OVERVIEW

This introductory calculus course builds up on the mathematics learnt during the high school. The main topics of the course are differentiation and integration of functions of one variable.

AIMS AND CONTENT

LEARNING OUTCOMES

The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Students should demonstrate an understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.

AIMS AND LEARNING OUTCOMES

At the end of this course the students are expected:

• to master the mathematical notation;
• to know the properties of the elementary functions and their graph;
• to be able to follow the mathematical arguments;
• and to solve simple exercises, and discuss the results obtained.

PREREQUISITES

Sets, equalities and inequalities, analytic geometry, trigonometry.

TEACHING METHODS

Both theory and exercises are presented by the teachers. Some tutorials will be carried out during the semester.

SYLLABUS/CONTENT

The real numbers - The real numbers, maxima, minima, supremum, infimum.

Functions - Elementary functions, composite function, inverse function.

Limits and continuity - Limits of functions. Continuity. Global properties of continuous functions. The intermediate value theorem and the extreme value theorem.

Differentiation - Derivative of a function. Tangent line. Derivative of the composite function and of the inverse function. The theorems of Rolle, Chauchy and Lagrange. De l’Hôpital's rule. 

Integration - Riemann and Cauchy sums. Indefinite integral. Area of a planar region. Mean value theorem. Integral functions. The fundamental theorem of calculus. Calculating primitives.

RECOMMENDED READING/BIBLIOGRAPHY

Some notes and exercises are available.

Recommended books

• C. Canuto, A. Tabacco, Mathematical Analysis 1, ISBN: 9788891931115 
• M. Oberguggenberger, A. Ostermann, Analysis for Computer Scientists: Foundations, Methods, and Algorithms, Springer-Verlag, 
  ISBN 978-0-85729-445-6
• M. Baronti, M., F. De Mari,  R. van der Putten, I. Venturi,  Calculus Problems, Springer-Verlag, ISBN: 978-3-319-15427-5

TEACHERS AND EXAM BOARD

Exam Board

GIOVANNI ALBERTI (President)

TOMMASO BRUNO

FEDERICO BENVENUTO (President Substitute)

LESSONS

LESSONS START

According to the accademic calendar

EXAMS

EXAM DESCRIPTION

The exam consists of one written test, composed of two parts:

• A test made of multiple-choice questions, about the theory seen during the course and with simple exercises
• Written test with more complex problems.

The exam is passed if both parts are sufficient. The final mark is given by

                 (first part)*1/3 + (second part)*2/3

ASSESSMENT METHODS

• The first part of the exam allows us to verify the ability of the students to handle the mathematical notation and to make simple deductive reasonings.
• The second part allows us to verify the ability to solve simple calculations and the knowledge of the main tools related to differentiation and integration.

Exam schedule