MATEMATICA E METODI STATISTICI

The course has a dual objective: on the one hand, it aims to provide basic computational tools useful for the major subjects; on the other, it aims to develop or consolidate reasoning and problem-solving skills, which are typical of a mathematical education and have cross-curricular utility.

We will always begin with a practical problem and then provide the mathematical knowledge needed to solve the problem posed.

Exercises will be particularly aimed at developing problem-solving skills.

Expected Learning Outcomes (AAR) according to the Dublin Descriptors:

a. Knowledge and understanding: knowledge of functions, trigonometry and trigonometry, differential calculus, basics of integral calculus, basics of statistics.

b. Applied knowledge and understanding: ability to work with the fundamentals of trigonometry, ability to work with functions, ability to interpret function graphs.

c. Independent judgment: the ability to provide a mathematical interpretation of real-world problems, the ability to deduce information about real-world problems from mathematical data, and the ability to make judgments about real-world facts based on mathematical considerations.

d. Communication skills: the ability to rigorously communicate the mathematical concepts studied, the ability to effectively communicate the mathematical meanings under study.

e. Learning skills: the ability to study and understand both in a group and independently, the ability to connect topics covered in the course, the ability to grasp connections between the mathematical topics covered and other disciplines (lateral transfer), and the ability to understand even more complex mathematical topics not covered in the course (vertical transfer).

The course includes twice the number of hours of practical exercises compared to the number of lectures: 42 hours of practical exercises and 21 hours of practical lessons, for a total of 63 hours.

If the course is taught in a blended format (distance learning - in-person learning), in-person lessons will be primarily devoted to practical exercises (RAA b.).

Mathematical concepts will be introduced starting from real-world problems (RAA c. ​​and e.) using a visual and practical approach, including the use of high-impact educational software (RAA a. and b.), and then moving on to a true formalism (RAA d.) through participatory lessons (RAA d.).

If the course is taught in a blended or distance learning format, any necessary changes to the previously stated curriculum may be made in order to comply with the planned program and the syllabus.

Information for students with disabilities and/or learning disabilities (LD)

To ensure equal opportunities and in compliance with applicable laws, interested students may request a personal interview to plan any compensatory and/or dispensatory measures, based on their educational objectives and specific needs.

You may also contact our Department's CInAP (Center for Active and Participatory Inclusion - Services for Disabilities and/or LD) contact, Professor Anna De Angelis.

Essential basic mathematics knowledge:

Arithmetic (numbers, operations, percentages, approximations);

Geometry (polygons, Pythagorean theorem);

Algebra (polynomials, first-degree equations and inequalities, second-degree equations and inequalities).

Course attendance is strongly recommended, especially for the exercises, which will actively engage students and enhance their learning.

Attendance will be recorded for statistical and course evaluation purposes only.

INTRODUCTION
What is a mathematical model?
SETS AND RELATIONS.
Sets and subsets. Intersection, union, difference, and Cartesian product. Relations between sets.
Equivalence relations.
NUMERICAL SETS.
N, Z, Q, R. Fractions and percentages; proportions; scientific notation. Whole and fractional algebraic equations and inequalities.
MATRICES, DETERMINANTS, AND SYSTEMS OF LINEAR EQUATIONS.
Matrices: General information. Matrix operations. Determinants. Cramer's theorem. Applications.
ELEMENTS OF EUCLIDEAN GEOMETRY.
Triangles and special points. Polygons and their properties. Circumference and circle. Pythagorean theorem. Measurements of length, area, and volume.
ELEMENTS OF ANALYTIC GEOMETRY.
The Cartesian plane. Distance between two points. Midpoint of a segment. Equation
of a line. Explicit equation of a line. Lines passing through a given point. Intersection of lines.
Parallelism between lines. Line passing through a point and parallel to a given line. Geometric meaning of the slope of a line. Perpendicularity between lines and distance from a point to a line. General information on conic sections. Circumference. Intersections of a conic section with a line.
GONIOMETRY, TRIGONOMETRY, AND GONIOMETRIC FUNCTIONS.
Angle measurement in degrees and radians; length of an arc. Sine, cosine, tangent of an angle and their notable values ​​(30°, 45°, 60°, 90°, 180°, associated arcs). Fundamental relation of goniometry. Trigonometry applied to right-angled triangles: First and second theorems of right-angled triangles and their applications. Basics of trigonometry applied to any triangle: Area of ​​any triangle. Chord theorem. Theorem of sines. Cosine theorem.
REAL FUNCTION OF A REAL VARIABLE.
Introduction to the concept of function: real functions of a real variable. Injective functions. Increasing and decreasing functions. Even and odd functions. Invertible functions. Composite functions.
Quadratic functions: parabolas. Cube functions and cubic functions. Hyperbolas. Root functions. Absolute value function.
Exponential functions. Logarithms. The number "e" and the natural logarithm. Logarithmic functions.
LIMITS
Neighborhoods of a point. Asymptotic functions. Definition of a finite limit as x tends to a finite value. Right- and left-hand limits. Calculating limits and indeterminate forms. Comparison between infinities and the hierarchy of infinities. Study of a function.
DERIVATIVES
Geometric meaning of the derivative at a point: limit of the incremental ratio. Derivative function. Elementary derivatives. Monotonicity intervals of a function. Extrema of a function: Fermat's theorem (statement only).
NOTES ON INTEGRALS
Indefinite integral as the inverse of the derivative. Finding the primitives of a function. Meaning of the definite integral. Calculating areas.
NOTES ON NUMERICAL SEQUENCES AND SERIES.
Arithmetic and geometric progressions. Limits of sequences. Numerical series. Geometric series. Harmonic series
STATISTICS AND PROBABILITY
Statistical investigation. Population, statistical unit, qualitative and quantitative nature. Graphical representations. Indices of central tendency: mode, median, arithmetic mean, geometric mean. Indices of dispersion: range, variance, standard deviation. Joint distribution of two characteristics: double frequency distributions. Linear regression. Correlation. Introduction to combinatorics. Elements of probability calculus. Some notable probability distributions.

[1] Dario Benedetto, Mirko Degli Esposti, Carlotta Maffei. Mathematics for Life Sciences. Third edition, Ambrosiana Publishing House. Exclusive distribution by Zanichelli.

[2] M. Gionfriddo, Institutions of Mathematics, Tringale, Catania.

[3] V. Villani, Mathematics for Biomedical Disciplines, McGraw-Hill

During the course, multiple-choice tests will be administered.

Procedures for those taking the ongoing tests:

The ongoing tests are worth 10 points out of 30 each, and the final written test (with exercises) is worth 10 points out of 30. Students who achieve a score of at least 18 between the ongoing tests and the written test, taken in one of the two sessions of the first session, may choose to retain their written grade for the oral exam.

Students who have passed the ongoing tests and the final written test (at least 18 out of 30) and decide to take the oral exam will have the opportunity to earn an additional 3 points by answering the questions correctly.

Procedures for those not taking the ongoing tests:

The final exam consists of a written test (with exercises) and an oral exam.

The assessment of learning may also be administered online, if conditions require.

The final written exam consists of standard exercises designed to assess basic skills and exercises to test modeling abilities.

Grading Criteria:

Failure to Pass

Knowledge and Understanding of the Subject Matter: Significant Deficiencies. Significant Inaccuracies

Applied Knowledge: Poor ability to manipulate mathematical objects.

Making Judgments: Poor or absent ability to make judgments about real or realistic facts based on mathematical considerations.

Communication Skills: Poor ability to communicate mathematical meanings. Absent or inappropriate use of mathematical language.

Learning Skills: Poor ability to connect topics studied. Poor problem-solving ability.

18-20

Knowledge and Understanding of the Subject Matter: Sufficient. Significant Inaccuracies.

Applied Knowledge: Barely adequate ability to manipulate mathematical objects.

Independent judgment: Sufficient ability to make judgments about real or realistic facts based on mathematical considerations.

Communication skills: Sufficient ability to communicate mathematical meanings. Sufficient use of appropriate mathematical language.

Ability to learn: Sufficient ability to connect studied topics. Sufficient ability to solve problems with guidance.

21-23

Knowledge and understanding of topic: Good. Some inaccuracies.

Applied knowledge: Ability to manipulate mathematical objects.

Independent judgment: Ability to make judgments about real or realistic facts based on mathematical considerations.

Communication skills: Ability to communicate mathematical meanings. Appropriate use of mathematical language.

Ability to learn: Ability to connect studied topics. Ability to solve problems with partial guidance.

24-26

Knowledge and understanding of topic: Good. Some inaccuracies.

Applied knowledge: Good ability to manipulate mathematical objects.

Making judgments: Good ability to make judgments about real or realistic facts based on mathematical considerations.

Communication skills: Good ability to communicate mathematical meanings. Appropriate use of mathematical language.

Ability to learn: Good ability to connect topics studied. Ability to solve problems independently.

27-29

Knowledge and understanding of the topic: Excellent. Without inaccuracies.

Applied knowledge: More than good ability to manipulate mathematical objects.

Making judgments: More than good ability to make judgments about real or realistic facts based on mathematical considerations and to make mathematical considerations based on real facts.

Communication skills: More than good ability to communicate mathematical meanings. Appropriate and rigorous use of mathematical language.

Ability to learn: More than good ability to connect topics studied. Ability to solve problems independently.

30-30 cum laude

Knowledge and understanding of the topic: Excellent. With in-depth analysis.

Applied knowledge: Excellent ability to manipulate mathematical objects. Ability to solve problems using correct, non-standard procedures.

Making judgments: Excellent ability to make judgments about real or realistic facts based on mathematical considerations and to make mathematical considerations based on real facts.

Communication skills: Excellent ability to communicate mathematical meanings. Appropriate and rigorous use of mathematical language.

Ability to learn: Excellent ability to connect topics studied. Ability to pose and solve problems independently.

Examples of Frequently Asked Questions and/or Exercises

EXAMPLES OF WRITTEN EXERCISES

Find the domain of the following function: f(x) = ln(4-x2)

Calculate the limit as x tends to infinity of the function f(x) = (x3-1)/(x3)

Study the behavior and graph the function f(x) = xex

A population of bacteria follows the behavior of the logistic function p(t) = 1000/(1 + e(-2t)), with t>=0, where the variable t expresses the time in days and p(t) the size of the bacterial colony. Calculate the limit. What can we deduce from the calculated limit?

Describe the behavior of the function p(t) = 1 / [1 + e -2t], which expresses the behavior of a population.

In the first three math tests, Giovanni's grades were 25, 20, and 26. The grade for the fourth and final test did not change the (arithmetic) mean. What is the grade for the fourth test?

What can be deduced from a set of data if the arithmetic mean is less than the median?

Write a set of data whose arithmetic mean is both greater than the median.

Determine the mean and median of the following set of data: 1, 2, 4, 4, 7, 7, 8, 15.

EXAMPLES OF ORAL QUESTIONS

Define the trigonometric functions sine, cosine, and tangent.

State the law of sines.

How do you solve a right-angled triangle?

How can a linear phenomenon be represented on the Cartesian plane? What type of equation will be associated with the representation on the plane?

How do you define a function? In what cases is it appropriate to use a function to model a real phenomenon? Provide examples

What phenomenon could an exponential function describe? What about a logarithmic function?

Define the finite limit of a real function of one real variable?

What is the purpose of calculating an infinite limit for a real function of one real variable?

Define the derivative of a real function of one real variable and explain its geometric meaning.

What is the purpose of understanding the behavior of the derivative of a function? Provide examples.

What is the purpose of integral calculus?

Explain what the mean and median of a data set are. When is it better to use one or the other?

What is the standard deviation?