### Objectives

General skills
This course concerns the applications of the fundamentals tools of calculus, probability and statistics to the solution of problems emerging within Natural Sciences, with a specific reference to Biological Sciences.
The main goal is for learners to understand the basic concepts of linear algebra, differential and integral calculus, discrete and continuous probability, as well as their application to the analysis of empirical data in biological sciences, in particular in the realm of genetics and evolution sciences.
The fact that students actually have the mentioned pre-knowledge is certified by the entrance test or, when appropriate, by the OFA course and exam.

The course includes both lectures and exercise sessions, aiming to test the ability of the students to apply the theoretical knowledge to the solution of concrete problems.

Specific skills

A) Knowledge and understanding
Knowledge and understanding of the basic notions of linear algebra (vectors, matrices, linear systems).
Knowledge and understanding of the concept of limit and of the fundamentals of differential and integral calculus.
Knowledge and understanding of the fundamentals of probability theory and of some tools of inferential and descriptive statistics.
Knowledge and understanding of data mining and of the diagrammatic representation of row data.
Knowledge and understanding of statistical tests for the analysis of empirical data.

B) Applying knowledge and understanding
Ability to properly use the specific terminology of mathematics and statistics.
Ability to translate a concrete problem, appearing e.g. in the context of Biological Sciences, to a corresponding mathematical problem, by a suitable procedure involving approximation, abstraction, and modeling.
Ability to use deductive reasoning in an abstract setting.
Ability to recognize the mathematical tools and concepts appearing within other courses (specifically: Physics, Chemistry, Genetics, Ecology) and to properly use them.
Ability to find the most convenient procedure to solve simple mathematical problems.
Ability to use appropriate software to treat and analyze empirical data.

C) Making judgements
Ability to autonomously judge the validity of a mathematical statement, through a critical analysis of the hypotheses and of the deductive steps leading to the proof of the statement itself.
Ability to autonomously formulate counterexamples to mathematical statements, whenever one of the hypotheses is denied.
Ability to self-questioning.
Ability to autonomously evaluate the validity of a theoretical model, through suitable statistical tests on the empirical data collected in a laboratory.

D) Communication skills
Ability to communicate what has been learned through written themes and oral exams.
Ability to formulate a logically structured speech, clearly distinguishing between hypotheses, deduction and thesis.

E) Learning skills
Learning the specific terminology.
Ability to make the logical connections between the topics covered.
Ability to identify the most relevant topic

### GIANLUCA PANATI   Teacher profile

#### Programme

Basic Mathematics. Numbers and algebraic operations; equations and inequalities; geometric representation of real numbers, fundamentals of analytic geometry (cartesian coordinates, distance between points in the cartesian plane).

Differential and Integral Calculus. The concept of function, graph and properties of elementary functions: polynomial, exponential, logarithmic, trigonometric functions; asymptotic behavior of sequences and functions (horizontal and vertical asymptotes); the notion of continuity of a function; incremental ratio; fundamentals of differential calculus (definition and geometric meaning of the derivative, derivatives of elementary functions, Leibnitz rule, chain rule); differentiability imlies continuity; maxima and minima; fundamental concepts concerning the integral calculus (definition and geometric meaning of integrals; Torricelli-Barrow theorem; integration by parts).

Probability Theory. Experiments and sample spaces; events (definition, intersection and union of events, incompatible events); probability of an event; probability of the union of two events; conditioned probability and independent events; elements of combinatorial calculus (dispositions, permutations and combinations); discrete random variables, expectation values and variance, random variables with binomial or Poisson distribution. Continuous random variables (density and distribution, expected value and variance); random variables with uniform, exponential or normal distribution.

As for the Calculus part, we suggest:

[BDM] D. Benedetto, M. Degli Esposti, C. Maffei, Matematica per le scienze della vita, Casa Editrice Ambrosiana, Milano, 2012.
[LMN1] L. Lamberti, L. Mereu, A. Nanni. Nuovo Lezioni di Matematica, vol. D. Etas RCS, Milano, 2012. [ISBN 978 88 451 6513 9] oppure [ISBN 978 88 451 6515 3].
[LMN2] L. Lamberti, L. Mereu, A. Nanni. Nuovo Lezioni di Matematica, vol. E. Etas RCS, Milano, 2012. [ISBN 978 88 451 6508 5] oppure [ISBN 978 88 451 6510 8].

For the part concerning Biostatistics, we suggest the following textbook, which will be useful also for Part 2 (MMIB):

[Ross] Sheldon M. Ross, Probabilità e statistica per l'ingegneria e per le scienze. Apogeo, Milano, 2003.

Further information concerning the course, including the “Lecture Calendar” and a wide selection of problems and exercises, is available through the course website one the e-Learning platform: https://elearning2.uniroma1.it

#### Prerequisites

It is assumed that students entering the course know the basics of elementary mathematics from the high school. The fact that students actually have the mentioned pre-knowledge is certified by the entrance test or, when appropriate, by the OFA course and exam.

#### Exam modes

Part 1: Calculus and Biostatistics

The exam consists of a written part, to evaluate the ability of the student to solve simple exercises and problems, and of an oral part. The written part contains some open-question problems, and its duration is at least 120 minutes.
The oral part consists in a short colloquium, and defines – together with the written part – the final mark for Part 1.

Part 2: Mathematical and informatic methods for biology

The exam consist of unified evaluation, including a written part and, possibly, a second part involving the use of a statistical software in the computer lab.
Final mark

The final mark (for both Part 1 + Part 2) is computed as the weighted average of the marks of the two parts, where weights are proportional to the number of CFUs (9+3). To achieve the level “cum laude”, the student is required to obtain “cum laude” in Part 1, and to pass Part 2 with at least 30/30.

Exam reservation date start Exam reservation date end Exam date
25/01/2021 26/03/2021 07/04/2021
25/01/2021 06/06/2021 09/06/2021
25/01/2021 07/07/2021 09/07/2021
25/01/2021 24/09/2021 27/09/2021
23/11/2021 28/01/2022 31/01/2022

### ELENA AGLIARI   Teacher profile

#### Programme

CALCULUS
Natural, integer, rational, real numbers. The real line, intervals, absolute value.
Linear algebra. Vectors in R^2. Linear combination of vectors. Geometric representation.
Scalar product. Matrices, determinants. Linear equation systems. Theorem of Cramer. Geometric applications.
Functions. Domain and codomain of a function. Composition of functions and inverse function.
Linear, polynomial, rational functions, trigonometric, exponential, logarithmic: main properties and graphic representation.
Limits of functions. Remarkable limits. Asymptotes.
Derivatives. Incremental ratio and derivative of a function; geometric meaning of derivative, straight tangent.
Derivation rules: sum, product, quotient, compound function, inverse and derivatives of fundamental functions.
Linear approximation of functions.
Growing and decreasing of a function. Maxima and minima. Fermat's theorem.
Stationary points. Function graphs.
Integrals. Integrals of scale functions. Definite integrals, geometric meaning and their properties.
Primitives. Immediate indefinite integrals and notes on integration methods.
Basic rule of integral calculus.
BIOSTATISTICS
Descriptive statistics: Arithmetic mean, geometric mean, median.
Standard deviation. Histogram, mode. Correlation. Linear regression.
Probabilities and random variables. Definitions and fundamental properties.
Probability of repeated tests. Independent events. Discrete random variables.
Binomial variable. Average of a random variable.
Continuous random variables: uniform, exponential, Gaussian.

- D. Benedetto, C. Maffei, M. Degli Esposti: “Matematica per le scienze della vita”, II edizione, CEA
- C. Cammarota: Elementi di Calcolo e di Statistica - Libreria Scientifica Dias

#### Bibliography

- L. Lamberti, L. Mereu, A. Nanni. Nuovo Lezioni di Matematica, vol. D . Etas RCS, Milano, 2012. - L. Lamberti, L. Mereu, A. Nanni. Nuovo Lezioni di Matematica, vol. E . Etas RCS, Milano, 2012. - D. S. Moore: “Statistica di base”, II edizione, APOGEO - Sheldon M. Ross, Probabilità e statistica per l'ingegneria e per le scienze. Apogeo, Milano, 2003

#### Exam modes

The written test involves the resolution of exercises related to the program of the course.

Exam reservation date start Exam reservation date end Exam date
15/12/2020 27/01/2021 28/01/2021
28/12/2020 28/01/2021 29/01/2021
15/01/2021 28/02/2021 01/03/2021
15/03/2021 08/04/2021 09/04/2021
15/06/2021 29/07/2021 30/07/2021
15/07/2021 29/09/2021 30/09/2021
15/11/2021 25/11/2021 26/11/2021

### VITO CRISMALE   Teacher profile

#### Programme

CALCULUS
Natural, integer, rational, real numbers. The real line, intervals, absolute value.
Linear algebra. Vectors in R^2. Linear combination of vectors. Geometric representation.
Scalar product. Matrices, determinants. Linear equation systems. Theorem of Cramer. Geometric applications.
Functions. Domain and codomain of a function. Composition of functions and inverse function.
Linear, polynomial, rational functions, trigonometric, exponential, logarithmic: main properties and graphic representation.
Limits of functions. Remarkable limits. Asymptotes.
Derivatives. Incremental ratio and derivative of a function; geometric meaning of derivative, straight tangent.
Derivation rules: sum, product, quotient, compound function, inverse and derivatives of fundamental functions.
Linear approximation of functions.
Growing and decreasing of a function. Maxima and minima. Fermat's theorem.
Stationary points. Function graphs.
Integrals. Integrals of scale functions. Definite integrals, geometric meaning and their properties.
Primitives. Immediate indefinite integrals and notes on integration methods.
Basic rule of integral calculus.
BIOSTATISTICS
Descriptive statistics: Arithmetic mean, geometric mean, median.
Standard deviation. Histogram, mode. Correlation. Linear regression.
Probabilities and random variables. Definitions and fundamental properties.
Probability of repeated tests. Independent events. Discrete random variables.
Binomial variable. Average of a random variable.
Continuous random variables: uniform, exponential, Gaussian.

- D. Benedetto, C. Maffei, M. Degli Esposti: “Matematica per le scienze della vita”, II edizione, CEA
- C. Cammarota: Elementi di Calcolo e di Statistica - Libreria Scientifica Dias

#### Bibliography

- L. Lamberti, L. Mereu, A. Nanni. Nuovo Lezioni di Matematica, vol. D . Etas RCS, Milano, 2012. - L. Lamberti, L. Mereu, A. Nanni. Nuovo Lezioni di Matematica, vol. E . Etas RCS, Milano, 2012. - D. S. Moore: “Statistica di base”, II edizione, APOGEO - Sheldon M. Ross, Probabilità e statistica per l'ingegneria e per le scienze. Apogeo, Milano, 2003

#### Prerequisites

CALCOLO E BIOSTATISTICA - 9 CFU It is assumed that students entering the course know the basics of elementary mathematics from the high school. The fact that students actually have the mentioned pre-knowledge is certified by the entrance test or, when appropriate, by the OFA course and exam. Basic Mathematics: numbers and algebraic operations; equations and inequalities; geometric representation of real numbers, fundamental notions of analytic geometry (Cartesian line and plane, two-point distance in the Cartesian plane, form concept). Powers, logarithms, trigonometry.

#### Exam modes

The written exam consists of both exercises and questions concerning the theory. There are two intermediate checkpoints, which can substitute the written exam.
An oral exam could be requested and is necessary to obtain full marks evaluation.

### CAMILLO CAMMAROTA   Teacher profile

#### Programme

Basics of Mathematics: numbers and basic algebra; equations and inqualities; geometric representation of real numbers; basics of geometry on the plane: straight lines, distance of two points and modulus.

CALCOLO E BIOSTATISTICA - 9 CFU

Linear Algebra: vectors (definition, operations, linear combination); matrices (definition, operations); systems of linear equations (geometric interpretation).

Differential and Integral Calculus: functions; graphs and properties of basic functions: polynomials, exponential, logarithm, periodic functions; asymptotics of sequences and functions (asymptotes); continuity; derivative (definition, geometric meaning, derivatives of basic functions, basic rules of differential calculus); monotonicity; extreme points, minima and maxima; second derivative; primitive function; Riemann integral (definition, geometric meaning, Torricelli-Barrow theorem, basic rules of integral calculus).

Theory of Probability: events (definition, incompatible events, union and intersection of events); probability; basic rules of probability; combinatorics (dispositions, permutations and combinations); discrete random variables; binomial and Poisson probability distributions, expectation value and variance. Continuous random variables (probability distribution, mean value and variance); uniform, exponential and gaussian probability distributions.

CALCOLO E BIOSTATISTICA - 9 CFU
Cammarota: Elementi di Calcolo e di Statistica. Libreria Scientifica Dias

#### Bibliography

CALCOLO E BIOSTATISTICA - 9 CFU Further information concerning the course, including a wide selection of problems and exercises, is available through the course website one the e-Learning platform: https://elearning2.uniroma1.it

#### Prerequisites

CALCOLO E BIOSTATISTICA - 9 CFU It is assumed that students entering the course know the basics of elementary mathematics from the high school. The fact that students actually have the mentioned pre-knowledge is certified by the entrance test or, when appropriate, by the OFA course and exam. Basic Mathematics: numbers and algebraic operations; equations and inequalities; geometric representation of real numbers, fundamental notions of analytic geometry (Cartesian line and plane, two-point distance in the Cartesian plane, form concept). Powers, logarithms, trigonometry.

#### Frequency modes

CALCOLO E BIOSTATISTICA - 9 CFU Lectures, exercise sessions are not mandatory.

#### Exam modes

Modalità di valutazione ENG

Part 1: Calculus and Biostatistics - 9 CFU

The exam consists of a written part, to evaluate the ability of the student to solve simple exercises and problems, and of an oral part. The written part contains some open-question problems, and its duration is at least 120 minutes.
The oral part consists in a short colloquium, and defines – together with the written part – the final mark for Part 1.

Part 2: Mathematical and informatic methods for biology - 3 CFU

The exam consist of unified evaluation, including a written part and, possibly, a second part involving the use of a statistical software in the computer lab.
Final mark

The final mark (for both Part 1 + Part 2) is computed as the weighted average of the marks of the two parts, where weights are proportional to the number of CFUs (9+3).
To achieve the level “cum laude”, the student is required to obtain “cum laude” in Part 1, and to pass Part 2 with at least 30/30.

Exam reservation date start Exam reservation date end Exam date
08/01/2021 19/01/2021 22/01/2021
11/07/2021 23/07/2021 27/07/2021
12/09/2021 24/09/2021 28/09/2021
04/11/2021 14/11/2021 18/11/2021
Course sheet
• Curriculum: Biosanitario
• Year: First year
• Semester: First semester
• Parent course:
1041434 - CALCULATION, BIOSTATISTICS and MATHEMATICAL METHODS for BIOLOGY
• SSD: MAT/07
• CFU: 9
Activities
• Attività formative di base
• Ambito disciplinare: Discipline matematiche, fisiche e informatiche
• Lecture (Hours): 54.0
• CFU: 6.0
• SSD: MAT/07
• Attività formative affini ed integrative
• Ambito disciplinare: Attività formative affini o integrative
• Exercise (Hours): 12.0
• Lecture (Hours): 18.0
• CFU: 3.0
• SSD: MAT/07