Mathematics
A.Y. 2022/2023
Learning objectives
The course aims to introduce some mathematical concepts and methods, with particular attention to the development of the most useful aspects of the discipline for a real understanding of the topics covered in the courses characterizing the agricultural and environmental degree courses.
The aim is to provide students with an adequate theoretical understanding of the topics covered, together with an adequate ability to perform the calculation procedures involved and provide the theoretical and technical tools to formulate and solve in a rigorous way simple applicative problems.
The course has as a transversal objective to help students develop an effective method of study, useful not only to face other exams during the academic career but also for the self-training required in many working situations.
The aim is to provide students with an adequate theoretical understanding of the topics covered, together with an adequate ability to perform the calculation procedures involved and provide the theoretical and technical tools to formulate and solve in a rigorous way simple applicative problems.
The course has as a transversal objective to help students develop an effective method of study, useful not only to face other exams during the academic career but also for the self-training required in many working situations.
Expected learning outcomes
At the end of the course, the student will be able to:
· manipulate formulas containing algebraic expressions, percentages and proportions, radicals, logarithms and exponentials, solve equations and inequalities, use the main tools and techniques of analytical geometry, plane and solid geometry and trigonometry;
· plot and interpret graphs of functions of one variable in different contexts, calculate limits, derivatives and integrals, and use these concepts to describe and solve real problems;
· understand and independently perform some simple mathematical steps commonly used in the scientific literature of agricultural and environmental sciences;
· critically use some simple software to describe and address applicative problems;
· acquire some additional new mathematical knowledge independently.
· manipulate formulas containing algebraic expressions, percentages and proportions, radicals, logarithms and exponentials, solve equations and inequalities, use the main tools and techniques of analytical geometry, plane and solid geometry and trigonometry;
· plot and interpret graphs of functions of one variable in different contexts, calculate limits, derivatives and integrals, and use these concepts to describe and solve real problems;
· understand and independently perform some simple mathematical steps commonly used in the scientific literature of agricultural and environmental sciences;
· critically use some simple software to describe and address applicative problems;
· acquire some additional new mathematical knowledge independently.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
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 2/Ed. con MyLab
(ISBN 9788891910615, Euro 29,00)
Exercises and interactive activities on Ariel
(ISBN 9788891910615, Euro 29,00)
Exercises and interactive activities on Ariel
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 Moodle 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 Moodle course website.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 6
Practicals: 48 hours
Lessons: 24 hours
Lessons: 24 hours
Professor:
Morando Paola
Professor(s)