CALCULUS
Syllabus
EN
IT
Learning Objectives
The purpose of this course is to introduce students to those basic notions in mathematics that are essential to describe, understand and analyze possibly different models of quantitative phenomena.
LEARNING OUTCOMES:
The course contributes to the achievement of the objectives of the degree course, in line with the professional profiles and employment outlets envisaged, providing students with notions useful for an in-depth and critical understanding of some major tools and topics in differential and integral calculus.
KNOWLEDGE AND UNDERSTANDING:
At the end of the course, students will get to know and understand the main concepts and tools of differential and integral calculus, and will become familiar with functions of real variables, notions of growth, limits, rate of change, optimization, time evolution, all of them being necessary to approach mathematical models in applied sciences as well as to pursue further studies in probability and statistics.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The course provides tools to investigate and analyze possibly different models of quantitative phenomena, and to understand the use of mathematical language in applied models of real life.
MAKING JUDGEMENTS: the ability to draw independent judgments and conclusions about quantititative phenomena appearing in economics and real-life models, is stimulated by highlighting the connections between the concepts developed during the course, the notions acquired in previous courses and the links between these notions and the major contemporary economic problems.
COMMUNICATION SKILLS: By attending the course and interacting with the lecturer and the T.A.s, students will develop their communication skills and their ability to organise and share articulate mathematical and logical reasonings.
LEARNING SKILLS: Through the study of the main concepts and tools of differential and integral calculus, students will acquire the ability to independently analyse and investigate specific topics related to the course contents.
LEARNING OUTCOMES:
The course contributes to the achievement of the objectives of the degree course, in line with the professional profiles and employment outlets envisaged, providing students with notions useful for an in-depth and critical understanding of some major tools and topics in differential and integral calculus.
KNOWLEDGE AND UNDERSTANDING:
At the end of the course, students will get to know and understand the main concepts and tools of differential and integral calculus, and will become familiar with functions of real variables, notions of growth, limits, rate of change, optimization, time evolution, all of them being necessary to approach mathematical models in applied sciences as well as to pursue further studies in probability and statistics.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The course provides tools to investigate and analyze possibly different models of quantitative phenomena, and to understand the use of mathematical language in applied models of real life.
MAKING JUDGEMENTS: the ability to draw independent judgments and conclusions about quantititative phenomena appearing in economics and real-life models, is stimulated by highlighting the connections between the concepts developed during the course, the notions acquired in previous courses and the links between these notions and the major contemporary economic problems.
COMMUNICATION SKILLS: By attending the course and interacting with the lecturer and the T.A.s, students will develop their communication skills and their ability to organise and share articulate mathematical and logical reasonings.
LEARNING SKILLS: Through the study of the main concepts and tools of differential and integral calculus, students will acquire the ability to independently analyse and investigate specific topics related to the course contents.
ALFONSO SORRENTINO
Prerequisites
No formal pre-requisites.
Students are expected to have adequate knowledge and practice of the mathematics which is taught in standard high school programs in Italy. This includes, in particular: elementary rules of computation involving numbers, fractions and polynomials, representation of points in the plane through Cartesian coordinates, basic notions concerning lines and parabolas and their representation in the plane, methods of resolution of first and second order equations and inequalities, definition and basic properties of logarithms, exponentials and trigonometric functions. All the arguments in the above list are included among the topics of the math Pre-courses so that all students can go through a quick review of those notions while attending the Pre-courses. However, a quick review may be not enough for all students who have a weak - or even very weak - background in mathematics. In this case, students are strongly encouraged to work hard even before the beginning of Pre-courses in order to catch up with the expected level. Any school textbook can be used for reviewing the above mentioned topics.
Students are expected to have adequate knowledge and practice of the mathematics which is taught in standard high school programs in Italy. This includes, in particular: elementary rules of computation involving numbers, fractions and polynomials, representation of points in the plane through Cartesian coordinates, basic notions concerning lines and parabolas and their representation in the plane, methods of resolution of first and second order equations and inequalities, definition and basic properties of logarithms, exponentials and trigonometric functions. All the arguments in the above list are included among the topics of the math Pre-courses so that all students can go through a quick review of those notions while attending the Pre-courses. However, a quick review may be not enough for all students who have a weak - or even very weak - background in mathematics. In this case, students are strongly encouraged to work hard even before the beginning of Pre-courses in order to catch up with the expected level. Any school textbook can be used for reviewing the above mentioned topics.
Program
The course programme is divided into 8 inter-related parts as follows:
1) Real numbers, elementary functions and graphs
2) Sequences and limits
3) Recurrence, discrete time models: exponentials and logarithms, log scales
4) Derivatives: rules and applications, rate of change in applied models
5) Optimization: maxima and minima, convexity, curve sketching
6) Integration: areas, antiderivatives, Fundamental Theorem of Calculus
7) Differential equations and growth models: equilibrium points, stability
8) Multivariable calculus: partial derivatives, optimization.
During each lecture, the lecturer presents the planned content at the blackboard and invites students to critical reflection and dialogue
1) Real numbers, elementary functions and graphs
2) Sequences and limits
3) Recurrence, discrete time models: exponentials and logarithms, log scales
4) Derivatives: rules and applications, rate of change in applied models
5) Optimization: maxima and minima, convexity, curve sketching
6) Integration: areas, antiderivatives, Fundamental Theorem of Calculus
7) Differential equations and growth models: equilibrium points, stability
8) Multivariable calculus: partial derivatives, optimization.
During each lecture, the lecturer presents the planned content at the blackboard and invites students to critical reflection and dialogue
Books
Lawrence D. Hoffmann, Gerald L. Bradley, Dave Sobecki, Michael Price: Applied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition, ed. Mc Graw-Hill, 2012-2013.
Claudia Neheauser:_Calculus for Biology and Medicine, 3rd ed. Pearson International, 2011.
Claudia Neheauser:_Calculus for Biology and Medicine, 3rd ed. Pearson International, 2011.
Teaching methods
The course is taught in the traditional way through classroom lectures, where theory and practice become intrinsically linked. Additional tutorials will be devoted to extra practice, based on exercises from the used textbook as well as from previous exams.
Exam Rules
Course assessment
Two mid-term written examinations are given, roughly corresponding to Topics 1-5 and 6-8 respectively, although variations may occur according to the updated lesson program. These exams will consist of multiple-choice questions.
At the end of the lecture period, a final written examination covering Topics 1 to 8 is given. The final written exam consists of two parts. The first part requires students to answer a set of multiple-choice questions, which will account for 60% of the final grade. The second part requires them to solve exercises through detailed written explanations containing all necessary steps and computations, which will account for up to 40% of the final grade. Achieving a sufficient score on both parts is necessary to pass the exam. Students who successfully pass the midterms will be exempt from taking the multiple-choice part of the final. Attendance at the final is compulsory to pass the exam.
The mid-term exam is open to both attending and non-attending students.
Students who fail or reject the grade of the final examination will need to take a new complete examination on different exam dates and forfeit their previous mid-term results. Students who do not attend the first final examination can retake it without forfeiting their mid-term results in the second exam of the winter session. Students who retake the exam in the September session must forfeit any results from the winter session and take a new complete examination.
The examination assesses the student's overall preparation, ability to integrate the knowledge of the different parts of the program, consequentiality of reasoning, analytical ability and clarity of presentation, in accordance with the Dublin descriptors (1. knowledge and understanding; 2. applying knowledge and understanding; 3. making judgements; 4. learning skills; 5. communication skills).
The examination will be graded according to the following criteria:
Unsuitable: important deficiencies and/or inaccuracies in the knowledge and understanding of the topics; the topics are exposed in an incoherent manner and with inappropriate language.
18-20: barely sufficient knowledge and understanding of most of the topics, with some missing items; sufficient capacity for analysis; the topics are sometimes exposed in an inconsistent manner and with inappropriate/technical language;
21-23: basic knowledge and understanding of most of the topics; ability to analyze and synthesize correctly with sufficiently coherent logical argumentation, with possibly some inaccuracy in the technical language.
24-26: good knowledge and understanding of most of the topics; good analytical and synthetic skills with rigorously expressed arguments, though with possibly a few inaccuracies in the technical language.
27-29: complete knowledge and understanding of the topics; good capacity for analysis and synthesis. Arguments presented in a rigorous manner and with appropriate/technical language, with only minor inaccuracies.
30-30L: very good level of knowledge and thorough understanding of topics. Excellent analytical and synthetic skills and independent judgement. Arguments expressed in an original manner and in appropriate technical language.
Two mid-term written examinations are given, roughly corresponding to Topics 1-5 and 6-8 respectively, although variations may occur according to the updated lesson program. These exams will consist of multiple-choice questions.
At the end of the lecture period, a final written examination covering Topics 1 to 8 is given. The final written exam consists of two parts. The first part requires students to answer a set of multiple-choice questions, which will account for 60% of the final grade. The second part requires them to solve exercises through detailed written explanations containing all necessary steps and computations, which will account for up to 40% of the final grade. Achieving a sufficient score on both parts is necessary to pass the exam. Students who successfully pass the midterms will be exempt from taking the multiple-choice part of the final. Attendance at the final is compulsory to pass the exam.
The mid-term exam is open to both attending and non-attending students.
Students who fail or reject the grade of the final examination will need to take a new complete examination on different exam dates and forfeit their previous mid-term results. Students who do not attend the first final examination can retake it without forfeiting their mid-term results in the second exam of the winter session. Students who retake the exam in the September session must forfeit any results from the winter session and take a new complete examination.
The examination assesses the student's overall preparation, ability to integrate the knowledge of the different parts of the program, consequentiality of reasoning, analytical ability and clarity of presentation, in accordance with the Dublin descriptors (1. knowledge and understanding; 2. applying knowledge and understanding; 3. making judgements; 4. learning skills; 5. communication skills).
The examination will be graded according to the following criteria:
Unsuitable: important deficiencies and/or inaccuracies in the knowledge and understanding of the topics; the topics are exposed in an incoherent manner and with inappropriate language.
18-20: barely sufficient knowledge and understanding of most of the topics, with some missing items; sufficient capacity for analysis; the topics are sometimes exposed in an inconsistent manner and with inappropriate/technical language;
21-23: basic knowledge and understanding of most of the topics; ability to analyze and synthesize correctly with sufficiently coherent logical argumentation, with possibly some inaccuracy in the technical language.
24-26: good knowledge and understanding of most of the topics; good analytical and synthetic skills with rigorously expressed arguments, though with possibly a few inaccuracies in the technical language.
27-29: complete knowledge and understanding of the topics; good capacity for analysis and synthesis. Arguments presented in a rigorous manner and with appropriate/technical language, with only minor inaccuracies.
30-30L: very good level of knowledge and thorough understanding of topics. Excellent analytical and synthetic skills and independent judgement. Arguments expressed in an original manner and in appropriate technical language.
CHRISTOPH HEINRICH LHOTKA
Books
Lawrence D. Hoffmann, Gerald L. Bradley, Dave Sobecki, Michael Price: Applied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition, ed. Mc Graw-Hill, 2012-2013.
Claudia Neheauser:_Calculus for Biology and Medicine, 3rd ed. Pearson International, 2011.
Claudia Neheauser:_Calculus for Biology and Medicine, 3rd ed. Pearson International, 2011.
Teaching methods
The course is taught in the traditional way through classroom lectures, where theory and practice become intrinsically linked. Additional tutorials will be devoted to extra practice, based on exercises from the used textbook as well as from previous exams.