Biology

GIULIA BASTI

The course concerns the application of mathematical methods from calculus, statistics and
probability to the resolution of problems that arise in the field of natural sciences, with
particular reference to biological sciences. The main objective of the course is to provide the
student with fundamental knowledge of linear algebra, differential and integral calculus,
discrete and continuous probability, which will later be applied in the analysis of empirical
data in biology, particularly in the field of genetics, evolution and ecology.
The course includes lectures and classroom exercises, useful for verifying knowledge for the
purposes of the students personal preparation.
Specific objectives
A) Knowledge and understanding
Knowledge and understanding of the fundamental concepts of linear algebra (vectors, linear
systems).
Knowledge and understanding of the concept of limit and the foundations of differential and
integral calculus.
Knowledge and understanding of the foundations of probability and some descriptive
statistics tools.
B) Ability to apply knowledge and understanding
Ability to properly use the specific terminology of mathematics and statistics.
Ability to translate a concrete problem, for example related to biological sciences, into a
corresponding mathematical problem, through appropriate approximation, abstraction and
modeling procedures.
Ability to use deductive reasoning in abstract contexts.
Ability to recognize the mathematical tools used in other courses (Physics, Chemistry,
Genetics, Ecology) and to use them with mastery and confidence.
Ability to identify the most convenient procedures for solving mathematical problems.

C) Autonomy of judgment
Ability to judge the rigorousness of the formulation and proof of a mathematical statement.
Ability to ask questions aimed at developing and deepening the knowledge acquired.
Ability to independently evaluate the validity of a quantitative model for the description of a
given phenomenon of the natural and biological sciences.
D) Communication skills
Ability to communicate what has been learned in the drafting of written exam essays and in
the presentation during the oral exam.
Ability to articulate a speech in a logically structured way, clearly distinguishing between
hypotheses, deductive procedure and conclusions.
E) Learning ability
Learn specific terminology.
Connect the knowledge acquired in a logical way.
Identify the most relevant themes of the subjects covered.

The course concerns the application of mathematical methods from calculus, statistics and probability to the resolution of problems that arise in the field of natural sciences, with particular reference to biological sciences. The main objective of the course is to provide the student with fundamental knowledge of linear algebra, differential and integral calculus, discrete and continuous probability, which will later be applied in the analysis of empirical data in biology, particularly in the field of genetics, evolution and ecology.

The course includes lectures and classroom exercises, useful for verifying knowledge for the purposes of the student's personal preparation.

Specific objectives:

A) Knowledge and understanding

Knowledge and understanding of the fundamental concepts of linear algebra (vectors, linear systems).
Knowledge and understanding of the concept of limit and the foundations of differential and integral calculus.
Knowledge and understanding of the foundations of probability and some descriptive statistics tools.

B) Ability to apply knowledge and understanding

Ability to properly use the specific terminology of mathematics and statistics.
Ability to translate a concrete problem, for example related to biological sciences, into a corresponding mathematical problem, through appropriate approximation, abstraction and modeling procedures.
Ability to use deductive reasoning in abstract contexts.
Ability to recognize the mathematical tools used in other courses (Physics, Chemistry, Genetics, Ecology) and to use them with mastery and confidence.
Ability to identify the most convenient procedures for solving mathematical problems.

C) Autonomy of judgment

Ability to judge the rigorousness of the formulation and proof of a mathematical statement.
Ability to ask questions aimed at developing and deepening the knowledge acquired.
Ability to independently evaluate the validity of a quantitative model for the description of a given phenomenon of the natural and biological sciences.

D) Communication skills

Ability to communicate what has been learned in the drafting of written exam essays and in the presentation during the oral exam.
Ability to articulate a speech in a logically structured way, clearly distinguishing between hypotheses, deductive procedure and conclusions.

E) Learning ability

Learn specific terminology.
Connect the knowledge acquired in a logical way.
Identify the most relevant themes of the subjects covered.

It is assumed that students entering the course know the basics of elementary mathematics learned in 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.

Linear Algebra. Vectors, geometric representation, operations between vectors. Systems of linear equations and their geometric interpretation.

Calculus. Functions. Domain and image of a function. Composition of functions and inverse function. Reading the graph of a function. Linear, exponential, logarithmic functions: main properties, graph, use in biological modeling. Log-scale graphs. Power-law functions and log-log-scale graphs. Concept of limit. Asymptotic behavior. Computation of limits. Difference quotient and derivative of a function; geometric interpretation of a derivative, tangent line. Linear approximation of a function. Derivation rules. Increasing and decreasing functions. Maxima and minima. Definite integrals, geometric interpretation and their properties. Primitives. Theorem of Torricelli & Barrow (fundamental theorem of calculus). Computation of integrals by substitution and by parts.

Probability and biostatistics. Random events and probability: definitions and fundamental properties, incompatible events. Uniform probability. Conditional probability, independent events, Bayes' formula and diagnostic tests. Iteration of independent trials and binomial formula. Random variables: expectation value, variance, covariance. Poisson random variables. Continuous random variables, probability density, probability distribution / cumulant. Uniform, exponential, gaussian variables.

D. Benedetto, M. Degli Esposti, C. Maffei "Matematica per le scienze della vita" Ed. CEA
Sheldon M. Ross "Probabilità e statistica per l'ingegneria e le scienze" , Apogeo Education

Other material, including the "Diary of the lectures" and a selection of exercises, will be made available on the course’s webpage curated by the teacher.

C. Cammarota, "Elementi di Calcolo e di Statistica", LSD, 2008
Sheldon M. Ross "Introduzione alla Statistica" , Apogeo Education, 2008
C. Neuhauser , M. Roper "Calculus For Biology And Medicine" Pearson Education, 2018.

Lectures in class and activities with tutors.

Recommended attendance of lectures and exercise sessions.

The exam includes a written test, aimed at assessing basic skills, and an oral test. The written test, lasting 120 minutes, consists of open-ended questions, possibly preceded by a pre-selection through multiple-choice questions. The oral test consists of an interview of variable length that leads - together with the written test - to the definition of the overall grade.
There are two intermediate written tests that, if passed successfully, allow direct access to the oral test.

derivative of a function and its geometric interpretation; asymptotic behaviour of a function; study of the properties of a function; exercises on classical probability (throw of a dice); conditional probability