Calculus
A.Y. 2024/2025
Learning objectives
The educational objective of the course is to provide students with theoretical and practical matematical tools for their application in the basic and the characteristics courses of the CdS.
Expected learning outcomes
At the end of the course the students will be able to apply the main concepts of infinitesimal calculus to the resolution of exercises related to the topics covered in class.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Number Sets: N, Z, Q, R. Ordering of the real line and the "infinite" symbols. Absolute value, n-th roots, logarithms and exponentials: definitions and properties. Percentages, averages, and proportions and their use in solving real problems (0.5 CFU).
2. Equations and Inequalities: first and second degree and reducible to them, rational fractional, irrational, exponential and logarithmic, with absolute values; systems of equations and inequalities (0.5 CFU).
3. The Orthogonal Cartesian Plane: coordinates, equations of the line, orthogonality, parallelism, distance between points, point-line distance, midpoint and axis of a segment. Linear functions and their applications to real problems. Systems of inequalities in two variables for the description of appropriate regions of the plane (1 CFU).
4. Goniometry and Trigonometry: definitions and main properties, orthogonal projections of a segment, trigonometric equations and inequalities, applications to real problems (1 CFU).
5. Real Functions of a Real Variable: The concept of a function: domain, codomain, graph. Monotone and invertible functions, composition of functions. Elementary functions and their graphs: linear functions, powers and roots, exponentials, logarithms, trigonometric functions, modulus and sign: definitions, properties, graphs. Basic operations on graphs (translations, reflections, symmetries, absolute values) (1 CFU).
6. Limits: definition, indeterminate forms and their resolution, notable limits, hierarchy of infinities and infinitesimals, asymptotic estimates for the resolution of indeterminate forms. Horizontal, vertical, and oblique asymptotes. Continuous functions (1 CFU).
7. Derivatives: Derivatives of elementary functions, differentiation rules, derivatives of composite functions. Relationships between continuity and differentiability. Geometric meaning of the first derivative and its applications; tangent lines; monotonicity and finding maximum and minimum points; De l'Hopital's theorem. Second derivative, concavity, and inflection points. Qualitative study of the graph of a function (1.5 CFU).
8. Integrals: Indefinite integrals: the notion of an antiderivative, antiderivatives of elementary functions, finding antiderivatives. Integration methods (immediate integrals, reducible to immediate integrals, by substitution, by parts, integration of rational functions). Definite integrals: the Fundamental Theorem of Calculus and its applications. Calculation of areas of plane regions (1.5 CFU).
2. Equations and Inequalities: first and second degree and reducible to them, rational fractional, irrational, exponential and logarithmic, with absolute values; systems of equations and inequalities (0.5 CFU).
3. The Orthogonal Cartesian Plane: coordinates, equations of the line, orthogonality, parallelism, distance between points, point-line distance, midpoint and axis of a segment. Linear functions and their applications to real problems. Systems of inequalities in two variables for the description of appropriate regions of the plane (1 CFU).
4. Goniometry and Trigonometry: definitions and main properties, orthogonal projections of a segment, trigonometric equations and inequalities, applications to real problems (1 CFU).
5. Real Functions of a Real Variable: The concept of a function: domain, codomain, graph. Monotone and invertible functions, composition of functions. Elementary functions and their graphs: linear functions, powers and roots, exponentials, logarithms, trigonometric functions, modulus and sign: definitions, properties, graphs. Basic operations on graphs (translations, reflections, symmetries, absolute values) (1 CFU).
6. Limits: definition, indeterminate forms and their resolution, notable limits, hierarchy of infinities and infinitesimals, asymptotic estimates for the resolution of indeterminate forms. Horizontal, vertical, and oblique asymptotes. Continuous functions (1 CFU).
7. Derivatives: Derivatives of elementary functions, differentiation rules, derivatives of composite functions. Relationships between continuity and differentiability. Geometric meaning of the first derivative and its applications; tangent lines; monotonicity and finding maximum and minimum points; De l'Hopital's theorem. Second derivative, concavity, and inflection points. Qualitative study of the graph of a function (1.5 CFU).
8. Integrals: Indefinite integrals: the notion of an antiderivative, antiderivatives of elementary functions, finding antiderivatives. Integration methods (immediate integrals, reducible to immediate integrals, by substitution, by parts, integration of rational functions). Definite integrals: the Fundamental Theorem of Calculus and its applications. Calculation of areas of plane regions (1.5 CFU).
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
Lectures, exercises, application of examples to concrete cases, use of an e-learning platform associated with the textbook, use of educational software, group work, use of educational games as a motivational lever for learning the subject and as a tool for verification and self-assessment on curricular topics.
The course uses the e-learning platform MyAriel, where exercise sheets and other educational materials related to the topics covered in the lectures are uploaded on a weekly basis.
Attendance of the course, although not mandatory, is highly recommended.
The course uses the e-learning platform MyAriel, where exercise sheets and other educational materials related to the topics covered in the lectures are uploaded on a weekly basis.
Attendance of the course, although not mandatory, is highly 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)
(ISBN 9788891910615, Euro 29,00)
Assessment methods and Criteria
The exam consists of a mandatory written test, lasting 120 minutes, which allows for a grade of up to 30/30, followed by a mandatory oral test, accessible only to students who have achieved a score of 16/30 or higher on the written test. The oral test can only be taken in the same session as the written test.
To take the exam, students must be regularly registered through the online exam registration service (also available in Unimia) and must show up in front of the classroom fifteen minutes before the start of the written test, with a photo ID and exam paper. The written test consists of some short response exercises and some others with open responses, where all solution steps must be explicitly justified. The purpose of this test is to evaluate whether the student has the minimum required skills and has acquired the calculation tools that they have been practicing in the semester. During the written test, consulting books, notes, using any type of calculators, computers, tablets, and cell phones is not allowed. Communication with peers is also prohibited, under penalty of immediate expulsion from the classroom. It is also not allowed to leave the classroom during the entire written test. At the end of the first hour, students who wish may submit their work or withdraw. The score of the written test is communicated individually through an automated email from the university.
The oral test consists of a short interview on the program topics, aimed at completing the assessment of the acquired tools and skills. The exam begins with a discussion of the written test, where the student can explain the procedures used in solving the exercises and clarify any unclear steps. Subsequently, knowledge and understanding of some topics covered in class will be checked, as well as the student's ability to apply this knowledge and understanding to simple exercises and the correct use of specific language of the subject.
The final grade is expressed out of thirty and will take into account both tests. The exam is passed if the final grade is 18/30 or higher.
Students have the chance to take an optional midterm test consisting of a written test covering the minimum required skills and topics discussed in the first weeks of the course. This midterm test allows for a grade of up to 10/10 and is passed with a score of 6/10 or higher. Students who pass the midterm can use the obtained score as part of the written test grade. This option can only be used in the first written test attempt after the midterm and only in the January-February exam session.
Students with Specific Learning Disabilities (DSA) and disabilities are requested to contact the instructor via email at least 10 days before the scheduled exam date to agree on any individualized measures. In the email addressed to the instructor, it is necessary to CC the respective University Services: [email protected] (for students with DSA) and [email protected] (for students with disabilities).
To take the exam, students must be regularly registered through the online exam registration service (also available in Unimia) and must show up in front of the classroom fifteen minutes before the start of the written test, with a photo ID and exam paper. The written test consists of some short response exercises and some others with open responses, where all solution steps must be explicitly justified. The purpose of this test is to evaluate whether the student has the minimum required skills and has acquired the calculation tools that they have been practicing in the semester. During the written test, consulting books, notes, using any type of calculators, computers, tablets, and cell phones is not allowed. Communication with peers is also prohibited, under penalty of immediate expulsion from the classroom. It is also not allowed to leave the classroom during the entire written test. At the end of the first hour, students who wish may submit their work or withdraw. The score of the written test is communicated individually through an automated email from the university.
The oral test consists of a short interview on the program topics, aimed at completing the assessment of the acquired tools and skills. The exam begins with a discussion of the written test, where the student can explain the procedures used in solving the exercises and clarify any unclear steps. Subsequently, knowledge and understanding of some topics covered in class will be checked, as well as the student's ability to apply this knowledge and understanding to simple exercises and the correct use of specific language of the subject.
The final grade is expressed out of thirty and will take into account both tests. The exam is passed if the final grade is 18/30 or higher.
Students have the chance to take an optional midterm test consisting of a written test covering the minimum required skills and topics discussed in the first weeks of the course. This midterm test allows for a grade of up to 10/10 and is passed with a score of 6/10 or higher. Students who pass the midterm can use the obtained score as part of the written test grade. This option can only be used in the first written test attempt after the midterm and only in the January-February exam session.
Students with Specific Learning Disabilities (DSA) and disabilities are requested to contact the instructor via email at least 10 days before the scheduled exam date to agree on any individualized measures. In the email addressed to the instructor, it is necessary to CC the respective University Services: [email protected] (for students with DSA) and [email protected] (for students with disabilities).
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8
Practicals: 48 hours
Lessons: 40 hours
Lessons: 40 hours
Professor:
Metere Giuseppe
Professor(s)