## MATHEMATICS MATEMATICA

A.Y. Credits
2018/2019 12
Lecturer Email Office hours for students
Alessia Elisabetta Kogoj
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

### Assigned to the Degree Course

Biological Sciences (L-13)
Curriculum: PERCORSO COMUNE
Date Time Classroom / Location
Date Time Classroom / Location

### Learning Objectives

Aim of the course is to give to the students some basic tools and topics in mathematical analysis and linear algebra.

### Program

01. Numbers:
01.01 Natural numbers, integers, rational numbers and real numbers.
01.02 Summations, factorials, binomial coefficients and Newton's binomial formula.
01.03 Algebraic properties and geometric representation of rational numbers.
01.04 From rational numbers to real ones.
01.05 Absolute value and distance in the real line.
01.06 Intervals in the real line. Bounded and unbounded sets in the real line. Maximum and minimum of a subset of the real line. Supremum and infimum of a subset of the real line.
01.07 Mathematical induction and its applications.

02. Functions of one variable:
02.01 What is a function.
02.02 Real functions of one real variable: preliminaries, bounded functions, even and odd functions, monotone functions, periodic functions.
02.03 Elementary functions.
02.04 Scaling and shifting a graph.
02.05 Piecewise defined functions.
02.06 Composition of functions.
02.07 Inverse function.
02.08 Inverse trigonometric functions.

03. Limits of functions:
03.01 Finite limits at a point.
03.02 Uniqueness theorem for limits*.
03.03 Finite limits at infinity.
03.04 Horizontal asymptotes.
03.05 Infinite limits at infinity.
03.06 Oblique asymptotes. Infinite limits at a point.
03.07 One-sided limits.
03.08 Vertical asymptotes.
03.09 Non-existence of a limit.
03.10 Algebra of limits and indeterminate forms.
03.11 Sign-stability theorem*.
03.12 The squeeze theorem*.
03.13 Change of variable in the limits.
03.14 Asymptotic functions.
03.15 Notable special limits.
03.16 Comparison between infinite functions.

04. Sequences:
04.01 Definition of sequence.
04.02 Convergent, divergent and indefinite sequences.
04.03 Monotone sequences.

05. Continuity:
05.01 Continuous functions.
05.02 Algebra of continuous functions.
05.03 Continuity of elementary functions.
05.04 Continuity of the composition of functions.
05.05 Limits for polynomial functions.
05.06 Limits for rational functions.
05.07 Notable special limits.
05.08 Points of discontinuity.
05.09 Continuous functions on an interval: Bolzano Theorem (bisection method)*, Weierstrass Theorem and Intermediate-Value Theorem*.
05.10 Continuity of the inverse function.

06. Differential calculus for functions of one variable:
06.01 Derivative of a function.
06.02 Geometric interpretation of the derivative.
06.03 Tangent line to the graph of a function.
06.04 Derivatives of elementary functions.
06.05 Relations between differentiability and continuity*.
06.06 Algebra of derivatives*.
06.07 Differentiation rules for product and quotient*.
06.08 The chain rule*.
06.09 Derivative of the inverse function*.
06.10 One-sided derivative and non-differentiability points.
06.11 Critical points, local and global maxima and minima.
06.12 Fermat Theorem*.
06.13 Mean-Value Theorem* and its applications: monotonicity test and characterization of functions with zero derivative in an interval.
06.14 Finding maxima and minima of a function.
06.15 De L'Hospital Rule.
06.16 Second-order derivatives.
06.17 Concavity and convexity of a function.
06.18 Inflection points.
06.19 Sketching the graph of a function.

07. Integral calculus for functions of one variable:
07.01 Antiderivative and indefinite integral of a function.
07.02 Antiderivatives of elementary functions.
07.03 Areas of plane regions.
07.04 Definition of definite integral.
07.05 Integrable functions.
07.06 Properties of definite integral.
07.07 A Mean-Value Theorem for integrals*.
07.08 The fundament theorem of integral calculus*.
07.09 Techniques of integration: scomposition and substitution.
07.10 Integrals of rational functions.
07.11 Integration by parts*.
07.12 Integrals of trigonometric functions.
07.13 Integration of irrational functions.
07.14 Improper integrals.
07.15 Integrability criteria: comparison and limit comparison*.

08. Complex numbers:
08.01 Algebraic definition of complex number and operations with complex numbers.
08.02 Gauss plane.
08.03 Conjugate and modulus of a complex number.
08.04 Trigonometric representation of a complex number and De Moivre's Theorem.
08.05 Roots of complex numbers.
08.06 Complex zeros of algebraic equations.

09. Approximation of functions and Taylor's formula:
09.01 Differential of a function and linear approximation.
09.02 "small-o" notation.
09.03 Approximating values of functions and application to the calculus of limits.
09.04 Taylor polynomials.
09.05 Taylor's formula with Peano remainder*.
09.06 Taylor's formula for elementary functions.
09.07 Taylor's formula with Lagrange remainder and integral remainder.
09.08 Applications: approximations of functions, error estimates and calculus of limits.
09.09 Taylor series and Taylor series of elementary functions.

10. Ordinary differential equations:
10.01 Preliminaries.
10.02 Separable equations.
10.03 Linear differential equations: preliminaries and superposition principle.
10.04 First-order linear equations.
10.05 Homogeneous constant-coefficients linear equations of higher-order: solution.
10.06 Non-homogeneous constant-coefficients linear equations of higher-order: solution.
10.07 Euler equation and Bernoulli equation.
10.08 Non-linear differential equations.
10.09 Cauchy problem.
10.10 Local existence and uniqueness for the Cauchy problem.
10.11 Boundary value problems.

11. Linear algebra:
11.01 Vectors and operations on them: sum, difference, scalar multiplication, scalar product and vector product.
11.02 Vector spaces: definition and properties*.
11.03 Vector subspaces. Necessary and sufficient condition for a subset to a subspace*.
11.04 Linear combination of vectors. Subspaces spanned by a set of vectors.
11.05 Linear dependence and linear independence of vectors.
11.06 Basis and dimension of a vector space. Uniqueness of the representation of a vector as linear combination of the elements of a basis*.
11.07 Operations on subspaces: sum and intersection. Grassmann Theorem.
11.08 Direct sum of vector subspaces. Necessary and sufficient condition for a vector space to be direct sum of its subspaces.
11.09 Preliminaries and operations with matrices: sum, scalar multiplication, product and their properties. Vector spaces of matrices (m,n).
11.10 The determinant of a square matrix and its properties.
11.11 Sarrus rule and Laplace Theorem for the determinant.
11.12 Invertible matrices.
11.13 Binet Theorem and its consequences.
11.14 Rank of a matrix. Kronecker Theorem.
11.15 Eigenvalues and eigenfunctions of a matrix. Characteristic polynomial and characteristic equation.
11.16 Linear systems: preliminaries and methods of solution.
11.17 Cramer Theorem.
11.18 Rouché-Capelli Theorem.
11.19 Linear systems depending on parameters.
11.20 Linear maps. Necessary and sufficient condition for a map to be linear*. Image of the zero vector by a linear map*.
11.21 Kernel and image of a linear map and their properties*.
11.22 Rank-nullity theorem.
11.23 Matrix associated with a linear map.

* : this means that the proof is required.

### Bridging Courses

There are no mandatory prerequisites. It is recommended to take the exam of Calculus during the first year of the Laurea Degree Program in Biology and of the Laurea Degree Program in Geology.

### Learning Achievements (Dublin Descriptors)

Knowledge and understanding:
At the end of the course the student will learn the basic notions of mathematical analysis and linear algebra.

Applying knowledge and understanding:
At the end of the course the student will learn the methodologies of mathematical analysis and linear algebra and will be able to apply them to the study of various problems.

Making judgements:
At the end of the course the student will be able to apply the techniques of mathematical analysis and linear algebra in order to solve new problems, also coming from real-world applications.

Communications skills:
At the end of the course the student will have the ability to express the fundamental notions of mathematical analysis and linear algebra using a rigorous terminology.

Learning skills:
During the course the student will learn the ability to study the notions of mathematical analysis and linear algebra, also in order to use it in solving different kind of problems.

### Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

### Supporting Activities

The teaching material and specific communications from the lecturer can be found, together with other supporting activities, inside the Moodle platform › blended.uniurb.it

### Didactics, Attendance, Course Books and Assessment

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Abate, Matematica e statistica. Le basi per le scienze della vita, McGraw-Hill

Bramanti - Pagani - Salsa, Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli

Salsa - Squellati, Esercizi di Analisi matematica 1, Zanichelli
Salsa - Squellati, Esercizi di Analisi matematica 2, Zanichelli

Assessment

The exam of Mathematics consists of a written exam and an oral one, both of them mandatory.
The written exam, to carry out in three hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.
The oral exam consists of a discussion related to the topics of the course. The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

### Additional Information for Non-Attending Students

Didactics

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Abate, Matematica e statistica. Le basi per le scienze della vita, McGraw-Hill

Bramanti - Pagani - Salsa, Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli

Salsa - Squellati, Esercizi di Analisi matematica 1, Zanichelli
Salsa - Squellati, Esercizi di Analisi matematica 2, Zanichelli

Assessment

The exam of Mathematics consists of a written exam and an oral one, both of them mandatory.
The written exam, to carry out in three hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.
The oral exam consists of a discussion related to the topics of the course. The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

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