotients. Limit uniqueness theorem; Sign permanence theorem, Comparison theorem. Significant limits. Continuous functions. Singular points; Continuous function theorems: Weierstrass theorem, intermediate values, and zeros. Asymptotes.
DIFFERENTIAL CALCULUS: Derivative of a function:definition, geometric meaning. Sum, product, quotient, Derivative functions continuity theorem; cuspidities, angular points, vertical tangent inflection. Derivative techniques applied to the main functions. Equation of the tangent line at a point on the graph. Derivative of a composite function and inverse function. Stationary points. Fermat theorem. Rolle theorem and Lagrange theorem . Monotony and sign of the derivative. Second Derivative,convexity, concavity, inflections . Function Studio. Cauchy and De l'Hopital theorems.

INTEGRATION : the primitive of a function, the indefinite integration; integration by parts and replacement. Integration of rational functions. The problem of calculating the areas ; The media theorem. The fundamental theorem of integral calculus, defined integrals.

Teaching format

Lessons of frontal type alternating with collective discussion; lessons in which the General proof will go to special, alternating with lessons conducted for problems.The activity of a tutor is foreseen . The tutor will correct the exercises proposed weekly by the teacher and carried out independently by the students and will manage group work sessions.

Extended Programme

SET THEORY: Sets; union, intersection, difference; subsets; powerset; complementary to a set; Cartesian product.
LOGICA: propositions; conjunction, disjunction, negation, implication and property; logical equivalence; truth tables; de Morgan's laws. Logical Predicates, quantifiers and basic properties. theorems; assumptions; thesis; proof by contradiction.
NUMERICAL SETS: The numbers: N,Z, Q, R. Real numbers axioms.
GEOMETRY IN THE PLANE AND SPACE.Coordinate Cartesian plane. Points and vectors. Distance in metric space; Euclidean distance on the line, plane and space. Absolute value. Density of Q in R. D Equation of circumference: from the point of the point to the equation; equation of a spherical shell. The conic as a locus; parable with axis parallel to the y axis equation , ellipse with fires on the x axis.
LINEAR ALGEBRA Vector space; sum of vectors, produced of a vector and a scalar, dot product of vectors in the plane and space. Condition of perpendicularity between vectors. Linear subspace; definition of linearly independent and dependent vectors; vector space generator system; base of a vector space; base of R2 and R3; a condition of parallelism between vectors. Parametric equation of the line, passage from parametric to Cartesian, Cartesian equation, equation in explicit form. perpendicularity conditions. Point-to-Point Distance. Plane equation known a normal vector, plane equation note a point and two vectors, plan equation known three points; special plans; equation of a line; directors parameters; fractional equation; straight as a two-planes intersection; relationship between plane and straight coefficients; distance of a point from one plane.
FUNCTIONS. Domain, range, function; gtsph of a function; vertical translation, exponential, logarithm of a function known to the graph. upper and lower extremities, maximum and minimum of a subset of R and of a function. Open and closed sets, intersections, accumulation points and isolated points, definitions, and examples. Limit definition. Limit operations, polynomial limits, and polynomial quotients. Limit uniqueness theorem; Sign permanence theorem, Comparison theorem. Significant limits. Continuous functions. Singular points; Continuous function theorems: Weierstrass theorem, intermediate values, and zeros. Asymptotes.
DIFFERENTIAL CALCULUS: Derivative of a function:definition, geometric meaning. Sum, product, quotient, Derivative functions continuity theorem; cuspidities, angular points, vertical tangent inflection. Derivative techniques applied to the main functions. Equation of the tangent line at a point on the graph. Derivative of a composite function and inverse function. Stationary points. Fermat theorem. Rolle theorem and Lagrange theorem . Monotony and sign of the derivative. Second Derivative,convexity, concavity, inflections . Function Studio. Cauchy and De l'Hopital theorems.

INTEGRATION : the primitive of a function, the indefinite integration; integration by parts and replacement. Integration of rational functions. The problem of calculating the areas ; The media theorem. The fundamental theorem of integral calculus, defined integrals.

End-of-course test

The exam consists of a written test where will be formulated theoretical questions and required application exercises.
If written test will get a rating grater than or equal to 16/30, it may be supplemented by an oral examination in the same session.
Oral examination could raise the rating upto five points.

For students attending the course it is planned to take the exam in three partial tests on the topics gradually developed

Texts/Books

Robert A. Adams Calcolo Differenziale I, ed. CEA (Casa EditriceAmbrosiana) http://www.cds.caltech.edu/~marsden/volume/Calculus/

https://www.math.uh.edu/~dlabate/settheory_Ashlock.pdf

Bramanti, Pagani, Salsa “Analisi Matematica 1. Con elementi di geometria e algebra lineare”, Zanichelli

Giovanni Naldi, Lorenzo Pareschi, Giacomo Aletti “Calcolo differenziale e algebra lineare”, Mc Graw-Hill

Gli studenti che hanno frequentato negli anni passati il corso facciano tranquillamente riferimento ai testi in loro possesso.

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