FUNDAMENTALS OF MATHEMATICS AND PHYSICS

Module MATHEMATICS

The mathematics' module has a twofold target: on the one hand, it aims to provide basic calculus tools, useful for other courses, on the other hand it aims to train the skills of reasoning and problem solving, typical skills of a mathematical education. Exercises (both on paper and on software) will be aimed, in a special way, at developing a problem-solving aptitude.

EXPECTED RESULTS (ER):

Knowledge and understanding abilities: Lines and linear functions. knowledge on univariate and bivariate statistics. Goniometry and trigonometry. Exponentials and exponential functions. Logarithms and logarithmic functions.

Ability to apply knowledge and understanding abilities: knowing how to describe and interpret a huge set of data. Knowing how to grasp relationships between two variables. Knowing how to manipulate and to interpret graphs of linear, exponential and logarithmic functions. Knowing how to manipulate goniometric and trigonometric basis.

Making judgements: knowing how to give mathematical interpretations of real problems, knowing how to deduce information on real problems starting from mathematical data, knowing how to make judgments on real facts starting from mathematical considerations.

Communication skills: knowing how to communicate effectively mathematical meanings, showing a deep understanding. Knowing how to communicate in a rigorous way mathematical topics.

Learning skills: ability to study and understand both in groups and independently, being able to connect the studied topics among them. Ability to grasp connections between mathematical topics and other disciplines (lateral transfer). Ability to understand also more complex mathematical topics (vertical transfer).

The class hours (42) are divided into lectures and exercises.

Topics will be mediated by a visual and practical approach, also with the aid of software with a great didactic impact to reach later on formal definitions, through participated lessons. Examples of real applications will be provided.

Information for students with disabilities and/or learning disorders.

As a guarantee of equal opportunities and in compliance with current regulations, students can ask for a personal interview in order to plan any compensatory and/or dispensatory measures, based on their specific needs and on learning objectives of the discipline. It is also possible to refer to the departmental contacts of CInAP (Center for Active and Participatory Inclusion - Services for Disabilities and/or learning disorders)

If conditions require teaching given in a hybrid mode or remotely, necessary changes may be introduced to what previously stated in order to comply with the program.

Basic math cultural prerequisites: 

Arithmetic (numbers, operations, percentages, approximations);

Geometry (polygons, Pythagorean theorem, similarity of triangles);

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

Attending the classes is not mandatory, though strongly advised

The module aims to provide foundational knowledge of some mathematical concepts with practical applications, useful for the analysis of typical phenomena in experimental sciences, with a particular focus on descriptive statistics and selected real functions of a real variable. The covered topics include:

-       Univariate descriptive statistics, introducing graphs, measures of central tendency (mean, median, mode), and measures of dispersion (variance, standard deviation, mean absolute deviation, interquartile range);

-       Bivariate statistics, introducing the concept of correlation between two variables, deviations, covariance, and Pearson's correlation coefficient. Theoretical concepts will be accompanied by exercises (also involving the use of digital spreadsheet and dynamic software tools).

The course will also address cases where two variables are correlated because they are functionally related, specifically in the context of:

-       Linear functions (lines and slope);

-       Exponential functions (increasing or decreasing);

-       Logarithmic functions (increasing or decreasing), with particular attention to phenomena such as bacterial or cellular duplication.

Finally, the course will introduce:

-       Goniometric functions (angles in radians, sine, cosine, and tangent of an angle, and the fundamental goniometric identities), with particular emphasis on applications to triangles (trigonometry: first and second theorems on right triangles, Law of Sines).

[1] Dario Benedetto, Mirko Degli Esposti, Carlotta Maffei. Matematica per scienze della vita. Quarta edizione. Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli.

[2] Lecture notes provided by the teacher on STUDIUM platform.

A written and an oral test are planned.

The vote follows the following scheme:

Negative:

Knowledge and understanding of mathematical topics: Important shortcomings. Significant inaccuracies

Ability to analyze and synthesize: Irrelevant. Frequent generalizations. Inability to synthesize mathematical topics.

Use of references: Completely inappropriate

18-20:

Knowledge and understanding of mathematical topics: Just sufficient. Obvious imperfections

Analysis and synthesis skills: Just sufficient skills

Use of references: just appropriate

21-23:

Knowledge and understanding of mathematical topics: Routine knowledge

Ability to analyze and synthesize: ability of correct analysis and synthesis. Argue logically and consistently

Using references: Use standard references

24-26:

Knowledge and understanding of mathematical topics: Good knowledge

Analysis and synthesis skills: good analysis and synthesis skills. The arguments are expressed consistently

Using references: Use of standard references

27-29:

Knowledge and understanding of mathematical topics: Knowledge more than good

Ability to analyze and synthesize: considerable abilities of analysis and synthesis

Use of references: the topic has been explored in depth

30-30 cum laude:

Knowledge and understanding of mathematical topics: Excellent knowledge

Ability to analyze and synthesize: excellent abilities of analysis and synthesis.

Use of references: Important insights.

EXAMPLES OF EXERCISES FOR THE WRITTEN TEST

-       In the first three exams Liliana's grades were 24, 18, 27. The grade on the fourth exam did not change the (arithmetic) mean . What is the grade of the fourth test?

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

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

-       Write down a set of 7 data representing the daily number of customers who made purchases at our store (open every day of the week) in the last week, so that the arithmetic mean is 15 and the median is 10. Then calculate the standard deviation. What can be deduced from the data, with respect to the frequency of purchases during the week?

-       Determine mean and median of the following data set: 1, 2, 3, 6, 7, 8, 9.

-       We carried out a study on the yield of a vine production. The aim was to see if there was a correlation between yield and dry matter percentage. Here are the values for the 5 samples A, B, C, D and E, which were the subject of the study, described as Sample: (Dry matter biomass in %, Yield in tonnes per hectare)

A: (18, 25). B: (24, 20). C: (12, 28). D: (21, 19). E: (25, 28).

-       Find the domain of the following function: f(x)= log(1-x^2).

-       Find the linear function r(x) passing through the two points A = (2, 6) and B = (0, 4) and represent it graphically. Find the image of x = 3 and the inverse image of y = 7.

-       You are considering a kitchen design following the ‘work triangle’ layout: a right-angled triangle whose vertices are the refrigerator area, the washing area and the cooking area. To optimise kitchen efficiency, it is advisable that the distance between the refrigerator and the cooking area is no more than 4 m. Will this requirement be met considering that in our design the refrigerator is at the right angle, the distance between the sink and the refrigerator is 2 m, and the angle at the corner of the sink area is 60°? Justify your answer.

-       In the laboratory, we begin an experiment starting with a bacterial colony of approximately 1000 individuals growing exponentially. We need to have a colony of at least 1 million bacteria. Assuming that the bacteria double every 3 hours, after how many hours will we reach our target?

-       Describe the behaviour of the p(t) = 1 / [1+ e ^ (-2t)] that expresses the trend of a population.

 EXAMPLES OF ORAL QUESTIONS

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

-       What are the dispersion indices of a data set and how we calculate them?

-       What is the standard deviation?

-       How do we calculate the correlation between two data sets?

-       What is and how do we calculate the Pearson correlation coefficient?

-       Define the goniometric functions sine, cosine, tangent.

-       Enunciate and prove the law of sines.

-       How do you solve a right triangle?

-       How can we represent a phenomenon with a linear behaviour on the Cartesian plane? What kind of equation will be associated with the representation on the plane?

-       What is the meaning and how do we calculate the slope of a line?

-       What phenomenon might an exponential function describe? What about a logarithmic function?