CALCULUS - 1

Calculus 1 is a mathematics and logic course that forms part of the Bachelor's degree in Computer Science. This course covers the properties of real functions with real variable and serves as a foundation for Calculus mod. 2.
Topics covered include infinitesimal calculus, which is used to study the behaviour of a quantity (a real variable) as another variable varies. This involves characterising its asymptotic behaviour and behaviour at singular points, as well as its properties of variation and accumulation.
This topic is broken down into four sections: the preliminary definitions and contextual issues; the study of limits for asymptotic behaviour and at singular points; the study of derivatives to determine function variation; and integral calculus with regard to accumulation problems. The course aims to equip students with the conceptual and computational tools required to study the characteristics of real function of real variable and to calculate its primitives and definite integrals.

By the end of the ‘Calculus mod. 1’ course, students will understand how to use calculus to analyse the characteristics and properties of real functions of a real variable. Through a problem-based approach, they will learn the definitions and procedures for calculating limits, differentiation and integration. They will also complete exercises applying their knowledge and understanding of the covered concepts in order to consolidate the procedures and techniques for solving problems.
To consolidate their learning method and develop their written communication skills in mathematics and mathematical problem solving, a number of specific metacognitive activities are planned.

There are no prerequisites; you simply need to have been admitted to the Computer Science degree programme and fulfilled any additional educational requirements. This means that the basic maths skills acquired in secondary school are sufficient. In particular, you should have knowledgeand skills regarding Cartesian coordinates, trigonometric, logarithmic and exponential functions, and the resolution of equations and inequalities of all types.

1. The concept of function
Knowledge and understanding
- Definition of a function from R to R: domain, codomain and image
- Definition of a function from N to R
- Definition of a numerical series
Invertibility of a function and definition of an inverse function
Ability to solve basic problems
- Determine the domain and codomain of a function
Problems for developing skills
- Study the domain and sign of functions, and define the areas of the plane on which their graphs lie
Problems for developing personal, study and updating skills
- Build and adapt formula sheets and summaries for learning and training purposes

2. Asymptotic behaviour of a function and singularities of the first kind
Knowledge and understanding
- Definition of neighbourhood and accumulation point
- Definition of a limit and algebra of limits
- Notable limits
- Infinitesimals and infinities
- Uniqueness theorem, sign permanence theorem, comparison theorem
Ability to solve basic problems
- Calculate the limit of a function in a neighbourhood of an accumulation point
- Solve indeterminate forms by reducing them to notable limits
Problems for developing skills
- Study the limits of a function
- Represent the asymptotic behaviour of a function on the Cartesian plane
Problems to develop personal, study and updating skills
- Agree on the schemes and conventions to be followed when taking a written mathematics test
- Build and adapt formula sheets and summaries for learning and training purposes

3.A. Variation of a function
Knowledge and understanding
- Definitions of continuity and differentiability of a function
- Definition of the first derivative and higher-order derivatives
- Derivation rules
- The derivative of the inverse function of a given function
- Continuity and differentiability
- Rolle's theorem, Lagrange's theorem, Cauchy's theorem and De l'Hospital's rule
Ability to solve basic problems
- Calculate the derivatives of a function
- Calculate the tangent to the graph of a function at a point
Problems for developing skills
- Study the variation of a function using the first derivative
- Study the concavity of a function using the second derivative
- Plot the graph of a function
Problems for developing personal, study and updating skills
- Write a mathematical paper on the study of functions
- Build and adapt formula sheets and summaries for learning and training purposes

3.B. Approximate a function with a polynomial
Knowledge
- Taylor polynomial
Ability to solve basic problems
- Calculate the Taylor polynomial for a given function
Problems for developing skills
- Use the Taylor expansion to calculate the limits of functions

4. Integral Calculus.
Knowledge
- Definite integral: calculation of areas and volumes
- Indefinite integral: calculation of primitives of a function
- Torricelli–Barrow theorem
- Integration methods
Ability to solve basic problems
- Calculate definite and indefinite integrals
Problems for developing skills
- Calculating the volume of solids of revolution
Problems for developing personal, study and updating skills
- Build and adapt formula sheets and summaries for learning and training purposes

Study materials are provided via the Moodle platform for this course.

The following handouts are available online:
- Luciano Battaia, 'Introduction to Differential Calculus': http://www.batmath.it/matematica/0-appunti_uni/testo_analisi.pdf
For integral calculus: Luciano Battaia, 'Notes for a Mathematics Course' (Chapter 7) http://www.batmath.it/matematica/0-appunti_uni/corso-ve.pdf

The following texts are useful for reference:
- Bramanti, Pagani, Salsa: Calculus 1, Zanichelli
Salsa Squellati: Exercises in Calculus 1, Zanichelli

Guidelines for the examination and assessment of Module 1 of Calculus.
The two-hour written test consists of open-ended questions on Calculus for functions of one variable. There are no mid-term tests.
During the test, students may refer to an A4 sheet containing formula sheets and summaries of theory that they have compiled during the course. Students must achieve a minimum score of 60% to pass the written exam. The maximum mark for the written exam is 30.
Students who have passed the written exam may take an oral exam if they wish. In the oral exam, students must demonstrate their knowledge of the basic theoretical concepts and main theorems introduced in the course, and explain them formally.
The overall exam result may be lower (leading to failure), the same as, or at most 3 marks higher than, the written exam mark. Honours are awarded only to students who pass the written exam with at least 28 marks and demonstrate mastery of the theoretical content in the oral exam.

Guidelines for the examination and overall assessment of Calculus.
Only students who have passed Module 1, as intermediate exam, may take Module 2. Both modules may be taken on the same day, but strictly in that order. The final grade for the Calculus exam will be the average of the two module grades (rounded up). Honours are awarded only with the unanimous approval of both teachers. The grade for Module 1 remains valid until Module 2 is passed; however, students are strongly advised to pass both Analysis modules in the same academic year.

written and oral

A score between 18 and 22 indicates a basic knowledge of the content and the ability to apply this knowledge to solve of the same level. It demonstrates an overall satisfactory performance in the test and the development of basic skills.

A score between 23 and 26 indicates an intermediate level of knowledge and competence, including a certain rigour in conducting the exercises. It indicates an overall good performance in the test and the development of intermediate-level skills.

A score between 27 and 30 indicates an advanced level of knowledge and an advanced ability to apply knowledge to solve problems. Excellent rigour in conducting the exercises is also evident. This indicates an excellent performance in the test and advanced skill development.

Honours will be awarded for excellent knowledge and understanding in terms of both content and application to problem solving. It indicates achievement of the most advanced levels of skill development.

Lectures and classroom exercises. Some additionary materials and exercises are available on the university's Moodle e-learning platform.