#
Mathematics

A.Y. 2020/2021

Learning objectives

The goal of the course is to introduce some mathematical concepts and tools with particular reference to the topics which can be useful for applications to Agricultural and Food Sciences. The course aims at helping students to gain an adequate theoretical understanding of the matter, as well as good computational skills. At the end of the course students should be able to exploit their math knowledge in order to set and solve simple applied problem in a rigorous way.

Expected learning outcomes

Knowledge and understanding concepts of basic mathematics and elementary Mathematical Analysis. In particular, with regard to basic mathematics, the student will be able to manipulate formulas containing algebraic expressions, percentages and proportions, radicals, logarithms and exponentials, to solve equations and inequalities, to use the main tools and techniques of analytical geometry, plane and solid geometry and trigonometry. As far as elementary Mathematical Analysis is concerned, the student will be able to draw and use graphics of real functions of one variable in many different frameworks, to calculate limits, derivatives and integrals and to use these concepts for describing and solving real problems. Moreover, students will be able to understand and execute autonomously simple mathematical steps commonly used in the scientific literature of his own sector.

**Lesson period:** First semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

Program:

During the emergency teaching phase, the program will not change.

Teaching methods:

The lessons involve the use of the asynchronous mode, with recorded lesson modules, for the most transmissive part and the synchronous mode for interacting with students. Classroom meetings are scheduled to be held frequently, during which the teacher will alternate the transmission of disciplinary content with active learning experiences through brain-storming, questioning and peer instruction activities. During the course, e-learning platforms (Ariel, Moodle) will be used to assign activities and share teaching materials.

The schedule of lessons, the methods and criteria for participating in face-to-face lessons (which require a booking with the appropriate app) and all the details of the activities will be published on the Ariel website of the course by the beginning of the lessons. Any updates will also be communicated through notices (often consult mail @ students.unimi.it)

Reference materials:

In addition to the bibliography already reported in the program, students will be able to refer to all the lessons, materials and resources published in the online course and to what will be communicated and published on the Ariel and Moodle platforms.

Exames and evaluation criteria:

If the regulations concerning social distancing allow it, the examination will take place in the presence. In this case, for the methods, refer to what is illustrated in the course sheet.

If the regulations concerning social distancing do not allow it, or for students who have objective, serious and documented impossibility of movement (e.g. people in quarantine, or with family frailty) a remote exam session will be organized.

In the case of remote mode, the exam will consist of a short written test followed (only in case of a positive outcome) by a 45-minutes oral test. During the written test, the use of the calculator is not allowed.

The written test will consist of 6 short open-ended questions, which span the entire course program. The questions will be formulated in order to verify a solid preparation and a good understanding of the topics covered throughout the course, but will not require long or complex calculations. The questions will be dictated (one at a time) and the student will have 5 minutes to answer after any question. Students who answer correctly at least 4 questions will be summoned to take the 45-minute oral test. During the oral test the student will have to demonstrate that he / she has the disciplinary content required by the program. Moreover, in order to pass the oral part of the exam students must be able to operate with mathematical symbolism, to formalize the path of solution of a problem through algebraic and graphical models, to use methods, tools and mathematical models in different situations, to identify the appropriate strategies for solving problems, to use appropriate language and symbols and to analyze and interpret the results obtained.

During the emergency teaching phase, the program will not change.

Teaching methods:

The lessons involve the use of the asynchronous mode, with recorded lesson modules, for the most transmissive part and the synchronous mode for interacting with students. Classroom meetings are scheduled to be held frequently, during which the teacher will alternate the transmission of disciplinary content with active learning experiences through brain-storming, questioning and peer instruction activities. During the course, e-learning platforms (Ariel, Moodle) will be used to assign activities and share teaching materials.

The schedule of lessons, the methods and criteria for participating in face-to-face lessons (which require a booking with the appropriate app) and all the details of the activities will be published on the Ariel website of the course by the beginning of the lessons. Any updates will also be communicated through notices (often consult mail @ students.unimi.it)

Reference materials:

In addition to the bibliography already reported in the program, students will be able to refer to all the lessons, materials and resources published in the online course and to what will be communicated and published on the Ariel and Moodle platforms.

Exames and evaluation criteria:

If the regulations concerning social distancing allow it, the examination will take place in the presence. In this case, for the methods, refer to what is illustrated in the course sheet.

If the regulations concerning social distancing do not allow it, or for students who have objective, serious and documented impossibility of movement (e.g. people in quarantine, or with family frailty) a remote exam session will be organized.

In the case of remote mode, the exam will consist of a short written test followed (only in case of a positive outcome) by a 45-minutes oral test. During the written test, the use of the calculator is not allowed.

The written test will consist of 6 short open-ended questions, which span the entire course program. The questions will be formulated in order to verify a solid preparation and a good understanding of the topics covered throughout the course, but will not require long or complex calculations. The questions will be dictated (one at a time) and the student will have 5 minutes to answer after any question. Students who answer correctly at least 4 questions will be summoned to take the 45-minute oral test. During the oral test the student will have to demonstrate that he / she has the disciplinary content required by the program. Moreover, in order to pass the oral part of the exam students must be able to operate with mathematical symbolism, to formalize the path of solution of a problem through algebraic and graphical models, to use methods, tools and mathematical models in different situations, to identify the appropriate strategies for solving problems, to use appropriate language and symbols and to analyze and interpret the results obtained.

**Course syllabus**

1. Numerical sets: the sets N, Z, Q, R. The real line and the symbols of ± ∞. Absolute value, nth roots, logarithms and exponentials: definitions and properties. Percentages, averages and proportions and their use in solving real problems (1/2 CFU).

2. Equations and inequalities: I and II degree and reducible to them, fractional, irrational, exponential and logarithmic, trigonometric, irrational, with absolute values; inequality systems (1/2 CFU).

3. Real functions of a real variable: The concept of function: Domain, codomain, graph, injective and surjective functions, monotone and invertible functions, composition of functions, symmetries (1/2 CFU)

4. The Cartesian plane: coordinates, straight line equations, orthogonality, parallelism, distance between points, distance of a point and a straight line, midpoint and axis of a segment. Linear functions and their applications to real problems. Two-variable inequality systems for the description of suitable regions of the plane. Goniometry and trigonometry: definitions and main properties, sinus theorem and Carnot theorem, applications to real problems (1CFU).

5. Elementary functions and their graphs: linear functions, powers and roots, exponentials, logarithms, goniometric functions, modulus and sign: definitions, properties, graphs. Elementary operations on graphs (translations, symmetries, absolute values) (1CFU)

6. Limits: definition, indeterminate forms and their resolution, significant limits, hierarchy of infinites and infinitesimals, asymptotic approximations for the resolution of indeterminate forms. Horizontal, vertical and oblique asymptotes. Continuous functions (1/2 CFU)

7. Derivatives: derivatives of elementary functions, derivation rules, derivatives of composition of functions. Relationship between continuity and derivability. Geometric meaning of the first derivative and its applications; tangent lines; monotony and search for points of maximum and minimum; change rates; application to optimization problems. Second derivative, concavity and inflection points. Qualitative study of the graph of a function (1 CFU)

8. Integrals: Indefinite integrals: notion of primitive function, primitives of elementary functions, search for primitives. Integration methods (immediate integrals, by substitution, by parts, integration of rational functions). Definite integrals: Fundamental Theorem of Integral Calculus and its applications. Calculation of areas of flat regions. (1CFU)

2. Equations and inequalities: I and II degree and reducible to them, fractional, irrational, exponential and logarithmic, trigonometric, irrational, with absolute values; inequality systems (1/2 CFU).

3. Real functions of a real variable: The concept of function: Domain, codomain, graph, injective and surjective functions, monotone and invertible functions, composition of functions, symmetries (1/2 CFU)

4. The Cartesian plane: coordinates, straight line equations, orthogonality, parallelism, distance between points, distance of a point and a straight line, midpoint and axis of a segment. Linear functions and their applications to real problems. Two-variable inequality systems for the description of suitable regions of the plane. Goniometry and trigonometry: definitions and main properties, sinus theorem and Carnot theorem, applications to real problems (1CFU).

5. Elementary functions and their graphs: linear functions, powers and roots, exponentials, logarithms, goniometric functions, modulus and sign: definitions, properties, graphs. Elementary operations on graphs (translations, symmetries, absolute values) (1CFU)

6. Limits: definition, indeterminate forms and their resolution, significant limits, hierarchy of infinites and infinitesimals, asymptotic approximations for the resolution of indeterminate forms. Horizontal, vertical and oblique asymptotes. Continuous functions (1/2 CFU)

7. Derivatives: derivatives of elementary functions, derivation rules, derivatives of composition of functions. Relationship between continuity and derivability. Geometric meaning of the first derivative and its applications; tangent lines; monotony and search for points of maximum and minimum; change rates; application to optimization problems. Second derivative, concavity and inflection points. Qualitative study of the graph of a function (1 CFU)

8. Integrals: Indefinite integrals: notion of primitive function, primitives of elementary functions, search for primitives. Integration methods (immediate integrals, by substitution, by parts, integration of rational functions). Definite integrals: Fundamental Theorem of Integral Calculus and its applications. Calculation of areas of flat regions. (1CFU)

**Prerequisites for admission**

As a first semester course in the first year, there are no specific prerequisites other than those required for entrance to the degree course.

**Teaching methods**

Frontal lessons, exercises, use of e-learning platform associated with the textbook, use of educational software, group work, use of didactic games as a motivational lever for the learning of the subject and as a tool of verification and self-evaluation on curricular themes. The course uses e-learning platforms (Ariel and Moodle) where weekly exercises and other teaching materials related to the topics covered in the lesson are uploaded. Attendance at the course, although not compulsory, is strongly recommended.

**Teaching Resources**

Silvia Annaratone, Matematica sul campo. Metodi ed esempi per le scienze della vita con MyLab e eText (ISBN 9788891901422, Euro 29,00)

**Assessment methods and Criteria**

To attend the exam, students must be enrolled regularly through SIFA and must be in front of the classroom 15 minutes before the beginning of the written test, with photo ID and protocol sheets.

The examination consists of a written test and an oral test. The use of calculator if forbidden during the written test. The written test is organized in two parts:

· Part A, lasting 30 minutes, consists of 10 open questions concerning the prerequisites to the course. The extremely simple questions are intended to assess whether the student has the minimum skills to approach a university course of mathematics and is able to operate correctly with mathematical symbolism. Part A will be passed answering correctly at least 8 out of 10 questions. Passing Part A is a necessary (but not sufficient!) condition for passing the written test

· Part B, which lasts 90 minutes, consists of six exercises related to topics of the course and aims to test the student's ability to use mathematical methods and tools in different situations and to identify appropriate strategies for solving problems. The written test is passed if and only if both parts A and B are passed. The score obtained in Part A does not contribute to the final score of the written test.

The total duration of the written test is 2 hours. During the written test it is forbidden to consult books, notes, use calculators of any kind, computers and mobile phones. It is also forbidden to communicate with the companions. During all the written test it is also forbidden to leave the classroom: in particular, during the first hour of part B it will not be possible to leave the classroom for any reason. At the end of the first hour, students who wish to do so can either finish or withdraw. The oral test may be taken only if the written test has been passed with a score of 18/30 or more, and only at the same session of the written test. The oral test aims to evaluate the student's ability to use an appropriate language and symbolism, to focus the path of solving a problem through algebraic and graphic models and to analyze and interpret the results obtained. Students who, after passing the written test, did not show up for the oral part will fail the exam.

The final exam mark will be the arithmetic mean between the written mark and the oral mark and will be expressed over 30.

Examples of written tests from past years are available on the Ariel course website.

The examination consists of a written test and an oral test. The use of calculator if forbidden during the written test. The written test is organized in two parts:

· Part A, lasting 30 minutes, consists of 10 open questions concerning the prerequisites to the course. The extremely simple questions are intended to assess whether the student has the minimum skills to approach a university course of mathematics and is able to operate correctly with mathematical symbolism. Part A will be passed answering correctly at least 8 out of 10 questions. Passing Part A is a necessary (but not sufficient!) condition for passing the written test

· Part B, which lasts 90 minutes, consists of six exercises related to topics of the course and aims to test the student's ability to use mathematical methods and tools in different situations and to identify appropriate strategies for solving problems. The written test is passed if and only if both parts A and B are passed. The score obtained in Part A does not contribute to the final score of the written test.

The total duration of the written test is 2 hours. During the written test it is forbidden to consult books, notes, use calculators of any kind, computers and mobile phones. It is also forbidden to communicate with the companions. During all the written test it is also forbidden to leave the classroom: in particular, during the first hour of part B it will not be possible to leave the classroom for any reason. At the end of the first hour, students who wish to do so can either finish or withdraw. The oral test may be taken only if the written test has been passed with a score of 18/30 or more, and only at the same session of the written test. The oral test aims to evaluate the student's ability to use an appropriate language and symbolism, to focus the path of solving a problem through algebraic and graphic models and to analyze and interpret the results obtained. Students who, after passing the written test, did not show up for the oral part will fail the exam.

The final exam mark will be the arithmetic mean between the written mark and the oral mark and will be expressed over 30.

Examples of written tests from past years are available on the Ariel course website.

MAT/02 - ALGEBRA - University credits: 6

Practicals: 48 hours

Lessons: 24 hours

Lessons: 24 hours

Professor:
Morando Paola

Educational website(s)

Professor(s)